金声玉亮2.0模型评测报告

问题

You are given a question and its solution. The solution however has its formulae masked out using the tag <missing X> where X indicates the identifier for the missing tag. You are also given a list of formulae in latex in the format "<expression Y> = $<latex code>$" where Y is the identifier for the formula. Your task is to match the formulae to the missing tags in the solution. Think step by step out loud as to what the answer should be. If you are not sure, give your best guess. Your answer should be in the form of a list of numbers, e.g., 5, 22, 3, ..., corresponding to the expression identifiers that fill the missing parts. For example, if your answer starts as 5, 22, 3, ..., then that means expression 5 fills <missing 1>, expression 22 fills <missing 2>, and expression 3 fills <missing 3>.

The question is:
Let $ABC$ be an acute-angled triangle with $AB < AC$. Let $\Omega$ be the circumcircle of $ABC$. Let $S$ be the midpoint of the arc $CB$ of $\Omega$ containing $A$. The perpendicular from $A$ to $BC$ meets $BS$ at $D$ and meets $\Omega$ again at $E \neq A$. The line through $D$ parallel to $BC$ meets line $BE$ at $L$. Denote the circumcircle of triangle $BDL$ by $\omega$. Let $\omega$ meet $\Omega$ again at $P \neq B$. Prove that the line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$.

The solution is:
Denote the point diametrically opposite to a point $S$ through <missing 1> is the internal angle bisector of <missing 2> .

Denote the crosspoint of $BS$ and <missing 3> through <missing 4> \[AE \perp BC, SS' \perp BC \implies \overset{\Large\frown} {AS} = \overset{\Large\frown} {ES'} = 2\varphi \implies\]
\[\angle EAS' = \varphi = \angle ABS \implies \angle DAH = \angle ABH \implies\]\[\triangle AHD \sim \triangle BAH \implies \frac {AH}{BH} = \frac {DH}{AH} \implies AH^2 = BH \cdot DH.\]To finishing the solution we need only to prove that $PH = AH.$ Denote <missing 5> <missing 6> is incenter of <missing 7> Denote <missing 8> is the orthocenter of $\triangle TSS'.$ Denote $G = PS' \cap AE \implies \angle BPG = \angle BPS' = \angle BSS' = \angle BDG \implies B, L, P, D,$ and $G$ are concyclic.

2023 IMO 2 lemma.png <missing 9> points $B, G,$ and $F$ are collinear <missing 10> is symmetric to $AF$ with respect <missing 11> We use the lemma and complete the proof.

Lemma 1

2023 IMO 2b Lemma.png
Let acute triangle <missing 12> be given.

Let $H$ be the orthocenter of <missing 13> be the height.

Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15> The point $E$ is symmetric to $D$ with respect to <missing 16> The line $BE$ meets <missing 17> again at <missing 18> .

Prove that <missing 19> Proof

Let <missing 20> be the circle centered at $H$ with radius <missing 21> The <missing 22> meets $\Omega$ again at <missing 23> Let $\omega$ meets <missing 24> again at <missing 25> .

We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27> and get <missing 28> (this idea was recommended by Leonid Shatunov). <missing 29> The point <missing 30> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD.$

The formulae are:
<expression 1> \Omega.
<expression 2> BE'F'
<expression 3> T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H
<expression 4> E' \neq F'
<expression 5> \triangle ABF.
<expression 6> \omega
<expression 7> \omega, \Omega
<expression 8> =\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H
<expression 9> \Omega
<expression 10> \omega
<expression 11> E'
<expression 12> AH \perp BC \implies AH \perp E'D \implies
<expression 13> \implies GF
<expression 14> \triangle ABC, BHD
<expression 15> S' \implies AS'
<expression 16> \triangle ABC, AB > AC
<expression 17> F' \neq D, HD = HF'.
<expression 18> AS'
<expression 19> H, \angle ABS = \varphi.
<expression 20> TF.
<expression 21> AH.
<expression 22> F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =
<expression 23> HF = HD.
<expression 24> \angle BAC
<expression 25> E'D || BC
<expression 26> F \neq B
<expression 27> \Omega
<expression 28> HD.
<expression 29> BF'
<expression 30> \angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies

Your final answer should be STRICTLY in the format:

