| 模型 | 模型答案 | 正确答案 |
|---|---|---|
| 金声玉亮2.0 | <Detailed reasoning> To solve this problem, we need to identify which formula from the given list matches the missing tag in the solution. The solution involves proving a mathematical inequality using the AM-GM inequality, and it calculates \(a_{n+2}\) in terms of \(a_n\). The solution steps are as follows: 1. Start with the definition of \(a_{n+2}\) and expand it. 2. Use the AM-GM inequality to simplify the expanded expression. 3. Derive a relationship between \(a_{n+2}\) and \(a_n\). The missing tag <missing 1> appears in the context where the solution is expanding \(a_{n+2}^2\) and then applying the AM-GM inequality. The given formulae list only contains "<expression 1> = x", which does not directly match the context of the missing tag in the solution. Given the lack of a direct match and the context of the solution, it seems that the provided formula "<expression 1> = x" does not fit the missing part related to the AM-GM inequality and the calculation of \(a_{n+2}\). Since there is no clear match from the provided formulae to the missing tag in the solution, and based on the context of the solution, it appears that the missing part is not directly provided in the formulae list. However, since we are required to give an answer, and considering the context of the solution, it seems that the missing part might be implicitly related to the AM-GM inequality application, which is not explicitly listed in the provided formulae. Given these considerations, my best guess is that the missing part is related to the application of the AM-GM inequality, but since there is no direct match in the provided formulae, I will still list "1" as the answer, acknowledging that this is a guess due to the lack of a clear match. Answer: 1 |
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| deepseek-r1 | Answer: 1 |
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| gpto1 | Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}} | 1 |
| gpt-4o-2024-08-06 | To solve this problem, we need to identify the missing formula in the solution provided. The solution involves manipulating the expression for \( a_{n+2}^2 \) in terms of \( a_n^2 \) and other terms. Let's go through the solution step by step: 1. **Understanding the Problem:** - We have a sequence of positive real numbers \( x_1, x_2, \ldots, x_{2023} \) that are pairwise different. - The expression for \( a_n \) is given by: \[ a_n = \sqrt{(x_1 + x_2 + \cdots + x_n)\left(\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n}\right)} \] - We need to prove that \( a_{2023} \ge 3034 \). 2. **Analyzing the Solution:** - The solution starts by expressing \( a_{n+2}^2 \) in terms of \( a_n^2 \) and other terms. - The expression for \( a_{n+2}^2 \) is expanded as: \[ a_{n+2}^2 = \left(\sum_{k=1}^{n+2} x_k\right)\left(\sum_{k=1}^{n+2} \frac{1}{x_k}\right) \] - This is further expanded to include terms involving \( x_{n+1} \) and \( x_{n+2} \), and the sum up to \( n \). 3. **Identifying the Missing Formula:** - The missing formula is likely related to the expression for \( a_n^2 \) since it appears in the final expanded form. - The solution uses the AM-GM inequality to simplify the expression and eventually shows that: \[ a_{n+2}^2 \ge (a_n + 2)^2 \] - This implies that the missing formula should be the expression for \( a_n^2 \). 4. **Conclusion:** - The missing formula is the expression for \( a_n^2 \), which is: \[ a_n^2 = \left(\sum_{k=1}^{n} x_k\right)\left(\sum_{k=1}^{n} \frac{1}{x_k}\right) \] - This matches the context of the problem where \( a_n \) is defined as the square root of this product. Since there is only one formula provided in the list, and it is labeled as "x", it seems there might be a mistake in the problem statement regarding the list of formulae. However, based on the context, the missing formula is indeed the expression for \( a_n^2 \). **Detailed reasoning:** The missing formula in the solution is the expression for \( a_n^2 \), which is used to derive the inequality involving \( a_{n+2} \) and \( a_n \). The solution uses this expression to apply the AM-GM inequality and derive the result that \( a_{n+2} \ge a_n + 2 \). **Answer:** 1 |
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| DeepSeek-Coder-V2-Lite-Instruct | <Detailed reasoning> First, let's identify the missing tags in the solution. The solution uses the following expressions: 1. \( a_{n+2}^2 = (\sum^{n+2}_{k=1}x_k)(\sum^{n+2}_{k=1}\frac1{x_k}) \) 2. \( a_{n+2}^2 \ge 4 \sqrt[4]{(\frac{x_{n+1}}{x_{n+1}})(\frac{x_{n+1}}{x_{n+2}})(\frac{x_{n+2}}{x_{n+1}})(\frac{x_{n+2}}{x_{n+2}})} + 4\sqrt[4]{ (\frac{1}{x_{n+1}}\sum^{n}_{k=1}x_k)(x_{n+1}\sum^{n}_{k=1}\frac1{x_k})(\frac{1}{x_{n+2}}\sum^{n}_{k=1}x_k)(x_{n+2}\sum^{n}_{k=1}\frac1{x_k}) } + a_n^2 \) 3. \( a_{n+2} \ge a_n + 2 \) The missing tag <missing 1> corresponds to the expression: \[ a_{n+2}^2 = (\sum^{n+2}_{k=1}x_k)(\sum^{n+2}_{k=1}\frac1{x_k}) \] This matches with expression 1. Answer: 1 |
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| Meta-Llama-3.1-405B-Instruct-Turbo | To solve this problem, we need to match the formulae to the missing tags in the solution. The solution has only one missing tag, which is <missing 1>. The formulae provided are "<expression 1> = x", but this does not seem to match the context of the solution. However, upon closer inspection, we can see that the solution is using the AM-GM inequality, and the missing part is likely related to this inequality. Given the format of the solution and the formulae provided, it seems that none of the provided formulae match the missing tag. However, since we are required to give an answer, I will provide a guess. Since there is only one missing tag and one formula provided, my guess would be that the expression 1 fills the missing part, even though it does not seem to match the context. Answer: 1 |