<Detailed reasoning>

Answer: <comma separated list of numbers representing expression identifiers>

答案

模型	模型答案	正确答案
金声玉亮2.0	To match the formulae to the missing tags in the solution, let's go through the solution step by step and identify which expression fits each <missing X> tag. 1. <missing 1>: "Denote the point diametrically opposite to a point $S$ through <missing 1>" - This refers to the circumcircle of the triangle, so it should be <expression 1> \Omega. 2. <missing 2>: "is the internal angle bisector of <missing 2>" - This refers to the angles in the triangle, specifically the internal angle bisector of some angle. The relevant expression here seems to be <expression 24> \angle BAC. 3. <missing 3>: "Denote the crosspoint of $BS$ and <missing 3> through <missing 4>" - This refers to a point of intersection, and from the context, it seems to be related to the height or perpendicular from A to BC. The relevant expression here is <expression 12> AH \perp BC \implies AH \perp E'D \implies. 4. <missing 4>: Continuing from the previous point, the crosspoint is denoted through some point, which seems to be related to the height or perpendicular from A to BC. The relevant expression here is <expression 12> AH \perp BC \implies AH \perp E'D \implies. 5. <missing 5>: "To finishing the solution we need only to prove that $PH = AH.$ Denote <missing 5>" - This refers to a triangle where we need to prove some equality. The relevant expression here is <expression 5> \triangle ABF. 6. <missing 6>: "is incenter of <missing 7>" - This refers to the incenter of a triangle. The relevant expression here is <expression 7> \omega, \Omega. 7. <missing 7>: Continuing from the previous point, the incenter is of some triangle. The relevant expression here is <expression 7> \omega, \Omega. 8. <missing 8>: "is the orthocenter of $\triangle TSS'.$ Denote <missing 8>" - This refers to the orthocenter of a triangle. The relevant expression here is <expression 8> =\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H. 9. <missing 9>: "points $B, G,$ and $F$ are collinear <missing 10>" - This refers to the circumcircle where points are collinear. The relevant expression here is <expression 9> \Omega. 10. <missing 10>: Continuing from the previous point, the collinear points are related to some circle. The relevant expression here is <expression 10> \omega. 11. <missing 11>: "is symmetric to $AF$ with respect <missing 11>" - This refers to symmetry with respect to some point. The relevant expression here is <expression 11> E'. 12. <missing 12>: "Let acute triangle <missing 12> be given." - This refers to the given triangle. The relevant expression here is <expression 16> \triangle ABC, AB > AC. 13. <missing 13>: "Let $H$ be the orthocenter of <missing 13>" - This refers to the orthocenter of the triangle. The relevant expression here is <expression 16> \triangle ABC, AB > AC. 14. <missing 14>: "Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>" - This refers to the circle with BC as the diameter. The relevant expression here is <expression 15> S' \implies AS'. 15. <missing 15>: Continuing from the previous point, the diameter of the circle. The relevant expression here is <expression 15> S' \implies AS'. 16. <missing 16>: "The point $E$ is symmetric to $D$ with respect to <missing 16>" - This refers to symmetry with respect to some point. The relevant expression here is <expression 18> AS'. 17. <missing 17>: "The line $BE$ meets <missing 17> again at <missing 18>" - This refers to the circle where the line meets again. The relevant expression here is <expression 9> \Omega. 18. <missing 18>: Continuing from the previous point, the point where the line meets again. The relevant expression here is <expression 4> E' \neq F'. 19. <missing 19>: "Prove that <missing 19>" - This refers to the proof statement. The relevant expression here is <expression 23> HF = HD. 20. <missing 20>: "Let <missing 20> be the circle centered at $H$ with radius <missing 21>" - This refers to the circle centered at H. The relevant expression here is <expression 20> TF. 21. <missing 21>: Continuing from the previous point, the radius of the circle. The relevant expression here is <expression 21> AH. 22. <missing 22>: "The <missing 22> meets $\Omega$ again at <missing 23>" - This refers to the circle meeting $\Omega$ again. The relevant expression here is <expression 22> F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =. 23. <missing 23>: Continuing from the previous point, the point where the circle meets $\Omega$ again. The relevant expression here is <expression 23> HF = HD. 24. <missing 24>: "Let $\omega$ meets <missing 24> again at <missing 25>" - This refers to the circle $\omega$ meeting something again. The relevant expression here is <expression 10> \omega. 25. <missing 25>: Continuing from the previous point, the point where $\omega$ meets again. The relevant expression here is <expression 4> E' \neq F'. 26. <missing 26>: "We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27> and get <missing 28>" - This refers to some point in the theorem. The relevant expression here is <expression 26> F \neq B. 27. <missing 27>: Continuing from the previous point, the lines in the theorem. The relevant expression here is <expression 27> \Omega. 28. <missing 28>: Continuing from the previous point, the result of the theorem. The relevant expression here is <expression 28> HD. 29. <missing 29>: "The point <missing 30> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD." - This refers to symmetry with respect to some point. The relevant expression here is <expression 29> BF'. 30. <missing 30>: Continuing from the previous point, the point of symmetry. The relevant expression here is <expression 30> \angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies. Answer: 1, 24, 12, 12, 5, 7, 7, 8, 9, 10, 11, 16, 16, 15, 15, 18, 9, 4, 23, 20, 21, 22, 23, 10, 4, 26, 27, 28, 29, 30	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
deepseek-r1	Answer: 15,24,3,22,5,6,14,19,13,29,21,16,14,9,21,6,22,23,23,28,21,22,23,9,25,26,27,23,17,30	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
gpto1	To solve this problem, we'll match each <missing X> in the solution to the appropriate expression from the list provided. Here's the step-by-step reasoning: 1. Missing 1: The sentence begins with "Denote the point diametrically opposite to a point \( S \) through <missing 1>". Since the point diametrically opposite \( S \) is found through the circle \( \Omega \), we can conclude that <missing 1> is Expression 1: \( \Omega. \) 2. Missing 2: The sentence continues "...is the internal angle bisector of <missing 2>". The internal angle bisector is of angle \( \angle BAC \), so <missing 2> is Expression 24: \( \angle BAC \). 3. Missing 3: "Denote the crosspoint of \( BS \) and <missing 3> through <missing 4>". The crosspoint of \( BS \) and \( AS' \) is significant in the context, so <missing 3> is Expression 18: \( AS' \). 4. Missing 4: Since the crosspoint is through point \( T \) (the intersection of \( BS \) and \( AS' \)), and Expression 3 introduces \( T \), <missing 4> is Expression 3: \( T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H \). 5. Missing 5: After stating we need to prove \( PH = AH \), it says "Denote <missing 5>", referring back to \( H \) and an angle \( \varphi \). So, <missing 5> is Expression 19: \( H, \angle ABS = \varphi. \) 6. Missing 6: It continues "<missing 6> is incenter of <missing 7>". The incenter is of triangle \( ABC \), so <missing 6> is Expression 21: \( AH. \) 7. Missing 7: Since the incenter is of triangle \( ABC \), <missing 7> is Expression 16: \( \triangle ABC, AB > AC \). 8. Missing 8: "Denote <missing 8> is the orthocenter of \( \triangle TSS' \).” The orthocenter is point \( H \), so <missing 8> is Expression 12: \( AH \perp BC \implies AH \perp E'D \implies \). 9. Missing 9: Before mentioning points \( B, G, \) and \( F \) are collinear, it's logical to reference the main circle, so <missing 9> is Expression 9: \( \Omega \). 10. Missing 10: "Point \( F \) is symmetric to \( AF \) with respect...", considering symmetry and circles, <missing 10> is Expression 10: \( \omega \). 11. Missing 11: The symmetry is with respect to point \( E' \), so <missing 11> is Expression 11: \( E' \). 12. Missing 12: In Lemma 1, "Let acute triangle <missing 12> be given." The triangle is \( ABC \) with \( AB > AC \), so <missing 12> is Expression 16: \( \triangle ABC, AB > AC \). 13. Missing 13: "Let \( H \) be the orthocenter of <missing 13> be the height." This refers to \( AH \), so <missing 13> is Expression 12: \( AH \perp BC \implies AH \perp E'D \implies \). 14. Missing 14: "Let <missing 14> be the circle \( BCD \).” This circle is \( \omega \), so <missing 14> is Expression 6: \( \omega \). 15. Missing 15: " \( BC \) is the diameter of <missing 15>." The diameter is in circle \( \Omega \), so <missing 15> is Expression 9: \( \Omega \). 16. Missing 16: "The point \( E \) is symmetric to \( D \) with respect to <missing 16>." The symmetry is with respect to \( E' \), so <missing 16> is Expression 11: \( E' \). 17. Missing 17: "The line \( BE \) meets <missing 17> again at..." Since it meets \( \Omega \), <missing 17> is Expression 27: \( \Omega \). 18. Missing 18: The point it meets is \( HD = HF' \), so <missing 18> is Expression 23: \( HF = HD. \) 19. Missing 19: "Prove that <missing 19>." The goal is to prove \( HF = HD \), so <missing 19> is Expression 28: \( HD. \) 20. Missing 20: "Let <missing 20> be the circle centered at \( H \)..." This refers to \( AH \), so <missing 20> is Expression 21: \( AH. \) 21. Missing 21: The radius is \( AH \), so <missing 21> is Expression 24: \( \angle BAC \). 22. Missing 22: "The <missing 22> meets \( \Omega \) again at..." This refers to \( \Omega \), so <missing 22> is Expression 9: \( \Omega \). 23. Missing 23: It meets again at a point where \( E'D \|\| BC \), so <missing 23> is Expression 25: \( E'D \|\| BC \). 24. Missing 24: "Let \( \omega \) meets <missing 24> again at..." Since \( \omega \) and \( \Omega \) meet, <missing 24> is Expression 7: \( \omega, \Omega \). 25. Missing 25: It meets again at \( \Omega \), so <missing 25> is Expression 27: \( \Omega \). 26. Missing 26: "We use Reim’s theorem for <missing 26> and lines..." The circles involved are \( \omega \) and \( \Omega \), so <missing 26> is Expression 7: \( \omega, \Omega \). 27. Missing 27: "and lines \( CDD \) and <missing 27>..." Since it's another reference to \( \Omega \), <missing 27> is Expression 27: \( \Omega \). 28. Missing 28: "and get <missing 28>" refers back to \( HD \), so <missing 28> is Expression 28: \( HD. \) 29. Missing 29: The next step likely involves \( BF' \), so <missing 29> is Expression 29: \( BF' \). 30. Missing 30: Finally, relating angles and implying something about \( \angle DBG = \varphi \), so <missing 30> is Expression 30: Given angle relationships. Answer: 1, 24, 18, 3, 19, 21, 16, 12, 9, 10, 11, 16, 12, 6, 9, 11, 27, 23, 28, 21, 24, 9, 25, 7, 27, 23, 7, 27, 28, 30	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
gpt-4o-2024-08-06	To solve this problem, we need to match the missing parts in the solution with the given expressions. Let's go through the solution step by step and see which expressions fit best. 1. First Missing Part: - "Denote the point diametrically opposite to a point $S$ through <missing 1> is the internal angle bisector of <missing 2>." - We are looking for something related to a circle, possibly the circumcircle $\Omega$. - Expression 1: \(\Omega\) seems to fit here as it refers to the circumcircle. 2. Second Missing Part: - Continuing from the first sentence, we need something that is an angle or a line related to the internal angle bisector. - Expression 24: \(\angle BAC\) fits well here as it is the angle in question. 3. Third Missing Part: - "Denote the crosspoint of $BS$ and <missing 3> through <missing 4>." - We need something that is a line or a segment that $BS$ intersects. - Expression 3: \(T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H\) seems to describe a geometric configuration involving $BS$. 4. Fourth Missing Part: - This part seems to be a continuation of the previous sentence, possibly describing a condition or a result of the intersection. - Expression 19: \(H, \angle ABS = \varphi.\) could fit here as it describes an angle condition. 5. Fifth Missing Part: - "To finishing the solution we need only to prove that $PH = AH.$ Denote <missing 5>" - We are looking for a triangle or a geometric figure. - Expression 5: \(\triangle ABF.\) fits as it denotes a triangle. 6. Sixth Missing Part: - "Denote <missing 6> is incenter of <missing 7>" - We need something that can be an incenter. - Expression 6: \(\omega\) fits as it is a circle, and the incenter is often related to circles. 7. Seventh Missing Part: - We need something that can be a circle or a set of circles. - Expression 7: \(\omega, \Omega\) fits as it refers to both circles. 8. Eighth Missing Part: - "Denote <missing 8> is the orthocenter of $\triangle TSS'.$" - We need something that can be an orthocenter. - Expression 8: \(\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H\) fits as it refers to an orthocenter. 9. Ninth Missing Part: - "2023 IMO 2 lemma.png <missing 9> points $B, G,$ and $F$ are collinear" - We need something that refers to a geometric figure or a circle. - Expression 9: \(\Omega\) fits as it refers to the circumcircle. 10. Tenth Missing Part: - "points $B, G,$ and $F$ are collinear <missing 10> is symmetric to $AF$ with respect" - We need something that can be symmetric. - Expression 10: \(\omega\) fits as it is a circle, and symmetry is often discussed in terms of circles. 11. Eleventh Missing Part: - "is symmetric to $AF$ with respect <missing 11>" - We need something that can be a line or a point of symmetry. - Expression 11: \(E'\) fits as it refers to a point. 12. Twelfth Missing Part: - "Let acute triangle <missing 12> be given." - We need something that describes a triangle. - Expression 14: \(\triangle ABC, BHD\) fits as it describes a triangle. 13. Thirteenth Missing Part: - "Let $H$ be the orthocenter of <missing 13> be the height." - We need something that can be a triangle or a line. - Expression 16: \(\triangle ABC, AB > AC\) fits as it describes a triangle. 14. Fourteenth Missing Part: - "Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>" - We need something that describes a circle. - Expression 2: \(BE'F'\) fits as it describes a circle. 15. Fifteenth Missing Part: - "BC is the diameter of <missing 15>" - We need something that can be a circle. - Expression 9: \(\Omega\) fits as it refers to the circumcircle. 16. Sixteenth Missing Part: - "The point $E$ is symmetric to $D$ with respect to <missing 16>" - We need something that can be a line or a point of symmetry. - Expression 11: \(E'\) fits as it refers to a point. 17. Seventeenth Missing Part: - "The line $BE$ meets <missing 17> again at <missing 18>" - We need something that can be a circle or a line. - Expression 27: \(\Omega\) fits as it refers to the circumcircle. 18. Eighteenth Missing Part: - We need something that can be a point. - Expression 18: \(AS'\) fits as it refers to a point. 19. Nineteenth Missing Part: - "Prove that <missing 19> Proof" - We need something that can be a statement or a condition. - Expression 19: \(H, \angle ABS = \varphi.\) fits as it is a statement. 20. Twentieth Missing Part: - "Let <missing 20> be the circle centered at $H$ with radius <missing 21>" - We need something that can be a circle. - Expression 20: \(TF.\) fits as it refers to a circle. 21. Twenty-first Missing Part: - We need something that can be a radius or a line. - Expression 21: \(AH.\) fits as it refers to a line. 22. Twenty-second Missing Part: - "The <missing 22> meets $\Omega$ again at <missing 23>" - We need something that can be a line or a circle. - Expression 22: \(F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =\) fits as it describes a line. 23. Twenty-third Missing Part: - We need something that can be a point. - Expression 23: \(HF = HD.\) fits as it refers to a point. 24. Twenty-fourth Missing Part: - "Let $\omega$ meets <missing 24> again at <missing 25>" - We need something that can be a circle or a line. - Expression 27: \(\Omega\) fits as it refers to the circumcircle. 25. Twenty-fifth Missing Part: - We need something that can be a point. - Expression 25: \(E'D \|\| BC\) fits as it refers to a point. 26. Twenty-sixth Missing Part: - "We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27>" - We need something that can be a circle or a line. - Expression 26: \(F \neq B\) fits as it refers to a condition. 27. Twenty-seventh Missing Part: - We need something that can be a circle or a line. - Expression 27: \(\Omega\) fits as it refers to the circumcircle. 28. Twenty-eighth Missing Part: - "and get <missing 28> (this idea was recommended by Leonid Shatunov)." - We need something that can be a result or a condition. - Expression 28: \(HD.\) fits as it refers to a condition. 29. Twenty-ninth Missing Part: - "The point <missing 29> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD.$" - We need something that can be a point or a line. - Expression 29: \(BF'\) fits as it refers to a point. 30. Thirtieth Missing Part: - We need something that can be a condition or a statement. - Expression 30: \(\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF\) fits as it is a statement. After matching all the expressions, the final answer is: Answer: 1, 24, 3, 19, 5, 6, 7, 8, 9, 10, 11, 14, 16, 2, 9, 11, 27, 18, 19, 20, 21, 22, 23, 27, 25, 26, 27, 28, 29, 30	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
DeepSeek-Coder-V2-Lite-Instruct	<Detailed reasoning> First, we need to identify the missing tags in the solution. The tags are used to replace placeholders in the solution text. Let's go through the solution step by step and match the placeholders with the given formulae. 1. Identify the midpoint of the arc \(CB\) containing \(A\): - This is referred to as \(S'\). - The corresponding formula is: \(S' = \Omega.S'\). - Expression: 1. 2. Internal angle bisector of \(\angle BAC\): - This is referred to as \(AI\). - The corresponding formula is: \(AI = \Omega.AI\). - Expression: 2. 3. Crosspoint of \(BS\) and \(AE\): - This is referred to as \(D\). - The corresponding formula is: \(D = BS \cap AE\). - Expression: 3. 4. Crosspoint of \(BS\) and \(AE\): - This is referred to as \(D\). - The corresponding formula is: \(D = BS \cap AE\). - Expression: 3. 5. Incenter of \(\triangle ABC\): - This is referred to as \(I\). - The corresponding formula is: \(I = \triangle ABC.I\). - Expression: 4. 6. Orthocenter of \(\triangle TSS'\): - This is referred to as \(H\). - The corresponding formula is: \(H = \triangle TSS'.H\). - Expression: 5. 7. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 6. 8. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 6. 9. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 6. 10. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 6. 11. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 6. 12. Acute triangle \(ABC\): - This is referred to as \(\triangle ABC\). - The corresponding formula is: \(\triangle ABC = \triangle ABC\). - Expression: 7. 13. Height of \(\triangle ABC\): - This is referred to as \(AH\). - The corresponding formula is: \(AH = \triangle ABC.AH\). - Expression: 8. 14. Diameter of \(\triangle ABC\): - This is referred to as \(BC\). - The corresponding formula is: \(BC = \triangle ABC.BC\). - Expression: 9. 15. Point \(E\) is symmetric to \(D\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 10. 16. Line \(BE\) meets \(\Omega\) again at \(P \neq A\): - This is referred to as \(P\). - The corresponding formula is: \(P = \Omega.P\). - Expression: 11. 17. Line tangent to \(\omega\) at \(P\) meets line \(BS\) on the internal angle bisector of \(\angle BAC\): - This is referred to as \(PH\). - The corresponding formula is: \(PH = \triangle ABC.PH\). - Expression: 12. 18. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 13. 19. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 14. 20. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 15. 21. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 16. 22. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 17. 23. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 18. 24. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 19. 25. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 20. 26. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 21. 27. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 22. 28. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 23. 29. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 24. 30. Symmetric to \(AF\) with respect to \(AH\): - This is referred to as \(E'\). - The corresponding formula is: \(E' = \triangle ABC.E'\). - Expression: 25. Answer: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
Meta-Llama-3.1-405B-Instruct-Turbo	To solve this problem, we need to carefully read the solution and match the missing tags with the given formulae. The first missing tag is <missing 1>, which is described as "the point diametrically opposite to a point $S$". This matches with <expression 1>, which is "$\Omega$". The second missing tag is <missing 2>, which is described as "the internal angle bisector of". This matches with <expression 24>, which is "$\angle BAC$". The third missing tag is <missing 3>, which is described as "the crosspoint of $BS$ and". This matches with <expression 3>, which is "$T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$". The fourth missing tag is <missing 4>, which is described as "through". This matches with <expression 4>, which is "$E' \neq F'$". The fifth missing tag is <missing 5>, which is described as "Denote". This matches with <expression 5>, which is "$\triangle ABF$". The sixth missing tag is <missing 6>, which is described as "is incenter of". This matches with <expression 6>, which is "$\omega$". The seventh missing tag is <missing 7>, which is described as "Denote". This matches with <expression 7>, which is "$\omega, \Omega$". The eighth missing tag is <missing 8>, which is described as "is the orthocenter of". This matches with <expression 8>, which is "$=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$". The ninth missing tag is <missing 9>, which is described as "points $B, G,$ and $F$ are collinear". This matches with <expression 9>, which is "$\Omega$". The tenth missing tag is <missing 10>, which is described as "is symmetric to $AF$ with respect". This matches with <expression 10>, which is "$\omega$". The eleventh missing tag is <missing 11>, which is described as "We use the lemma and complete the proof.". This matches with <expression 11>, which is "$E'$". The twelfth missing tag is <missing 12>, which is described as "Let acute triangle". This matches with <expression 16>, which is "$\triangle ABC, AB > AC$". The thirteenth missing tag is <missing 13>, which is described as "be the height.". This matches with <expression 14>, which is "$\triangle ABC, BHD$". The fourteenth missing tag is <missing 14>, which is described as "be the circle $BCD$". This matches with <expression 15>, which is "$S' \implies AS'$". The fifteenth missing tag is <missing 15>, which is described as "$BC$ is the diameter of". This matches with <expression 18>, which is "$AS'$". The sixteenth missing tag is <missing 16>, which is described as "The point $E$ is symmetric to $D$ with respect to". This matches with <expression 12>, which is "$AH \perp BC \implies AH \perp E'D \implies$". The seventeenth missing tag is <missing 17>, which is described as "The line $BE$ meets". This matches with <expression 2>, which is "$BE'F'$". The eighteenth missing tag is <missing 18>, which is described as "again at". This matches with <expression 26>, which is "$F \neq B$". The nineteenth missing tag is <missing 19>, which is described as "Prove that". This matches with <expression 19>, which is "$H, \angle ABS = \varphi$". The twentieth missing tag is <missing 20>, which is described as "Let". This matches with <expression 20>, which is "$TF$". The twenty-first missing tag is <missing 21>, which is described as "with radius". This matches with <expression 21>, which is "$AH$". The twenty-second missing tag is <missing 22>, which is described as "The". This matches with <expression 22>, which is "$F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =$". The twenty-third missing tag is <missing 23>, which is described as "meets $\Omega$ again at". This matches with <expression 23>, which is "$HF = HD$". The twenty-fourth missing tag is <missing 24>, which is described as "meets". This matches with <expression 27>, which is "$\Omega$". The twenty-fifth missing tag is <missing 25>, which is described as "again at". This matches with <expression 25>, which is "$E'D \|\| BC$". The twenty-sixth missing tag is <missing 26>, which is described as "We use Reim’s theorem for". This matches with <expression 29>, which is "$BF'$". The twenty-seventh missing tag is <missing 27>, which is described as "and lines $CDD$ and". This matches with <expression 30>, which is "$\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies$". The twenty-eighth missing tag is <missing 28>, which is described as "and get". This matches with <expression 28>, which is "$HD$". The twenty-ninth missing tag is <missing 29>, which is described as "The point". This matches with <expression 13>, which is "$\implies GF$". The thirtieth missing tag is <missing 30>, which is described as "is symmetric to $D$ with respect to $AH$". This matches with <expression 17>, which is "$F' \neq D, HD = HF'$". Answer: 1, 24, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 14, 15, 18, 12, 2, 26, 19, 20, 21, 22, 23, 27, 25, 29, 30, 28, 13, 17	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
Meta-Llama-3.1-70B-Instruct-Turbo	To solve this problem, we need to carefully match the formulae to the missing tags in the solution. Here's my step-by-step reasoning: 1. The first missing tag is <missing 1>. The solution states that the point diametrically opposite to a point $S$ through <missing 1> is the internal angle bisector of <missing 2>. This suggests that <missing 1> is related to the circumcircle of triangle $ABC$. Looking at the formulae, <expression 1> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 1> is likely to be <expression 1>. 2. The next missing tag is <missing 2>. The solution states that the point diametrically opposite to a point $S$ through <missing 1> is the internal angle bisector of <missing 2>. This suggests that <missing 2> is related to the internal angle bisector of $\angle BAC$. Looking at the formulae, <expression 24> is $\angle BAC$, which is the internal angle bisector of $\angle BAC$. Therefore, <missing 2> is likely to be <expression 24>. 3. The next missing tag is <missing 3>. The solution states that the crosspoint of $BS$ and <missing 3> through <missing 4>. Looking at the formulae, <expression 3> is $T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$, which is related to the crosspoint of $BS$ and $T$. Therefore, <missing 3> is likely to be <expression 3>. 4. The next missing tag is <missing 4>. The solution states that the crosspoint of $BS$ and <missing 3> through <missing 4>. Looking at the formulae, <expression 4> is $E' \neq F'$, which is not directly related to the crosspoint of $BS$ and $T$. However, <expression 9> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 4> is likely to be <expression 9>. 5. The next missing tag is <missing 5>. The solution states that denote <missing 5> <missing 6> is incenter of <missing 7>. Looking at the formulae, <expression 5> is $\triangle ABF$, which is not directly related to the incenter of a triangle. However, <expression 14> is $\triangle ABC, BHD$, which is related to the incenter of triangle $ABC$. Therefore, <missing 5> is likely to be <expression 14>. 6. The next missing tag is <missing 6>. The solution states that denote <missing 5> <missing 6> is incenter of <missing 7>. Looking at the formulae, <expression 6> is $\omega$, which is the circumcircle of triangle $BDL$. Therefore, <missing 6> is likely to be <expression 6>. 7. The next missing tag is <missing 7>. The solution states that denote <missing 5> <missing 6> is incenter of <missing 7>. Looking at the formulae, <expression 7> is $\omega, \Omega$, which is related to the incenter of triangle $ABC$. Therefore, <missing 7> is likely to be <expression 7>. 8. The next missing tag is <missing 8>. The solution states that denote <missing 8> is the orthocenter of $\triangle TSS'$. Looking at the formulae, <expression 8> is $=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$, which is related to the orthocenter of triangle $TSS'$. Therefore, <missing 8> is likely to be <expression 8>. 9. The next missing tag is <missing 9>. The solution states that <missing 9> points $B, G,$ and $F$ are collinear. Looking at the formulae, <expression 10> is $\omega$, which is the circumcircle of triangle $BDL$. Therefore, <missing 9> is likely to be <expression 10>. 10. The next missing tag is <missing 10>. The solution states that <missing 10> is symmetric to $AF$ with respect <missing 11>. Looking at the formulae, <expression 11> is $E'$, which is symmetric to $AF$ with respect to $H$. Therefore, <missing 10> is likely to be <expression 11>. 11. The next missing tag is <missing 11>. The solution states that <missing 10> is symmetric to $AF$ with respect <missing 11>. Looking at the formulae, <expression 12> is $AH \perp BC \implies AH \perp E'D \implies$, which is related to the symmetry of $AF$ with respect to $H$. Therefore, <missing 11> is likely to be <expression 12>. 12. The next missing tag is <missing 12>. The solution states that Let acute triangle <missing 12> be given. Looking at the formulae, <expression 16> is $\triangle ABC, AB > AC$, which is an acute triangle. Therefore, <missing 12> is likely to be <expression 16>. 13. The next missing tag is <missing 13>. The solution states that Let $H$ be the orthocenter of <missing 13>. Looking at the formulae, <expression 13> is $\implies GF$, which is related to the orthocenter of triangle $ABC$. Therefore, <missing 13> is likely to be <expression 13>. 14. The next missing tag is <missing 14>. The solution states that Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>. Looking at the formulae, <expression 15> is $S' \implies AS'$, which is related to the circle $BCD$. Therefore, <missing 14> is likely to be <expression 15>. 15. The next missing tag is <missing 15>. The solution states that Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>. Looking at the formulae, <expression 9> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 15> is likely to be <expression 9>. 16. The next missing tag is <missing 16>. The solution states that The point $E$ is symmetric to $D$ with respect to <missing 16>. Looking at the formulae, <expression 12> is $AH \perp BC \implies AH \perp E'D \implies$, which is related to the symmetry of $E$ with respect to $H$. Therefore, <missing 16> is likely to be <expression 12>. 17. The next missing tag is <missing 17>. The solution states that The line $BE$ meets <missing 17> again at <missing 18>. Looking at the formulae, <expression 10> is $\omega$, which is the circumcircle of triangle $BDL$. Therefore, <missing 17> is likely to be <expression 10>. 18. The next missing tag is <missing 18>. The solution states that The line $BE$ meets <missing 17> again at <missing 18>. Looking at the formulae, <expression 11> is $E'$, which is the point where the line $BE$ meets the circumcircle of triangle $BDL$ again. Therefore, <missing 18> is likely to be <expression 11>. 19. The next missing tag is <missing 19>. The solution states that Prove that <missing 19>. Looking at the formulae, <expression 19> is $H, \angle ABS = \varphi$, which is the statement to be proved. Therefore, <missing 19> is likely to be <expression 19>. 20. The next missing tag is <missing 20>. The solution states that Let <missing 20> be the circle centered at $H$ with radius <missing 21>. Looking at the formulae, <expression 20> is $TF$, which is the circle centered at $H$ with radius $AH$. Therefore, <missing 20> is likely to be <expression 20>. 21. The next missing tag is <missing 21>. The solution states that Let <missing 20> be the circle centered at $H$ with radius <missing 21>. Looking at the formulae, <expression 21> is $AH$, which is the radius of the circle centered at $H$. Therefore, <missing 21> is likely to be <expression 21>. 22. The next missing tag is <missing 22>. The solution states that The <missing 22> meets $\Omega$ again at <missing 23>. Looking at the formulae, <expression 22> is $F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =$, which is related to the point where the circle meets $\Omega$ again. Therefore, <missing 22> is likely to be <expression 22>. 23. The next missing tag is <missing 23>. The solution states that The <missing 22> meets $\Omega$ again at <missing 23>. Looking at the formulae, <expression 23> is $HF = HD$, which is the point where the circle meets $\Omega$ again. Therefore, <missing 23> is likely to be <expression 23>. 24. The next missing tag is <missing 24>. The solution states that Let $\omega$ meets <missing 24> again at <missing 25>. Looking at the formulae, <expression 24> is $\angle BAC$, which is the angle where $\omega$ meets again. Therefore, <missing 24> is likely to be <expression 24>. 25. The next missing tag is <missing 25>. The solution states that Let $\omega$ meets <missing 24> again at <missing 25>. Looking at the formulae, <expression 25> is $E'D \|\| BC$, which is the point where $\omega$ meets again. Therefore, <missing 25> is likely to be <expression 25>. 26. The next missing tag is <missing 26>. The solution states that We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27>. Looking at the formulae, <expression 26> is $F \neq B$, which is related to the application of Reim's theorem. Therefore, <missing 26> is likely to be <expression 26>. 27. The next missing tag is <missing 27>. The solution states that We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27>. Looking at the formulae, <expression 27> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 27> is likely to be <expression 27>. 28. The next missing tag is <missing 28>. The solution states that We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27> and get <missing 28>. Looking at the formulae, <expression 28> is $HD$, which is the result of applying Reim's theorem. Therefore, <missing 28> is likely to be <expression 28>. 29. The next missing tag is <missing 29>. The solution states that <missing 29> The point <missing 30> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD.$ Looking at the formulae, <expression 29> is $BF'$, which is related to the symmetry of $D$ with respect to $AH$. Therefore, <missing 29> is likely to be <expression 29>. 30. The next missing tag is <missing 30>. The solution states that <missing 29> The point <missing 30> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD.$ Looking at the formulae, <expression 30> is $\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies$, which is related to the symmetry of $D$ with respect to $AH$. Therefore, <missing 30> is likely to be <expression 30>. Answer: 1, 24, 3, 9, 14, 6, 7, 8, 10, 11, 12, 16, 13, 15, 9, 12, 10, 11, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
Qwen2-72B-Instruct	<Detailed reasoning> The solution starts by defining the point diametrically opposite to a point \(S\) through <missing 1>. This must refer to a circle, and since the context is about points on circles, it likely refers to \(\Omega\), which is the circumcircle of triangle \(ABC\). Therefore, <missing 1> corresponds to <expression 1>. Next, it mentions the internal angle bisector of <missing 2>. Given the context of the problem, this likely refers to the angle bisector of \(\angle BAC\), which is mentioned later in the solution. Thus, <missing 2> corresponds to <expression 24>. The third missing part, <missing 3>, is described as the crosspoint of \(BS\) and something else, which is defined through <missing 4>. Since the context involves circles and intersections, this could refer to the intersection of \(BS\) with another circle or line. However, without more specific information, we can only guess based on the expressions provided. Given the context of the problem, it seems logical that this would involve either \(\Omega\) or \(\omega\), but since <missing 4> is not yet filled, we cannot determine this directly. However, since the next sentence refers to \(AE \perp BC\), it's likely that <missing 3> and <missing 4> are related to the construction involving \(AE\) and possibly \(\omega\). Looking at the expressions, <expression 6> refers to \(\omega\), which fits the context of being a circle that might intersect with \(BS\). Therefore, <missing 3> could correspond to <expression 6>. Following this logic, <missing 4> should be related to the intersection or definition of \(\omega\). Given the context of the problem and the expressions available, <missing 4> could correspond to <expression 7>, which refers to the intersection of \(\omega\) and \(\Omega\). The fifth missing part, <missing 5>, is denoted as a point that is introduced later in the solution. Given the context, this could be a point on a circle or a line. Looking at the expressions, <expression 18> refers to a point on \(\omega\) other than \(B\), which fits the context of introducing a new point related to the circles involved. Therefore, <missing 5> could correspond to <expression 18>. The sixth missing part, <missing 6>, is described as being in the context of a triangle. Given the context of the problem, this likely refers to a point within triangle \(ABC\) or a triangle formed by points related to the construction. Looking at the expressions, <expression 5> refers to \(\triangle ABF\), which could fit the context of a triangle related to the points and circles discussed. Therefore, <missing 6> could correspond to <expression 5>. The seventh missing part, <missing 7>, is described as being the incenter of something. Given the context of the problem, this likely refers to the incenter of triangle \(ABC\) or a triangle formed by points related to the construction. Looking at the expressions, there isn't a direct reference to the incenter, but considering the context, it could be implied in the description of a triangle. Since <expression 5> refers to \(\triangle ABF\), and assuming that the incenter is relevant to the construction, <missing 7> could correspond to <expression 5>. The eighth missing part, <missing 8>, is described as being the orthocenter of a triangle. Given the context of the problem, this likely refers to the orthocenter of triangle \(ABC\) or a triangle formed by points related to the construction. Looking at the expressions, <expression 3> refers to the orthocenter of a triangle, which fits the context. Therefore, <missing 8> could correspond to <expression 3>. The ninth missing part, <missing 9>, is described as points being collinear. Given the context of the problem, this likely refers to a statement about the alignment of points on a line. Looking at the expressions, there isn't a direct reference to collinearity, but considering the context, it could be implied in the description of points and their relationships. Since the context involves points on lines and circles, <missing 9> could correspond to a statement about points on a line or circle. Without a direct match, we can only guess based on the context, but none of the expressions directly state collinearity. However, given the context of the problem, it's likely related to points on a line or circle, so we'll have to make an educated guess here. Considering the context of the problem, <missing 9> could correspond to <expression 29>, which refers to a line related to points in the construction. The tenth missing part, <missing 10>, is described as being symmetric to something with respect to something else. Given the context of the problem, this likely refers to a point or line being symmetric to another point or line with respect to a line or axis. Looking at the expressions, <expression 12> refers to a symmetry with respect to a line, which fits the context. Therefore, <missing 10> could correspond to <expression 12>. The eleventh missing part, <missing 11>, is described as being with respect to something. Given the context of the problem, this likely refers to a line or axis with respect to which a symmetry or reflection is considered. Looking at the expressions, <expression 11> refers to a line that could serve as an axis of symmetry. Therefore, <missing 11> could correspond to <expression 11>. Answer: 1, 24, 6, 7, 18, 5, 5, 3, 29, 12, 11	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
baichuan4	Based on the context and the information provided, let's try to match the formulae to the missing tags in the solution. 1. <missing 1>: This refers to the circumcircle of triangle $ABC$, which is $\Omega$. So, it should be filled by <expression 1>. 2. <missing 2>: This refers to the angle $\angle BAC$, which is mentioned in <expression 24>. 3. <missing 3>: This refers to the line tangent to $\omega$ at $P$, which is mentioned in <expression 10>. 4. <missing 4>: This refers to the point $P$, which is mentioned in <expression 23>. 5. <missing 5>: This refers to the point $T$, which is mentioned in <expression 3>. 6. <missing 6>: This refers to the point $H$, which is mentioned in <expression 19>. 7. <missing 7>: This refers to the triangle $ABF$, which is mentioned in <expression 5>. 8. <missing 8>: This refers to the point $F$, which is mentioned in <expression 22>. 9. <missing 9>: This refers to the circumcircle $\Omega$, which is mentioned in <expression 9>. 10. <missing 10>: This refers to the circle $\omega$, which is mentioned in <expression 10>. 11. <missing 11>: This refers to the point $E'$, which is mentioned in <expression 11>. 12. <missing 12>: This refers to the triangle $ABC$, which is mentioned in <expression 14>. 13. <missing 13>: This refers to the triangle $BHD$, which is mentioned in <expression 14>. 14. <missing 14>: This refers to the circle $BCD$, which is mentioned in <expression 15>. 15. <missing 15>: This refers to the point $S'$, which is mentioned in <expression 15>. 16. <missing 16>: This refers to the triangle $ABC$, which is mentioned in <expression 16>. 17. <missing 17>: This refers to the point $F'$, which is mentioned in <expression 17>. 18. <missing 18>: This refers to the line $AS'$, which is mentioned in <expression 18>. 19. <missing 19>: This refers to the point $H$, which is mentioned in <expression 19>. 20. <missing 20>: This refers to the circle centered at $H$ with radius $AH$, which is mentioned in <expression 21>. 21. <missing 21>: This refers to the line $AH$, which is mentioned in <expression 21>. 22. <missing 22>: This refers to the point $F$, which is mentioned in <expression 22>. 23. <missing 23>: This refers to the point $H$, which is mentioned in <expression 23>. 24. <missing 24>: This refers to the circle $\omega$, which is mentioned in <expression 10>. 25. <missing 25>: This refers to the point $P$, which is mentioned in <expression 23>. 26. <missing 26>: This refers to the point $F$, which is mentioned in <expression 26>. 27. <missing 27>: This refers to the circumcircle $\Omega$, which is mentioned in <expression 27>. 28. <missing 28>: This refers to the line $HD$, which is mentioned in <expression 28>. 29. <missing 29>: This refers to the line $BF'$, which is mentioned in <expression 29>. 30. <missing 30>: This refers to the point $G$, which is mentioned in <expression 30>. Answer: 1, 24, 10, 23, 3, 19, 5, 22, 9, 10, 11, 14, 14, 15, 15, 16, 17, 18, 19, 21, 21, 22, 23, 10, 26, 27, 28, 29, 30	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
doubao	<Detailed reasoning>: In the solution, "through <missing 1> is the internal angle bisector of <missing 2>" likely refers to the circumcircle, so <missing 1> is likely <expression 1>. For "Denote the crosspoint of $BS$ and <missing 3> through <missing 4>", it seems <missing 3> could be <expression 5> and <missing 4> could be <expression 6>. For "Denote <missing 5> <missing 6> is incenter of <missing 7>", <missing 5> could be <expression 7> and <missing 6> could be <expression 8>. For "Denote <missing 8> is the orthocenter of $\triangle TSS'$", <missing 8> could be <expression 3>. Answer: 1,5,7,3	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
gpto1	To solve this problem, we'll match each <missing X> in the solution to the appropriate expression from the list provided. Here's the step-by-step reasoning: 1. Missing 1: The sentence begins with "Denote the point diametrically opposite to a point \( S \) through <missing 1>". Since the point diametrically opposite \( S \) is found through the circle \( \Omega \), we can conclude that <missing 1> is Expression 1: \( \Omega. \) 2. Missing 2: The sentence continues "...is the internal angle bisector of <missing 2>". The internal angle bisector is of angle \( \angle BAC \), so <missing 2> is Expression 24: \( \angle BAC \). 3. Missing 3: "Denote the crosspoint of \( BS \) and <missing 3> through <missing 4>". The crosspoint of \( BS \) and \( AS' \) is significant in the context, so <missing 3> is Expression 18: \( AS' \). 4. Missing 4: Since the crosspoint is through point \( T \) (the intersection of \( BS \) and \( AS' \)), and Expression 3 introduces \( T \), <missing 4> is Expression 3: \( T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H \). 5. Missing 5: After stating we need to prove \( PH = AH \), it says "Denote <missing 5>", referring back to \( H \) and an angle \( \varphi \). So, <missing 5> is Expression 19: \( H, \angle ABS = \varphi. \) 6. Missing 6: It continues "<missing 6> is incenter of <missing 7>". The incenter is of triangle \( ABC \), so <missing 6> is Expression 21: \( AH. \) 7. Missing 7: Since the incenter is of triangle \( ABC \), <missing 7> is Expression 16: \( \triangle ABC, AB > AC \). 8. Missing 8: "Denote <missing 8> is the orthocenter of \( \triangle TSS' \).” The orthocenter is point \( H \), so <missing 8> is Expression 12: \( AH \perp BC \implies AH \perp E'D \implies \). 9. Missing 9: Before mentioning points \( B, G, \) and \( F \) are collinear, it's logical to reference the main circle, so <missing 9> is Expression 9: \( \Omega \). 10. Missing 10: "Point \( F \) is symmetric to \( AF \) with respect...", considering symmetry and circles, <missing 10> is Expression 10: \( \omega \). 11. Missing 11: The symmetry is with respect to point \( E' \), so <missing 11> is Expression 11: \( E' \). 12. Missing 12: In Lemma 1, "Let acute triangle <missing 12> be given." The triangle is \( ABC \) with \( AB > AC \), so <missing 12> is Expression 16: \( \triangle ABC, AB > AC \). 13. Missing 13: "Let \( H \) be the orthocenter of <missing 13> be the height." This refers to \( AH \), so <missing 13> is Expression 12: \( AH \perp BC \implies AH \perp E'D \implies \). 14. Missing 14: "Let <missing 14> be the circle \( BCD \).” This circle is \( \omega \), so <missing 14> is Expression 6: \( \omega \). 15. Missing 15: " \( BC \) is the diameter of <missing 15>." The diameter is in circle \( \Omega \), so <missing 15> is Expression 9: \( \Omega \). 16. Missing 16: "The point \( E \) is symmetric to \( D \) with respect to <missing 16>." The symmetry is with respect to \( E' \), so <missing 16> is Expression 11: \( E' \). 17. Missing 17: "The line \( BE \) meets <missing 17> again at..." Since it meets \( \Omega \), <missing 17> is Expression 27: \( \Omega \). 18. Missing 18: The point it meets is \( HD = HF' \), so <missing 18> is Expression 23: \( HF = HD. \) 19. Missing 19: "Prove that <missing 19>." The goal is to prove \( HF = HD \), so <missing 19> is Expression 28: \( HD. \) 20. Missing 20: "Let <missing 20> be the circle centered at \( H \)..." This refers to \( AH \), so <missing 20> is Expression 21: \( AH. \) 21. Missing 21: The radius is \( AH \), so <missing 21> is Expression 24: \( \angle BAC \). 22. Missing 22: "The <missing 22> meets \( \Omega \) again at..." This refers to \( \Omega \), so <missing 22> is Expression 9: \( \Omega \). 23. Missing 23: It meets again at a point where \( E'D \|\| BC \), so <missing 23> is Expression 25: \( E'D \|\| BC \). 24. Missing 24: "Let \( \omega \) meets <missing 24> again at..." Since \( \omega \) and \( \Omega \) meet, <missing 24> is Expression 7: \( \omega, \Omega \). 25. Missing 25: It meets again at \( \Omega \), so <missing 25> is Expression 27: \( \Omega \). 26. Missing 26: "We use Reim’s theorem for <missing 26> and lines..." The circles involved are \( \omega \) and \( \Omega \), so <missing 26> is Expression 7: \( \omega, \Omega \). 27. Missing 27: "and lines \( CDD \) and <missing 27>..." Since it's another reference to \( \Omega \), <missing 27> is Expression 27: \( \Omega \). 28. Missing 28: "and get <missing 28>" refers back to \( HD \), so <missing 28> is Expression 28: \( HD. \) 29. Missing 29: The next step likely involves \( BF' \), so <missing 29> is Expression 29: \( BF' \). 30. Missing 30: Finally, relating angles and implying something about \( \angle DBG = \varphi \), so <missing 30> is Expression 30: Given angle relationships. Answer: 1, 24, 18, 3, 19, 21, 16, 12, 9, 10, 11, 16, 12, 6, 9, 11, 27, 23, 28, 21, 24, 9, 25, 7, 27, 23, 7, 27, 28, 30	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
kimi	<Detailed reasoning> 1. The first missing tag is referring to the diametrically opposite point to $S$. Since $S$ is the midpoint of the arc $CB$ containing $A$, the diametrically opposite point to $S$ would be on the circumcircle $\Omega$. So, the expression that fits here is <expression 1>. 2. The second missing tag is referring to the internal angle bisector of a certain angle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 24>. 3. The third missing tag is referring to the crosspoint of $BS$ and another line. Since the line through $D$ parallel to $BC$ meets line $BE$ at $L$, the expression that fits here is <expression 3>. 4. The fourth missing tag is referring to a point on the line through $D$ parallel to $BC$. Since this line meets line $BE$ at $L$, the expression that fits here is <expression 4>. 5. The fifth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 5>. 6. The sixth missing tag is referring to a circle. Since the circumcircle of triangle $BDL$ is denoted by $\omega$, the expression that fits here is <expression 6>. 7. The seventh missing tag is referring to two circles. Since the circumcircle of triangle $BDL$ is denoted by $\omega$ and the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 7>. 8. The eighth missing tag is referring to an angle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 8>. 9. The ninth missing tag is referring to a circle. Since the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 9>. 10. The tenth missing tag is referring to a circle. Since the circumcircle of triangle $BDL$ is denoted by $\omega$, the expression that fits here is <expression 10>. 11. The eleventh missing tag is referring to a point. Since the line through $D$ parallel to $BC$ meets line $BE$ at $L$, the expression that fits here is <expression 11>. 12. The twelfth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 12>. 13. The thirteenth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 13>. 14. The fourteenth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 14>. 15. The fifteenth missing tag is referring to a circle. Since the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 15>. 16. The sixteenth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 16>. 17. The seventeenth missing tag is referring to a point. Since the line $BE$ meets $\Omega$ again at $E \neq A$, the expression that fits here is <expression 17>. 18. The eighteenth missing tag is referring to a point. Since the line $BE$ meets $\Omega$ again at $E \neq A$, the expression that fits here is <expression 18>. 19. The nineteenth missing tag is referring to a point. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 19>. 20. The twentieth missing tag is referring to a circle. Since the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 20>. 21. The twenty-first missing tag is referring to a radius. Since the radius of the circle centered	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
minimax	To solve this problem, we need to match the missing parts of the solution with the corresponding formulae provided. Let's go through the solution step by step and match the missing tags with the expressions. 1. The first missing tag is about the point diametrically opposite to a point $S$ through some operation or line. The only formula that fits this description is <expression 1> $\Omega$, which indicates that the point is found through the circumcircle $\Omega$. 2. The second missing tag refers to the internal angle bisector of some angle. The only formula that fits this description is <expression 2> $BE'F'$, which suggests that the angle bisector is related to triangle $BE'F'$. 3. The third missing tag is about the crosspoint of $BS$ and some line. The only formula that fits this description is <expression 3> $T = S'B \cap SA$, which indicates that the crosspoint is found at the intersection of $S'B$ and $SA$. 4. The fourth missing tag is about the point $E'$ not being equal to some other point $F'$. The only formula that fits this description is <expression 4> $E' \neq F	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
wenxinyiyan	<Detailed reasoning> Let's go through the solution step by step and match the missing tags with the given formulae. 1. "<missing 1>" should be the circumcircle of triangle ABC, which is denoted as "\Omega" in the formulae. So, <missing 1> corresponds to expression 1. 2. "<missing 2>" refers to the internal angle bisector of a certain angle. From the context, it is clear that it refers to "\angle BAC". So, we do not have a direct formula for this, but we can identify it from the given information. 3. "<missing 3>" is the crosspoint of BS and a certain line. From the context, it seems to be the line AE, which is perpendicular to BC. So, <missing 3> corresponds to the point where BS intersects AE, which is denoted as "T" in the formulae (expression 3). 4. "<missing 4>" is related to the previous missing tag and seems to refer to the line through a certain point parallel to BC. From the context, it is clear that it refers to the line through D parallel to BC, which meets line BE at L. We do not have a direct formula for this line, but we can identify it from the given information. 5. "<missing 5>" and "<missing 6>" seem to refer to a point and a circle respectively. From the context, it is clear that <missing 5> refers to the point "H" (the orthocenter of triangle TSS') and <missing 6> refers to the circumcircle of triangle BDL, which is denoted as "\omega" in the formulae. So, <missing 5> corresponds to expression 3 (indirectly, as H is defined there) and <missing 6> corresponds to expression 6. 6. "<missing 7>" refers to a triangle whose incenter is mentioned. From the context, it is clear that it refers to "\triangle ABF". So, <missing 7> corresponds to expression 5. 7. "<missing 8>" to "<missing 11>" seem to be related to a lemma used in the solution. We can identify them from the lemma provided. Specifically, <missing 8> refers to the orthocenter of triangle TSS', which is "H" (expression 3). <missing 9> refers to the circumcircle "\Omega". <missing 10> refers to the circle "\omega". <missing 11> refers to the line "BE'", which is symmetric to "AF" with respect to "AH". So, <missing 8> corresponds to expression 3, <missing 9> corresponds to expression 1, <missing 10> corresponds to expression 6, and <missing 11> corresponds to the line BE' defined in the lemma (no direct formula, but can be identified). 8. "<missing 12>" to "<missing 30>" are all related to the lemma provided at the end of the solution. We can identify them from the lemma as follows: * <missing 12> refers to the acute triangle "ABC", so it corresponds to expression 16 (indirectly). * <missing 13> and <missing 14> refer to the triangle and its height mentioned in the lemma. From the context, it is clear that they refer to "\triangle ABC" and "AH" respectively. So, <missing 13> corresponds to expression 16 and <missing 14> corresponds to the height "AH" defined in the lemma (no direct formula, but can be identified). * <missing 15> to <missing 30> can be identified similarly from the lemma. Specifically, <missing 15> refers to the circle "BCD" with BC as the diameter (no direct formula), <missing 16> refers to the line "AH" (no direct formula), <missing 17> refers to the circle "\Omega", <missing 18> refers to the point "E", <missing 19> refers to the angle "\angle BAC" (expression 24), <missing 20> refers to the circle centered at H with a certain radius (no direct formula), <missing 21> refers to the radius "AH" (no direct formula), <missing 22> refers to the circle "\Omega", <missing 23> refers to a point where \omega meets \Omega (no direct formula), <missing 24> refers to the line "CD" (no direct formula), <missing 25> refers to a point where \omega meets a certain line (no direct formula), <missing 26> refers to a certain configuration of circles and lines mentioned in the lemma	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
zhipu	### Detailed Reasoning 1. <missing 1>: This should refer to a point or a property related to the circumcircle $\Omega$. The most relevant expression is `<expression 1> \Omega.` which seems to fit as it refers to the circumcircle. 2. <missing 2>: This should be an angle or a property related to the triangle. The most relevant expression is `<expression 24> \angle BAC` which fits as it refers to the angle $\angle BAC$. 3. <missing 3>: This should be a line or a property related to the triangle. The most relevant expression is `<expression 18> AS'` which fits as it refers to the line $AS'$. 4. <missing 4>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 3> T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H` which fits as it defines a point $T$ and relates it to $S'$ and $H$. 5. <missing 5>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 15> S' \implies AS'` which fits as it refers to the point $S'$. 6. <missing 6>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 14> \triangle ABC, BHD` which fits as it refers to the triangle $ABC$ and the point $H$. 7. <missing 7>: This should be a triangle or a property related to the triangle. The most relevant expression is `<expression 5> \triangle ABF.` which fits as it refers to the triangle $ABF$. 8. <missing 8>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 3> T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H` which fits as it defines a point $H$. 9. <missing 9>: This should be a property or a statement related to collinearity. The most relevant expression is `<expression 13> \implies GF` which fits as it implies a relationship between points $G$ and $F$. 10. <missing 10>: This should be a property or a statement related to symmetry. The most relevant expression is `<expression 11> E'` which fits as it refers to the symmetric point $E'$. 11. <missing 11>: This should be a property or a statement related to symmetry. The most relevant expression is `<expression 1> \Omega.` which fits as it refers to the circumcircle $\Omega$. 12. <missing 12>: This should be a triangle or a property related to the triangle. The most relevant expression is `<expression 5> \triangle ABF.` which fits as it refers to the triangle $ABF$. 13. <missing 13>: This should be a triangle or a property related to the triangle. The most relevant expression is `<expression 14> \triangle ABC, BHD` which fits as it refers to the triangle $ABC$. 14. <missing 14>: This should be a circle or a property related to the circle. The most relevant expression is `<expression 6> \omega` which fits as it refers to the circle $\omega$. 15. <missing 15>: This should be a circle or a property related to the circle. The most relevant expression is `<expression 9> \Omega` which fits as it refers to the circle $\Omega$. 16. <missing 16>: This should be a line or a property related to the triangle. The most relevant expression is `<expression 25> E'D \|\| BC` which fits as it refers to the line $E'D$. 17. <missing 17>: This should be a circle or a property related to the circle. The most relevant expression is `<expression 10> \omega` which fits as it refers to the circle $\omega$. 18. <missing 18>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 2> BE'F'` which fits as it refers to the point $F'$. 19. <missing 19>: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 19> H, \angle ABS = \varphi.` which fits as it refers to the angle $\angle ABS$. 20. <missing 20>: This should be a circle or a property related to the circle. The most relevant expression is `<expression 20> TF` which fits as it refers to the circle centered at $T$. 21. <missing 21>: This should be a length or a property related to the triangle. The most relevant expression is `<expression 21> AH` which fits as it refers to the length $AH$. 22. <missing 22>: This should be a circle or a property related to the circle. The most relevant expression is `<expression 6> \omega` which fits as it refers to the circle $\omega$. 23. <missing 23>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 23> HF = HD` which fits as it refers to the point $F$. 24. <missing 24>: This should be a circle or a property related to the circle. The most relevant expression is `<expression 10> \omega` which fits as it refers to the circle $\omega$. 25. <missing 25>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 2> BE'F'` which fits as it refers to the point $F'$. 26. <missing 26>: This should be a point or a property related to the triangle. The most relevant expression is `<expression 26> F \neq B` which fits as it refers to the point $F$. 27. <missing 27>: This should be a circle or a property related to the circle. The most relevant expression is `<expression 27> \Omega` which fits as it refers to the circle $\Omega$. 28. <missing 28>: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 28> HD` which fits as it refers to the length $HD$. 29. <missing 29>: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 29> BF'` which fits as it refers to the line $BF'$. 30. <missing 30>: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 30> \angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies` which fits as it refers to the angles involving $E$, $B$, $S'$, $L$, $D$, and $G$. ### Answer 1, 24, 18, 3, 15, 14, 5, 3, 13, 11, 1, 5, 14, 6, 9, 25, 10, 2, 19, 20, 21, 6, 23, 10, 2, 26, 27, 28, 29, 30	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11
deepseek-v3	Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-1e8ab241-29df-98cf-86a1-6a7f44d9394e', 'request_id': '1e8ab241-29df-98cf-86a1-6a7f44d9394e'}	15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

模型

模型答案

正确答案

金声玉亮2.0

To match the formulae to the missing tags in the solution, let's go through the solution step by step and identify which expression fits each <missing X> tag.

1. **<missing 1>**: "Denote the point diametrically opposite to a point $S$ through <missing 1>" - This refers to the circumcircle of the triangle, so it should be <expression 1> \Omega.

2. **<missing 2>**: "is the internal angle bisector of <missing 2>" - This refers to the angles in the triangle, specifically the internal angle bisector of some angle. The relevant expression here seems to be <expression 24> \angle BAC.

3. **<missing 3>**: "Denote the crosspoint of $BS$ and <missing 3> through <missing 4>" - This refers to a point of intersection, and from the context, it seems to be related to the height or perpendicular from A to BC. The relevant expression here is <expression 12> AH \perp BC \implies AH \perp E'D \implies.

4. **<missing 4>**: Continuing from the previous point, the crosspoint is denoted through some point, which seems to be related to the height or perpendicular from A to BC. The relevant expression here is <expression 12> AH \perp BC \implies AH \perp E'D \implies.

5. **<missing 5>**: "To finishing the solution we need only to prove that $PH = AH.$ Denote <missing 5>" - This refers to a triangle where we need to prove some equality. The relevant expression here is <expression 5> \triangle ABF.

6. **<missing 6>**: "is incenter of <missing 7>" - This refers to the incenter of a triangle. The relevant expression here is <expression 7> \omega, \Omega.

7. **<missing 7>**: Continuing from the previous point, the incenter is of some triangle. The relevant expression here is <expression 7> \omega, \Omega.

8. **<missing 8>**: "is the orthocenter of $\triangle TSS'.$ Denote <missing 8>" - This refers to the orthocenter of a triangle. The relevant expression here is <expression 8> =\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H.

9. **<missing 9>**: "points $B, G,$ and $F$ are collinear <missing 10>" - This refers to the circumcircle where points are collinear. The relevant expression here is <expression 9> \Omega.

10. **<missing 10>**: Continuing from the previous point, the collinear points are related to some circle. The relevant expression here is <expression 10> \omega.

11. **<missing 11>**: "is symmetric to $AF$ with respect <missing 11>" - This refers to symmetry with respect to some point. The relevant expression here is <expression 11> E'.

12. **<missing 12>**: "Let acute triangle <missing 12> be given." - This refers to the given triangle. The relevant expression here is <expression 16> \triangle ABC, AB > AC.

13. **<missing 13>**: "Let $H$ be the orthocenter of <missing 13>" - This refers to the orthocenter of the triangle. The relevant expression here is <expression 16> \triangle ABC, AB > AC.

14. **<missing 14>**: "Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>" - This refers to the circle with BC as the diameter. The relevant expression here is <expression 15> S' \implies AS'.

15. **<missing 15>**: Continuing from the previous point, the diameter of the circle. The relevant expression here is <expression 15> S' \implies AS'.

16. **<missing 16>**: "The point $E$ is symmetric to $D$ with respect to <missing 16>" - This refers to symmetry with respect to some point. The relevant expression here is <expression 18> AS'.

17. **<missing 17>**: "The line $BE$ meets <missing 17> again at <missing 18>" - This refers to the circle where the line meets again. The relevant expression here is <expression 9> \Omega.

18. **<missing 18>**: Continuing from the previous point, the point where the line meets again. The relevant expression here is <expression 4> E' \neq F'.

19. **<missing 19>**: "Prove that <missing 19>" - This refers to the proof statement. The relevant expression here is <expression 23> HF = HD.

20. **<missing 20>**: "Let <missing 20> be the circle centered at $H$ with radius <missing 21>" - This refers to the circle centered at H. The relevant expression here is <expression 20> TF.

21. **<missing 21>**: Continuing from the previous point, the radius of the circle. The relevant expression here is <expression 21> AH.

22. **<missing 22>**: "The <missing 22> meets $\Omega$ again at <missing 23>" - This refers to the circle meeting $\Omega$ again. The relevant expression here is <expression 22> F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =.

23. **<missing 23>**: Continuing from the previous point, the point where the circle meets $\Omega$ again. The relevant expression here is <expression 23> HF = HD.

24. **<missing 24>**: "Let $\omega$ meets <missing 24> again at <missing 25>" - This refers to the circle $\omega$ meeting something again. The relevant expression here is <expression 10> \omega.

25. **<missing 25>**: Continuing from the previous point, the point where $\omega$ meets again. The relevant expression here is <expression 4> E' \neq F'.

26. **<missing 26>**: "We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27> and get <missing 28>" - This refers to some point in the theorem. The relevant expression here is <expression 26> F \neq B.

27. **<missing 27>**: Continuing from the previous point, the lines in the theorem. The relevant expression here is <expression 27> \Omega.

28. **<missing 28>**: Continuing from the previous point, the result of the theorem. The relevant expression here is <expression 28> HD.

29. **<missing 29>**: "The point <missing 30> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD." - This refers to symmetry with respect to some point. The relevant expression here is <expression 29> BF'.

30. **<missing 30>**: Continuing from the previous point, the point of symmetry. The relevant expression here is <expression 30> \angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies.

Answer: 1, 24, 12, 12, 5, 7, 7, 8, 9, 10, 11, 16, 16, 15, 15, 18, 9, 4, 23, 20, 21, 22, 23, 10, 4, 26, 27, 28, 29, 30

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

deepseek-r1

Answer: 15,24,3,22,5,6,14,19,13,29,21,16,14,9,21,6,22,23,23,28,21,22,23,9,25,26,27,23,17,30

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

gpto1

To solve this problem, we'll match each <missing X> in the solution to the appropriate expression from the list provided. Here's the step-by-step reasoning:

1. **Missing 1:** The sentence begins with "Denote the point diametrically opposite to a point $ S $ through <missing 1>". Since the point diametrically opposite $ S $ is found through the circle $ \Omega $, we can conclude that **<missing 1>** is **Expression 1: $ \Omega. $**

2. **Missing 2:** The sentence continues "...is the internal angle bisector of <missing 2>". The internal angle bisector is of angle $ \angle BAC $, so **<missing 2>** is **Expression 24: $ \angle BAC $.**

3. **Missing 3:** "Denote the crosspoint of $ BS $ and <missing 3> through <missing 4>". The crosspoint of $ BS $ and $ AS' $ is significant in the context, so **<missing 3>** is **Expression 18: $ AS' $.**

4. **Missing 4:** Since the crosspoint is through point $ T $ (the intersection of $ BS $ and $ AS' $), and Expression 3 introduces $ T $, **<missing 4>** is **Expression 3: $ T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H $.**

5. **Missing 5:** After stating we need to prove $ PH = AH $, it says "Denote <missing 5>", referring back to $ H $ and an angle $ \varphi $. So, **<missing 5>** is **Expression 19: $ H, \angle ABS = \varphi. $**

6. **Missing 6:** It continues "<missing 6> is incenter of <missing 7>". The incenter is of triangle $ ABC $, so **<missing 6>** is **Expression 21: $ AH. $**

7. **Missing 7:** Since the incenter is of triangle $ ABC $, **<missing 7>** is **Expression 16: $ \triangle ABC, AB > AC $.**

8. **Missing 8:** "Denote <missing 8> is the orthocenter of $ \triangle TSS' $.” The orthocenter is point $ H $, so **<missing 8>** is **Expression 12: $ AH \perp BC \implies AH \perp E'D \implies $.**

9. **Missing 9:** Before mentioning points $ B, G, $ and $ F $ are collinear, it's logical to reference the main circle, so **<missing 9>** is **Expression 9: $ \Omega $.**

10. **Missing 10:** "Point $ F $ is symmetric to $ AF $ with respect...", considering symmetry and circles, **<missing 10>** is **Expression 10: $ \omega $.**

11. **Missing 11:** The symmetry is with respect to point $ E' $, so **<missing 11>** is **Expression 11: $ E' $.**

12. **Missing 12:** In Lemma 1, "Let acute triangle <missing 12> be given." The triangle is $ ABC $ with $ AB > AC $, so **<missing 12>** is **Expression 16: $ \triangle ABC, AB > AC $.**

13. **Missing 13:** "Let $ H $ be the orthocenter of <missing 13> be the height." This refers to $ AH $, so **<missing 13>** is **Expression 12: $ AH \perp BC \implies AH \perp E'D \implies $.**

14. **Missing 14:** "Let <missing 14> be the circle $ BCD $.” This circle is $ \omega $, so **<missing 14>** is **Expression 6: $ \omega $.**

15. **Missing 15:** " $ BC $ is the diameter of <missing 15>." The diameter is in circle $ \Omega $, so **<missing 15>** is **Expression 9: $ \Omega $.**

16. **Missing 16:** "The point $ E $ is symmetric to $ D $ with respect to <missing 16>." The symmetry is with respect to $ E' $, so **<missing 16>** is **Expression 11: $ E' $.**

17. **Missing 17:** "The line $ BE $ meets <missing 17> again at..." Since it meets $ \Omega $, **<missing 17>** is **Expression 27: $ \Omega $.**

18. **Missing 18:** The point it meets is $ HD = HF' $, so **<missing 18>** is **Expression 23: $ HF = HD. $**

19. **Missing 19:** "Prove that <missing 19>." The goal is to prove $ HF = HD $, so **<missing 19>** is **Expression 28: $ HD. $**

20. **Missing 20:** "Let <missing 20> be the circle centered at $ H $..." This refers to $ AH $, so **<missing 20>** is **Expression 21: $ AH. $**

21. **Missing 21:** The radius is $ AH $, so **<missing 21>** is **Expression 24: $ \angle BAC $.**

22. **Missing 22:** "The <missing 22> meets $ \Omega $ again at..." This refers to $ \Omega $, so **<missing 22>** is **Expression 9: $ \Omega $.**

23. **Missing 23:** It meets again at a point where $ E'D || BC $, so **<missing 23>** is **Expression 25: $ E'D || BC $.**

24. **Missing 24:** "Let $ \omega $ meets <missing 24> again at..." Since $ \omega $ and $ \Omega $ meet, **<missing 24>** is **Expression 7: $ \omega, \Omega $.**

25. **Missing 25:** It meets again at $ \Omega $, so **<missing 25>** is **Expression 27: $ \Omega $.**

26. **Missing 26:** "We use Reim’s theorem for <missing 26> and lines..." The circles involved are $ \omega $ and $ \Omega $, so **<missing 26>** is **Expression 7: $ \omega, \Omega $.**

27. **Missing 27:** "and lines $ CDD $ and <missing 27>..." Since it's another reference to $ \Omega $, **<missing 27>** is **Expression 27: $ \Omega $.**

28. **Missing 28:** "and get <missing 28>" refers back to $ HD $, so **<missing 28>** is **Expression 28: $ HD. $**

29. **Missing 29:** The next step likely involves $ BF' $, so **<missing 29>** is **Expression 29: $ BF' $.**

30. **Missing 30:** Finally, relating angles and implying something about $ \angle DBG = \varphi $, so **<missing 30>** is **Expression 30: Given angle relationships.**

**Answer:** 1, 24, 18, 3, 19, 21, 16, 12, 9, 10, 11, 16, 12, 6, 9, 11, 27, 23, 28, 21, 24, 9, 25, 7, 27, 23, 7, 27, 28, 30

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

gpt-4o-2024-08-06

To solve this problem, we need to match the missing parts in the solution with the given expressions. Let's go through the solution step by step and see which expressions fit best.

1. **First Missing Part:**
- "Denote the point diametrically opposite to a point $S$ through <missing 1> is the internal angle bisector of <missing 2>."
- We are looking for something related to a circle, possibly the circumcircle $\Omega$.
- **Expression 1**: $\Omega$ seems to fit here as it refers to the circumcircle.

2. **Second Missing Part:**
- Continuing from the first sentence, we need something that is an angle or a line related to the internal angle bisector.
- **Expression 24**: $\angle BAC$ fits well here as it is the angle in question.

3. **Third Missing Part:**
- "Denote the crosspoint of $BS$ and <missing 3> through <missing 4>."
- We need something that is a line or a segment that $BS$ intersects.
- **Expression 3**: $T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$ seems to describe a geometric configuration involving $BS$.

4. **Fourth Missing Part:**
- This part seems to be a continuation of the previous sentence, possibly describing a condition or a result of the intersection.
- **Expression 19**: $H, \angle ABS = \varphi.$ could fit here as it describes an angle condition.

5. **Fifth Missing Part:**
- "To finishing the solution we need only to prove that $PH = AH.$ Denote <missing 5>"
- We are looking for a triangle or a geometric figure.
- **Expression 5**: $\triangle ABF.$ fits as it denotes a triangle.

6. **Sixth Missing Part:**
- "Denote <missing 6> is incenter of <missing 7>"
- We need something that can be an incenter.
- **Expression 6**: $\omega$ fits as it is a circle, and the incenter is often related to circles.

7. **Seventh Missing Part:**
- We need something that can be a circle or a set of circles.
- **Expression 7**: $\omega, \Omega$ fits as it refers to both circles.

8. **Eighth Missing Part:**
- "Denote <missing 8> is the orthocenter of $\triangle TSS'.$"
- We need something that can be an orthocenter.
- **Expression 8**: $\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$ fits as it refers to an orthocenter.

9. **Ninth Missing Part:**
- "2023 IMO 2 lemma.png <missing 9> points $B, G,$ and $F$ are collinear"
- We need something that refers to a geometric figure or a circle.
- **Expression 9**: $\Omega$ fits as it refers to the circumcircle.

10. **Tenth Missing Part:**
- "points $B, G,$ and $F$ are collinear <missing 10> is symmetric to $AF$ with respect"
- We need something that can be symmetric.
- **Expression 10**: $\omega$ fits as it is a circle, and symmetry is often discussed in terms of circles.

11. **Eleventh Missing Part:**
- "is symmetric to $AF$ with respect <missing 11>"
- We need something that can be a line or a point of symmetry.
- **Expression 11**: $E'$ fits as it refers to a point.

12. **Twelfth Missing Part:**
- "Let acute triangle <missing 12> be given."
- We need something that describes a triangle.
- **Expression 14**: $\triangle ABC, BHD$ fits as it describes a triangle.

13. **Thirteenth Missing Part:**
- "Let $H$ be the orthocenter of <missing 13> be the height."
- We need something that can be a triangle or a line.
- **Expression 16**: $\triangle ABC, AB > AC$ fits as it describes a triangle.

14. **Fourteenth Missing Part:**
- "Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>"
- We need something that describes a circle.
- **Expression 2**: $BE'F'$ fits as it describes a circle.

15. **Fifteenth Missing Part:**
- "BC is the diameter of <missing 15>"
- We need something that can be a circle.
- **Expression 9**: $\Omega$ fits as it refers to the circumcircle.

16. **Sixteenth Missing Part:**
- "The point $E$ is symmetric to $D$ with respect to <missing 16>"
- We need something that can be a line or a point of symmetry.
- **Expression 11**: $E'$ fits as it refers to a point.

17. **Seventeenth Missing Part:**
- "The line $BE$ meets <missing 17> again at <missing 18>"
- We need something that can be a circle or a line.
- **Expression 27**: $\Omega$ fits as it refers to the circumcircle.

18. **Eighteenth Missing Part:**
- We need something that can be a point.
- **Expression 18**: $AS'$ fits as it refers to a point.

19. **Nineteenth Missing Part:**
- "Prove that <missing 19> Proof"
- We need something that can be a statement or a condition.
- **Expression 19**: $H, \angle ABS = \varphi.$ fits as it is a statement.

20. **Twentieth Missing Part:**
- "Let <missing 20> be the circle centered at $H$ with radius <missing 21>"
- We need something that can be a circle.
- **Expression 20**: $TF.$ fits as it refers to a circle.

21. **Twenty-first Missing Part:**
- We need something that can be a radius or a line.
- **Expression 21**: $AH.$ fits as it refers to a line.

22. **Twenty-second Missing Part:**
- "The <missing 22> meets $\Omega$ again at <missing 23>"
- We need something that can be a line or a circle.
- **Expression 22**: $F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =$ fits as it describes a line.

23. **Twenty-third Missing Part:**
- We need something that can be a point.
- **Expression 23**: $HF = HD.$ fits as it refers to a point.

24. **Twenty-fourth Missing Part:**
- "Let $\omega$ meets <missing 24> again at <missing 25>"
- We need something that can be a circle or a line.
- **Expression 27**: $\Omega$ fits as it refers to the circumcircle.

25. **Twenty-fifth Missing Part:**
- We need something that can be a point.
- **Expression 25**: $E'D || BC$ fits as it refers to a point.

26. **Twenty-sixth Missing Part:**
- "We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27>"
- We need something that can be a circle or a line.
- **Expression 26**: $F \neq B$ fits as it refers to a condition.

27. **Twenty-seventh Missing Part:**
- We need something that can be a circle or a line.
- **Expression 27**: $\Omega$ fits as it refers to the circumcircle.

28. **Twenty-eighth Missing Part:**
- "and get <missing 28> (this idea was recommended by Leonid Shatunov)."
- We need something that can be a result or a condition.
- **Expression 28**: $HD.$ fits as it refers to a condition.

29. **Twenty-ninth Missing Part:**
- "The point <missing 29> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD.$"
- We need something that can be a point or a line.
- **Expression 29**: $BF'$ fits as it refers to a point.

30. **Thirtieth Missing Part:**
- We need something that can be a condition or a statement.
- **Expression 30**: $\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF$ fits as it is a statement.

After matching all the expressions, the final answer is:

Answer: 1, 24, 3, 19, 5, 6, 7, 8, 9, 10, 11, 14, 16, 2, 9, 11, 27, 18, 19, 20, 21, 22, 23, 27, 25, 26, 27, 28, 29, 30

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

DeepSeek-Coder-V2-Lite-Instruct

<Detailed reasoning>

First, we need to identify the missing tags in the solution. The tags are used to replace placeholders in the solution text. Let's go through the solution step by step and match the placeholders with the given formulae.

1. **Identify the midpoint of the arc $CB$ containing $A$**:
- This is referred to as $S'$.
- The corresponding formula is: $S' = \Omega.S'$.
- Expression: 1.

2. **Internal angle bisector of $\angle BAC$**:
- This is referred to as $AI$.
- The corresponding formula is: $AI = \Omega.AI$.
- Expression: 2.

3. **Crosspoint of $BS$ and $AE$**:
- This is referred to as $D$.
- The corresponding formula is: $D = BS \cap AE$.
- Expression: 3.

4. **Crosspoint of $BS$ and $AE$**:
- This is referred to as $D$.
- The corresponding formula is: $D = BS \cap AE$.
- Expression: 3.

5. **Incenter of $\triangle ABC$**:
- This is referred to as $I$.
- The corresponding formula is: $I = \triangle ABC.I$.
- Expression: 4.

6. **Orthocenter of $\triangle TSS'$**:
- This is referred to as $H$.
- The corresponding formula is: $H = \triangle TSS'.H$.
- Expression: 5.

7. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 6.

8. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 6.

9. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 6.

10. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 6.

11. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 6.

12. **Acute triangle $ABC$**:
- This is referred to as $\triangle ABC$.
- The corresponding formula is: $\triangle ABC = \triangle ABC$.
- Expression: 7.

13. **Height of $\triangle ABC$**:
- This is referred to as $AH$.
- The corresponding formula is: $AH = \triangle ABC.AH$.
- Expression: 8.

14. **Diameter of $\triangle ABC$**:
- This is referred to as $BC$.
- The corresponding formula is: $BC = \triangle ABC.BC$.
- Expression: 9.

15. **Point $E$ is symmetric to $D$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 10.

16. **Line $BE$ meets $\Omega$ again at $P \neq A$**:
- This is referred to as $P$.
- The corresponding formula is: $P = \Omega.P$.
- Expression: 11.

17. **Line tangent to $\omega$ at $P$ meets line $BS$ on the internal angle bisector of $\angle BAC$**:
- This is referred to as $PH$.
- The corresponding formula is: $PH = \triangle ABC.PH$.
- Expression: 12.

18. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 13.

19. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 14.

20. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 15.

21. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 16.

22. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 17.

23. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 18.

24. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 19.

25. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 20.

26. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 21.

27. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 22.

28. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 23.

29. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 24.

30. **Symmetric to $AF$ with respect to $AH$**:
- This is referred to as $E'$.
- The corresponding formula is: $E' = \triangle ABC.E'$.
- Expression: 25.

Answer: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

Meta-Llama-3.1-405B-Instruct-Turbo

To solve this problem, we need to carefully read the solution and match the missing tags with the given formulae.

The first missing tag is <missing 1>, which is described as "the point diametrically opposite to a point $S$". This matches with <expression 1>, which is "$\Omega$".

The second missing tag is <missing 2>, which is described as "the internal angle bisector of". This matches with <expression 24>, which is "$\angle BAC$".

The third missing tag is <missing 3>, which is described as "the crosspoint of $BS$ and". This matches with <expression 3>, which is "$T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$".

The fourth missing tag is <missing 4>, which is described as "through". This matches with <expression 4>, which is "$E' \neq F'$".

The fifth missing tag is <missing 5>, which is described as "Denote". This matches with <expression 5>, which is "$\triangle ABF$".

The sixth missing tag is <missing 6>, which is described as "is incenter of". This matches with <expression 6>, which is "$\omega$".

The seventh missing tag is <missing 7>, which is described as "Denote". This matches with <expression 7>, which is "$\omega, \Omega$".

The eighth missing tag is <missing 8>, which is described as "is the orthocenter of". This matches with <expression 8>, which is "$=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$".

The ninth missing tag is <missing 9>, which is described as "points $B, G,$ and $F$ are collinear". This matches with <expression 9>, which is "$\Omega$".

The tenth missing tag is <missing 10>, which is described as "is symmetric to $AF$ with respect". This matches with <expression 10>, which is "$\omega$".

The eleventh missing tag is <missing 11>, which is described as "We use the lemma and complete the proof.". This matches with <expression 11>, which is "$E'$".

The twelfth missing tag is <missing 12>, which is described as "Let acute triangle". This matches with <expression 16>, which is "$\triangle ABC, AB > AC$".

The thirteenth missing tag is <missing 13>, which is described as "be the height.". This matches with <expression 14>, which is "$\triangle ABC, BHD$".

The fourteenth missing tag is <missing 14>, which is described as "be the circle $BCD$". This matches with <expression 15>, which is "$S' \implies AS'$".

The fifteenth missing tag is <missing 15>, which is described as "$BC$ is the diameter of". This matches with <expression 18>, which is "$AS'$".

The sixteenth missing tag is <missing 16>, which is described as "The point $E$ is symmetric to $D$ with respect to". This matches with <expression 12>, which is "$AH \perp BC \implies AH \perp E'D \implies$".

The seventeenth missing tag is <missing 17>, which is described as "The line $BE$ meets". This matches with <expression 2>, which is "$BE'F'$".

The eighteenth missing tag is <missing 18>, which is described as "again at". This matches with <expression 26>, which is "$F \neq B$".

The nineteenth missing tag is <missing 19>, which is described as "Prove that". This matches with <expression 19>, which is "$H, \angle ABS = \varphi$".

The twentieth missing tag is <missing 20>, which is described as "Let". This matches with <expression 20>, which is "$TF$".

The twenty-first missing tag is <missing 21>, which is described as "with radius". This matches with <expression 21>, which is "$AH$".

The twenty-second missing tag is <missing 22>, which is described as "The". This matches with <expression 22>, which is "$F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =$".

The twenty-third missing tag is <missing 23>, which is described as "meets $\Omega$ again at". This matches with <expression 23>, which is "$HF = HD$".

The twenty-fourth missing tag is <missing 24>, which is described as "meets". This matches with <expression 27>, which is "$\Omega$".

The twenty-fifth missing tag is <missing 25>, which is described as "again at". This matches with <expression 25>, which is "$E'D || BC$".

The twenty-sixth missing tag is <missing 26>, which is described as "We use Reim’s theorem for". This matches with <expression 29>, which is "$BF'$".

The twenty-seventh missing tag is <missing 27>, which is described as "and lines $CDD$ and". This matches with <expression 30>, which is "$\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies$".

The twenty-eighth missing tag is <missing 28>, which is described as "and get". This matches with <expression 28>, which is "$HD$".

The twenty-ninth missing tag is <missing 29>, which is described as "The point". This matches with <expression 13>, which is "$\implies GF$".

The thirtieth missing tag is <missing 30>, which is described as "is symmetric to $D$ with respect to $AH$". This matches with <expression 17>, which is "$F' \neq D, HD = HF'$".

Answer: 1, 24, 3, 4, 5, 6, 7, 8, 9, 10, 11, 16, 14, 15, 18, 12, 2, 26, 19, 20, 21, 22, 23, 27, 25, 29, 30, 28, 13, 17

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

Meta-Llama-3.1-70B-Instruct-Turbo

To solve this problem, we need to carefully match the formulae to the missing tags in the solution. Here's my step-by-step reasoning:

1. The first missing tag is <missing 1>. The solution states that the point diametrically opposite to a point $S$ through <missing 1> is the internal angle bisector of <missing 2>. This suggests that <missing 1> is related to the circumcircle of triangle $ABC$. Looking at the formulae, <expression 1> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 1> is likely to be <expression 1>.

2. The next missing tag is <missing 2>. The solution states that the point diametrically opposite to a point $S$ through <missing 1> is the internal angle bisector of <missing 2>. This suggests that <missing 2> is related to the internal angle bisector of $\angle BAC$. Looking at the formulae, <expression 24> is $\angle BAC$, which is the internal angle bisector of $\angle BAC$. Therefore, <missing 2> is likely to be <expression 24>.

3. The next missing tag is <missing 3>. The solution states that the crosspoint of $BS$ and <missing 3> through <missing 4>. Looking at the formulae, <expression 3> is $T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H$, which is related to the crosspoint of $BS$ and $T$. Therefore, <missing 3> is likely to be <expression 3>.

4. The next missing tag is <missing 4>. The solution states that the crosspoint of $BS$ and <missing 3> through <missing 4>. Looking at the formulae, <expression 4> is $E' \neq F'$, which is not directly related to the crosspoint of $BS$ and $T$. However, <expression 9> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 4> is likely to be <expression 9>.

5. The next missing tag is <missing 5>. The solution states that denote <missing 5> <missing 6> is incenter of <missing 7>. Looking at the formulae, <expression 5> is $\triangle ABF$, which is not directly related to the incenter of a triangle. However, <expression 14> is $\triangle ABC, BHD$, which is related to the incenter of triangle $ABC$. Therefore, <missing 5> is likely to be <expression 14>.

6. The next missing tag is <missing 6>. The solution states that denote <missing 5> <missing 6> is incenter of <missing 7>. Looking at the formulae, <expression 6> is $\omega$, which is the circumcircle of triangle $BDL$. Therefore, <missing 6> is likely to be <expression 6>.

7. The next missing tag is <missing 7>. The solution states that denote <missing 5> <missing 6> is incenter of <missing 7>. Looking at the formulae, <expression 7> is $\omega, \Omega$, which is related to the incenter of triangle $ABC$. Therefore, <missing 7> is likely to be <expression 7>.

8. The next missing tag is <missing 8>. The solution states that denote <missing 8> is the orthocenter of $\triangle TSS'$. Looking at the formulae, <expression 8> is $=\angle FCB + \varphi \implies \angle FBS = \angle ABS \implies H$, which is related to the orthocenter of triangle $TSS'$. Therefore, <missing 8> is likely to be <expression 8>.

9. The next missing tag is <missing 9>. The solution states that <missing 9> points $B, G,$ and $F$ are collinear. Looking at the formulae, <expression 10> is $\omega$, which is the circumcircle of triangle $BDL$. Therefore, <missing 9> is likely to be <expression 10>.

10. The next missing tag is <missing 10>. The solution states that <missing 10> is symmetric to $AF$ with respect <missing 11>. Looking at the formulae, <expression 11> is $E'$, which is symmetric to $AF$ with respect to $H$. Therefore, <missing 10> is likely to be <expression 11>.

11. The next missing tag is <missing 11>. The solution states that <missing 10> is symmetric to $AF$ with respect <missing 11>. Looking at the formulae, <expression 12> is $AH \perp BC \implies AH \perp E'D \implies$, which is related to the symmetry of $AF$ with respect to $H$. Therefore, <missing 11> is likely to be <expression 12>.

12. The next missing tag is <missing 12>. The solution states that Let acute triangle <missing 12> be given. Looking at the formulae, <expression 16> is $\triangle ABC, AB > AC$, which is an acute triangle. Therefore, <missing 12> is likely to be <expression 16>.

13. The next missing tag is <missing 13>. The solution states that Let $H$ be the orthocenter of <missing 13>. Looking at the formulae, <expression 13> is $\implies GF$, which is related to the orthocenter of triangle $ABC$. Therefore, <missing 13> is likely to be <expression 13>.

14. The next missing tag is <missing 14>. The solution states that Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>. Looking at the formulae, <expression 15> is $S' \implies AS'$, which is related to the circle $BCD$. Therefore, <missing 14> is likely to be <expression 15>.

15. The next missing tag is <missing 15>. The solution states that Let <missing 14> be the circle $BCD. BC$ is the diameter of <missing 15>. Looking at the formulae, <expression 9> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 15> is likely to be <expression 9>.

16. The next missing tag is <missing 16>. The solution states that The point $E$ is symmetric to $D$ with respect to <missing 16>. Looking at the formulae, <expression 12> is $AH \perp BC \implies AH \perp E'D \implies$, which is related to the symmetry of $E$ with respect to $H$. Therefore, <missing 16> is likely to be <expression 12>.

17. The next missing tag is <missing 17>. The solution states that The line $BE$ meets <missing 17> again at <missing 18>. Looking at the formulae, <expression 10> is $\omega$, which is the circumcircle of triangle $BDL$. Therefore, <missing 17> is likely to be <expression 10>.

18. The next missing tag is <missing 18>. The solution states that The line $BE$ meets <missing 17> again at <missing 18>. Looking at the formulae, <expression 11> is $E'$, which is the point where the line $BE$ meets the circumcircle of triangle $BDL$ again. Therefore, <missing 18> is likely to be <expression 11>.

19. The next missing tag is <missing 19>. The solution states that Prove that <missing 19>. Looking at the formulae, <expression 19> is $H, \angle ABS = \varphi$, which is the statement to be proved. Therefore, <missing 19> is likely to be <expression 19>.

20. The next missing tag is <missing 20>. The solution states that Let <missing 20> be the circle centered at $H$ with radius <missing 21>. Looking at the formulae, <expression 20> is $TF$, which is the circle centered at $H$ with radius $AH$. Therefore, <missing 20> is likely to be <expression 20>.

21. The next missing tag is <missing 21>. The solution states that Let <missing 20> be the circle centered at $H$ with radius <missing 21>. Looking at the formulae, <expression 21> is $AH$, which is the radius of the circle centered at $H$. Therefore, <missing 21> is likely to be <expression 21>.

22. The next missing tag is <missing 22>. The solution states that The <missing 22> meets $\Omega$ again at <missing 23>. Looking at the formulae, <expression 22> is $F = SS' \cap AC \implies \angle CBS = \frac {\overset{\Large\frown} {CS}}{2} = \frac {\overset{\Large\frown} {BS}}{2} = \frac {\overset{\Large\frown} {AB}}{2} + \frac {\overset{\Large\frown} {AS}}{2} =$, which is related to the point where the circle meets $\Omega$ again. Therefore, <missing 22> is likely to be <expression 22>.

23. The next missing tag is <missing 23>. The solution states that The <missing 22> meets $\Omega$ again at <missing 23>. Looking at the formulae, <expression 23> is $HF = HD$, which is the point where the circle meets $\Omega$ again. Therefore, <missing 23> is likely to be <expression 23>.

24. The next missing tag is <missing 24>. The solution states that Let $\omega$ meets <missing 24> again at <missing 25>. Looking at the formulae, <expression 24> is $\angle BAC$, which is the angle where $\omega$ meets again. Therefore, <missing 24> is likely to be <expression 24>.

25. The next missing tag is <missing 25>. The solution states that Let $\omega$ meets <missing 24> again at <missing 25>. Looking at the formulae, <expression 25> is $E'D || BC$, which is the point where $\omega$ meets again. Therefore, <missing 25> is likely to be <expression 25>.

26. The next missing tag is <missing 26>. The solution states that We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27>. Looking at the formulae, <expression 26> is $F \neq B$, which is related to the application of Reim's theorem. Therefore, <missing 26> is likely to be <expression 26>.

27. The next missing tag is <missing 27>. The solution states that We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27>. Looking at the formulae, <expression 27> is $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 27> is likely to be <expression 27>.

28. The next missing tag is <missing 28>. The solution states that We use Reim’s theorem for <missing 26> and lines $CDD$ and <missing 27> and get <missing 28>. Looking at the formulae, <expression 28> is $HD$, which is the result of applying Reim's theorem. Therefore, <missing 28> is likely to be <expression 28>.

29. The next missing tag is <missing 29>. The solution states that <missing 29> The point <missing 30> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD.$ Looking at the formulae, <expression 29> is $BF'$, which is related to the symmetry of $D$ with respect to $AH$. Therefore, <missing 29> is likely to be <expression 29>.

30. The next missing tag is <missing 30>. The solution states that <missing 29> The point <missing 30> is symmetric to $D$ with respect to $AH \implies E' = E \implies F' = F \implies HF = HD.$ Looking at the formulae, <expression 30> is $\angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies$, which is related to the symmetry of $D$ with respect to $AH$. Therefore, <missing 30> is likely to be <expression 30>.

Answer: 1, 24, 3, 9, 14, 6, 7, 8, 10, 11, 12, 16, 13, 15, 9, 12, 10, 11, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30.

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

Qwen2-72B-Instruct

<Detailed reasoning>

The solution starts by defining the point diametrically opposite to a point $S$ through <missing 1>. This must refer to a circle, and since the context is about points on circles, it likely refers to $\Omega$, which is the circumcircle of triangle $ABC$. Therefore, <missing 1> corresponds to <expression 1>.

Next, it mentions the internal angle bisector of <missing 2>. Given the context of the problem, this likely refers to the angle bisector of $\angle BAC$, which is mentioned later in the solution. Thus, <missing 2> corresponds to <expression 24>.

The third missing part, <missing 3>, is described as the crosspoint of $BS$ and something else, which is defined through <missing 4>. Since the context involves circles and intersections, this could refer to the intersection of $BS$ with another circle or line. However, without more specific information, we can only guess based on the expressions provided. Given the context of the problem, it seems logical that this would involve either $\Omega$ or $\omega$, but since <missing 4> is not yet filled, we cannot determine this directly. However, since the next sentence refers to $AE \perp BC$, it's likely that <missing 3> and <missing 4> are related to the construction involving $AE$ and possibly $\omega$. Looking at the expressions, <expression 6> refers to $\omega$, which fits the context of being a circle that might intersect with $BS$. Therefore, <missing 3> could correspond to <expression 6>.

Following this logic, <missing 4> should be related to the intersection or definition of $\omega$. Given the context of the problem and the expressions available, <missing 4> could correspond to <expression 7>, which refers to the intersection of $\omega$ and $\Omega$.

The fifth missing part, <missing 5>, is denoted as a point that is introduced later in the solution. Given the context, this could be a point on a circle or a line. Looking at the expressions, <expression 18> refers to a point on $\omega$ other than $B$, which fits the context of introducing a new point related to the circles involved. Therefore, <missing 5> could correspond to <expression 18>.

The sixth missing part, <missing 6>, is described as being in the context of a triangle. Given the context of the problem, this likely refers to a point within triangle $ABC$ or a triangle formed by points related to the construction. Looking at the expressions, <expression 5> refers to $\triangle ABF$, which could fit the context of a triangle related to the points and circles discussed. Therefore, <missing 6> could correspond to <expression 5>.

The seventh missing part, <missing 7>, is described as being the incenter of something. Given the context of the problem, this likely refers to the incenter of triangle $ABC$ or a triangle formed by points related to the construction. Looking at the expressions, there isn't a direct reference to the incenter, but considering the context, it could be implied in the description of a triangle. Since <expression 5> refers to $\triangle ABF$, and assuming that the incenter is relevant to the construction, <missing 7> could correspond to <expression 5>.

The eighth missing part, <missing 8>, is described as being the orthocenter of a triangle. Given the context of the problem, this likely refers to the orthocenter of triangle $ABC$ or a triangle formed by points related to the construction. Looking at the expressions, <expression 3> refers to the orthocenter of a triangle, which fits the context. Therefore, <missing 8> could correspond to <expression 3>.

The ninth missing part, <missing 9>, is described as points being collinear. Given the context of the problem, this likely refers to a statement about the alignment of points on a line. Looking at the expressions, there isn't a direct reference to collinearity, but considering the context, it could be implied in the description of points and their relationships. Since the context involves points on lines and circles, <missing 9> could correspond to a statement about points on a line or circle. Without a direct match, we can only guess based on the context, but none of the expressions directly state collinearity. However, given the context of the problem, it's likely related to points on a line or circle, so we'll have to make an educated guess here. Considering the context of the problem, <missing 9> could correspond to <expression 29>, which refers to a line related to points in the construction.

The tenth missing part, <missing 10>, is described as being symmetric to something with respect to something else. Given the context of the problem, this likely refers to a point or line being symmetric to another point or line with respect to a line or axis. Looking at the expressions, <expression 12> refers to a symmetry with respect to a line, which fits the context. Therefore, <missing 10> could correspond to <expression 12>.

The eleventh missing part, <missing 11>, is described as being with respect to something. Given the context of the problem, this likely refers to a line or axis with respect to which a symmetry or reflection is considered. Looking at the expressions, <expression 11> refers to a line that could serve as an axis of symmetry. Therefore, <missing 11> could correspond to <expression 11>.

Answer: 1, 24, 6, 7, 18, 5, 5, 3, 29, 12, 11

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

baichuan4

Based on the context and the information provided, let's try to match the formulae to the missing tags in the solution.

1. <missing 1>: This refers to the circumcircle of triangle $ABC$, which is $\Omega$. So, it should be filled by <expression 1>.

2. <missing 2>: This refers to the angle $\angle BAC$, which is mentioned in <expression 24>.

3. <missing 3>: This refers to the line tangent to $\omega$ at $P$, which is mentioned in <expression 10>.

4. <missing 4>: This refers to the point $P$, which is mentioned in <expression 23>.

5. <missing 5>: This refers to the point $T$, which is mentioned in <expression 3>.

6. <missing 6>: This refers to the point $H$, which is mentioned in <expression 19>.

7. <missing 7>: This refers to the triangle $ABF$, which is mentioned in <expression 5>.

8. <missing 8>: This refers to the point $F$, which is mentioned in <expression 22>.

9. <missing 9>: This refers to the circumcircle $\Omega$, which is mentioned in <expression 9>.

10. <missing 10>: This refers to the circle $\omega$, which is mentioned in <expression 10>.

11. <missing 11>: This refers to the point $E'$, which is mentioned in <expression 11>.

12. <missing 12>: This refers to the triangle $ABC$, which is mentioned in <expression 14>.

13. <missing 13>: This refers to the triangle $BHD$, which is mentioned in <expression 14>.

14. <missing 14>: This refers to the circle $BCD$, which is mentioned in <expression 15>.

15. <missing 15>: This refers to the point $S'$, which is mentioned in <expression 15>.

16. <missing 16>: This refers to the triangle $ABC$, which is mentioned in <expression 16>.

17. <missing 17>: This refers to the point $F'$, which is mentioned in <expression 17>.

18. <missing 18>: This refers to the line $AS'$, which is mentioned in <expression 18>.

19. <missing 19>: This refers to the point $H$, which is mentioned in <expression 19>.

20. <missing 20>: This refers to the circle centered at $H$ with radius $AH$, which is mentioned in <expression 21>.

21. <missing 21>: This refers to the line $AH$, which is mentioned in <expression 21>.

22. <missing 22>: This refers to the point $F$, which is mentioned in <expression 22>.

23. <missing 23>: This refers to the point $H$, which is mentioned in <expression 23>.

24. <missing 24>: This refers to the circle $\omega$, which is mentioned in <expression 10>.

25. <missing 25>: This refers to the point $P$, which is mentioned in <expression 23>.

26. <missing 26>: This refers to the point $F$, which is mentioned in <expression 26>.

27. <missing 27>: This refers to the circumcircle $\Omega$, which is mentioned in <expression 27>.

28. <missing 28>: This refers to the line $HD$, which is mentioned in <expression 28>.

29. <missing 29>: This refers to the line $BF'$, which is mentioned in <expression 29>.

30. <missing 30>: This refers to the point $G$, which is mentioned in <expression 30>.

Answer: 1, 24, 10, 23, 3, 19, 5, 22, 9, 10, 11, 14, 14, 15, 15, 16, 17, 18, 19, 21, 21, 22, 23, 10, 26, 27, 28, 29, 30

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

doubao

<Detailed reasoning>: In the solution, "through <missing 1> is the internal angle bisector of <missing 2>" likely refers to the circumcircle, so <missing 1> is likely <expression 1>. For "Denote the crosspoint of $BS$ and <missing 3> through <missing 4>", it seems <missing 3> could be <expression 5> and <missing 4> could be <expression 6>. For "Denote <missing 5> <missing 6> is incenter of <missing 7>", <missing 5> could be <expression 7> and <missing 6> could be <expression 8>. For "Denote <missing 8> is the orthocenter of $\triangle TSS'$", <missing 8> could be <expression 3>.
Answer: 1,5,7,3

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

gpto1

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

kimi

<Detailed reasoning>

1. The first missing tag is referring to the diametrically opposite point to $S$. Since $S$ is the midpoint of the arc $CB$ containing $A$, the diametrically opposite point to $S$ would be on the circumcircle $\Omega$. So, the expression that fits here is <expression 1>.

2. The second missing tag is referring to the internal angle bisector of a certain angle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 24>.

3. The third missing tag is referring to the crosspoint of $BS$ and another line. Since the line through $D$ parallel to $BC$ meets line $BE$ at $L$, the expression that fits here is <expression 3>.

4. The fourth missing tag is referring to a point on the line through $D$ parallel to $BC$. Since this line meets line $BE$ at $L$, the expression that fits here is <expression 4>.

5. The fifth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 5>.

6. The sixth missing tag is referring to a circle. Since the circumcircle of triangle $BDL$ is denoted by $\omega$, the expression that fits here is <expression 6>.

7. The seventh missing tag is referring to two circles. Since the circumcircle of triangle $BDL$ is denoted by $\omega$ and the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 7>.

8. The eighth missing tag is referring to an angle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 8>.

9. The ninth missing tag is referring to a circle. Since the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 9>.

10. The tenth missing tag is referring to a circle. Since the circumcircle of triangle $BDL$ is denoted by $\omega$, the expression that fits here is <expression 10>.

11. The eleventh missing tag is referring to a point. Since the line through $D$ parallel to $BC$ meets line $BE$ at $L$, the expression that fits here is <expression 11>.

12. The twelfth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 12>.

13. The thirteenth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 13>.

14. The fourteenth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 14>.

15. The fifteenth missing tag is referring to a circle. Since the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 15>.

16. The sixteenth missing tag is referring to a triangle. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 16>.

17. The seventeenth missing tag is referring to a point. Since the line $BE$ meets $\Omega$ again at $E \neq A$, the expression that fits here is <expression 17>.

18. The eighteenth missing tag is referring to a point. Since the line $BE$ meets $\Omega$ again at $E \neq A$, the expression that fits here is <expression 18>.

19. The nineteenth missing tag is referring to a point. Since the problem is about proving that the tangent line at $P$ meets the internal angle bisector of $\angle BAC$, the expression that fits here is <expression 19>.

20. The twentieth missing tag is referring to a circle. Since the circumcircle of $ABC$ is denoted by $\Omega$, the expression that fits here is <expression 20>.

21. The twenty-first missing tag is referring to a radius. Since the radius of the circle centered

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

minimax

To solve this problem, we need to match the missing parts of the solution with the corresponding formulae provided. Let's go through the solution step by step and match the missing tags with the expressions.

1. The first missing tag is about the point diametrically opposite to a point $S$ through some operation or line. The only formula that fits this description is <expression 1> $\Omega$, which indicates that the point is found through the circumcircle $\Omega$.

2. The second missing tag refers to the internal angle bisector of some angle. The only formula that fits this description is <expression 2> $BE'F'$, which suggests that the angle bisector is related to triangle $BE'F'$.

3. The third missing tag is about the crosspoint of $BS$ and some line. The only formula that fits this description is <expression 3> $T = S'B \cap SA$, which indicates that the crosspoint is found at the intersection of $S'B$ and $SA$.

4. The fourth missing tag is about the point $E'$ not being equal to some other point $F'$. The only formula that fits this description is <expression 4> $E' \neq F

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

wenxinyiyan

<Detailed reasoning>
Let's go through the solution step by step and match the missing tags with the given formulae.

1. "<missing 1>" should be the circumcircle of triangle ABC, which is denoted as "\Omega" in the formulae. So, <missing 1> corresponds to expression 1.

2. "<missing 2>" refers to the internal angle bisector of a certain angle. From the context, it is clear that it refers to "\angle BAC". So, we do not have a direct formula for this, but we can identify it from the given information.

3. "<missing 3>" is the crosspoint of BS and a certain line. From the context, it seems to be the line AE, which is perpendicular to BC. So, <missing 3> corresponds to the point where BS intersects AE, which is denoted as "T" in the formulae (expression 3).

4. "<missing 4>" is related to the previous missing tag and seems to refer to the line through a certain point parallel to BC. From the context, it is clear that it refers to the line through D parallel to BC, which meets line BE at L. We do not have a direct formula for this line, but we can identify it from the given information.

5. "<missing 5>" and "<missing 6>" seem to refer to a point and a circle respectively. From the context, it is clear that <missing 5> refers to the point "H" (the orthocenter of triangle TSS') and <missing 6> refers to the circumcircle of triangle BDL, which is denoted as "\omega" in the formulae. So, <missing 5> corresponds to expression 3 (indirectly, as H is defined there) and <missing 6> corresponds to expression 6.

6. "<missing 7>" refers to a triangle whose incenter is mentioned. From the context, it is clear that it refers to "\triangle ABF". So, <missing 7> corresponds to expression 5.

7. "<missing 8>" to "<missing 11>" seem to be related to a lemma used in the solution. We can identify them from the lemma provided. Specifically, <missing 8> refers to the orthocenter of triangle TSS', which is "H" (expression 3). <missing 9> refers to the circumcircle "\Omega". <missing 10> refers to the circle "\omega". <missing 11> refers to the line "BE'", which is symmetric to "AF" with respect to "AH". So, <missing 8> corresponds to expression 3, <missing 9> corresponds to expression 1, <missing 10> corresponds to expression 6, and <missing 11> corresponds to the line BE' defined in the lemma (no direct formula, but can be identified).

8. "<missing 12>" to "<missing 30>" are all related to the lemma provided at the end of the solution. We can identify them from the lemma as follows:
* <missing 12> refers to the acute triangle "ABC", so it corresponds to expression 16 (indirectly).
* <missing 13> and <missing 14> refer to the triangle and its height mentioned in the lemma. From the context, it is clear that they refer to "\triangle ABC" and "AH" respectively. So, <missing 13> corresponds to expression 16 and <missing 14> corresponds to the height "AH" defined in the lemma (no direct formula, but can be identified).
* <missing 15> to <missing 30> can be identified similarly from the lemma. Specifically, <missing 15> refers to the circle "BCD" with BC as the diameter (no direct formula), <missing 16> refers to the line "AH" (no direct formula), <missing 17> refers to the circle "\Omega", <missing 18> refers to the point "E", <missing 19> refers to the angle "\angle BAC" (expression 24), <missing 20> refers to the circle centered at H with a certain radius (no direct formula), <missing 21> refers to the radius "AH" (no direct formula), <missing 22> refers to the circle "\Omega", <missing 23> refers to a point where \omega meets \Omega (no direct formula), <missing 24> refers to the line "CD" (no direct formula), <missing 25> refers to a point where \omega meets a certain line (no direct formula), <missing 26> refers to a certain configuration of circles and lines mentioned in the lemma

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

zhipu

### Detailed Reasoning

1. **<missing 1>**: This should refer to a point or a property related to the circumcircle $\Omega$. The most relevant expression is `<expression 1> \Omega.` which seems to fit as it refers to the circumcircle.

2. **<missing 2>**: This should be an angle or a property related to the triangle. The most relevant expression is `<expression 24> \angle BAC` which fits as it refers to the angle $\angle BAC$.

3. **<missing 3>**: This should be a line or a property related to the triangle. The most relevant expression is `<expression 18> AS'` which fits as it refers to the line $AS'$.

4. **<missing 4>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 3> T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H` which fits as it defines a point $T$ and relates it to $S'$ and $H$.

5. **<missing 5>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 15> S' \implies AS'` which fits as it refers to the point $S'$.

6. **<missing 6>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 14> \triangle ABC, BHD` which fits as it refers to the triangle $ABC$ and the point $H$.

7. **<missing 7>**: This should be a triangle or a property related to the triangle. The most relevant expression is `<expression 5> \triangle ABF.` which fits as it refers to the triangle $ABF$.

8. **<missing 8>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 3> T = S'B \cap SA \implies SB \perp TS', S'A \perp TS \implies H` which fits as it defines a point $H$.

9. **<missing 9>**: This should be a property or a statement related to collinearity. The most relevant expression is `<expression 13> \implies GF` which fits as it implies a relationship between points $G$ and $F$.

10. **<missing 10>**: This should be a property or a statement related to symmetry. The most relevant expression is `<expression 11> E'` which fits as it refers to the symmetric point $E'$.

11. **<missing 11>**: This should be a property or a statement related to symmetry. The most relevant expression is `<expression 1> \Omega.` which fits as it refers to the circumcircle $\Omega$.

12. **<missing 12>**: This should be a triangle or a property related to the triangle. The most relevant expression is `<expression 5> \triangle ABF.` which fits as it refers to the triangle $ABF$.

13. **<missing 13>**: This should be a triangle or a property related to the triangle. The most relevant expression is `<expression 14> \triangle ABC, BHD` which fits as it refers to the triangle $ABC$.

14. **<missing 14>**: This should be a circle or a property related to the circle. The most relevant expression is `<expression 6> \omega` which fits as it refers to the circle $\omega$.

15. **<missing 15>**: This should be a circle or a property related to the circle. The most relevant expression is `<expression 9> \Omega` which fits as it refers to the circle $\Omega$.

16. **<missing 16>**: This should be a line or a property related to the triangle. The most relevant expression is `<expression 25> E'D || BC` which fits as it refers to the line $E'D$.

17. **<missing 17>**: This should be a circle or a property related to the circle. The most relevant expression is `<expression 10> \omega` which fits as it refers to the circle $\omega$.

18. **<missing 18>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 2> BE'F'` which fits as it refers to the point $F'$.

19. **<missing 19>**: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 19> H, \angle ABS = \varphi.` which fits as it refers to the angle $\angle ABS$.

20. **<missing 20>**: This should be a circle or a property related to the circle. The most relevant expression is `<expression 20> TF` which fits as it refers to the circle centered at $T$.

21. **<missing 21>**: This should be a length or a property related to the triangle. The most relevant expression is `<expression 21> AH` which fits as it refers to the length $AH$.

22. **<missing 22>**: This should be a circle or a property related to the circle. The most relevant expression is `<expression 6> \omega` which fits as it refers to the circle $\omega$.

23. **<missing 23>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 23> HF = HD` which fits as it refers to the point $F$.

24. **<missing 24>**: This should be a circle or a property related to the circle. The most relevant expression is `<expression 10> \omega` which fits as it refers to the circle $\omega$.

25. **<missing 25>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 2> BE'F'` which fits as it refers to the point $F'$.

26. **<missing 26>**: This should be a point or a property related to the triangle. The most relevant expression is `<expression 26> F \neq B` which fits as it refers to the point $F$.

27. **<missing 27>**: This should be a circle or a property related to the circle. The most relevant expression is `<expression 27> \Omega` which fits as it refers to the circle $\Omega$.

28. **<missing 28>**: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 28> HD` which fits as it refers to the length $HD$.

29. **<missing 29>**: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 29> BF'` which fits as it refers to the line $BF'$.

30. **<missing 30>**: This should be a property or a statement related to the triangle. The most relevant expression is `<expression 30> \angle EBS' = \varphi, \angle LBG = \angle LDG = 90^\circ = \angle DBS' \implies \angle DBG = \varphi = \angle SBF \implies` which fits as it refers to the angles involving $E$, $B$, $S'$, $L$, $D$, and $G$.

### Answer
1, 24, 18, 3, 15, 14, 5, 3, 13, 11, 1, 5, 14, 6, 9, 25, 10, 2, 19, 20, 21, 6, 23, 10, 2, 26, 27, 28, 29, 30

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11

deepseek-v3

Error code: 400 - {'error': {'code': 'RequestTimeOut', 'param': None, 'message': 'Request timed out, please try again later.', 'type': 'RequestTimeOut'}, 'id': 'chatcmpl-1e8ab241-29df-98cf-86a1-6a7f44d9394e', 'request_id': '1e8ab241-29df-98cf-86a1-6a7f44d9394e'}

15,24,18,19,22,8,5,3,30,13,20,16,14,27,1,21,9,26,23,6,28,10,17,29,4,7,2,25,12,11