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| Meta-Llama-3.1-70B-Instruct-Turbo | To solve this problem, we need to match the formulae to the missing tags in the solution. The first missing tag is <missing 1>. The solution states that "We solve for $a_{n+2}$ in terms of $a_n$ and <missing 1>". This suggests that <missing 1> is an expression that is used to solve for $a_{n+2}$. Looking at the formulae, we see that <expression 1> is simply "x". However, this does not seem to match the context of the solution. Since there is only one formula provided, we will have to make an educated guess. It seems that the solution is trying to express $a_{n+2}$ in terms of $a_n$ and some other variable or expression. Given the information provided, our best guess is that <missing 1> is indeed <expression 1>, which is "x". Answer: 1 |
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| Qwen2-72B-Instruct | The solution provided contains a single missing part which requires a formula to be filled in. However, the given list of formulae does not contain any actual formulas, it only contains "<expression 1> x." which doesn't provide any information. Since we have only one missing part and no usable formulae, we cannot directly match any formula to the missing part based on the provided information. However, from the context of the solution, it's clear that the missing part is supposed to be a reference to \(a_n\), which is defined earlier in the problem statement. Since we don't have a specific formula to match, and given the instruction to give our best guess, we can infer that the missing part is referring to the definition or usage of \(a_n\) in the context of the solution. Since there is only one missing part and no valid formulae provided, and considering the context of the solution, my best guess would be to assign the placeholder "<expression 1>" to the missing part, even though this is more of an educated guess than a direct match. Answer: 1 |
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| baichuan4 | To solve this problem, we need to identify the correct formula that matches each <missing X> tag in the solution. Let's go through the solution step by step and determine which formula fits where. First, we see that <missing 1> appears in the context of expressing $a_{n+2}^2$ in terms of $a_n$ and some additional terms. The solution then expands this expression and applies the AM-GM inequality. However, the specific formula for <missing 1> is not directly visible from the context provided. Given that the solution involves manipulating sums and products involving $x_k$ and their reciprocals, and it eventually simplifies to a recurrence relation for $a_{n+2}$ in terms of $a_n$, we can infer that <missing 1> likely represents a manipulation or simplification of the sum-product terms that leads to this recurrence relation. Since the formulae provided are limited and do not directly correspond to any specific step in the solution, we cannot definitively match <missing 1> with any of the given formulae. The same applies to any other potential missing tags that might be present in the full solution but are not visible in the excerpt provided. Therefore, without additional context or more explicit steps in the solution that align with the given formulae, we cannot provide a definitive answer to match the formulae to the missing tags. <Detailed reasoning> The solution manipulates sums and products involving $x_k$ and their reciprocals to derive a recurrence relation for $a_{n+2}$ in terms of $a_n$. However, the specific formula for <missing 1> is not discernible from the context provided, and the given formulae do not directly correspond to any visible step in the solution. Answer: It is not possible to provide a definitive answer without additional context or explicit steps in the solution that align with the given formulae. |
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| doubao | <Detailed reasoning>: In the solution, there is no place where the expression "x" seems to be directly relevant or needed to fill any of the missing parts. Answer: |
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| gpto1 | Error code: 400 - {'error': {'message': 'Invalid prompt: your prompt was flagged as potentially violating our usage policy. Please try again with a different prompt: https://platform.openai.com/docs/guides/reasoning/advice-on-prompting', 'type': 'invalid_request_error', 'param': None, 'code': 'invalid_prompt'}} | 1 |
| kimi | <Detailed reasoning> Looking at the solution, we need to find the expression that matches the missing part of the equation for $a_{n+2}^2$. The missing part is the term that represents $a_{n+1}^2$. The solution states that $a_{n+2}^2 = (\sum^{n+2}_{k=1}x_k)(\sum^{n+2}_{k=1}\frac1{x_k})$, which is the general formula for $a_n$ for any $n$. Since we are looking for $a_{n+1}^2$, we can replace $n+2$ with $n+1$ in the general formula to get $a_{n+1}^2 = (\sum^{n+1}_{k=1}x_k)(\sum^{n+1}_{k=1}\frac1{x_k})$. The only expression provided is <expression 1> x. However, this expression does not seem to match the missing part in the solution. It seems there might be an error or missing information in the provided expressions. Based on the given information, I cannot confidently determine the correct expression identifier to fill the missing part in the solution. Answer: <empty list> |
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| minimax | To solve this problem, we need to match the missing formula in the solution with the given list of formulae. The missing formula is used to derive the inequality $a_{n+2} \ge a_{n} + 2$. The solution uses the AM-GM inequality, which states that for non-negative real numbers, the arithmetic mean is greater than or equal to the geometric mean. The AM-GM inequality is given by: $$\frac{a_1 + a_2 + \cdots + a_n}{n} \ge \sqrt[n]{a_1 \cdot a_2 \cdots a_n}$$ Equality holds if and only if $a_1 = a_2 = \cdots = a_n$. In the solution, the AM-GM inequality is applied to two sets of terms in the expansion of $a_{n+2}^2$. The first set involves the terms $\frac{x_{n+1}}{x_{n+1}}, \frac{x_{n+1}}{x_{n+2}}, \frac{x_{n+2}}{x_{n+1}}, \frac{x_{n+2}}{x_{n+2}}$, and the second set |
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| wenxinyiyan | <Detailed reasoning> In the solution, <missing 1> is referred to as the relationship between $a_{n+2}$ and $a_n$. Specifically, the solution derives $a_{n+2}^2$ in terms of $a_n^2$ and other terms, eventually showing that $a_{n+2} \ge a_n + 2$. This relationship is not explicitly given in the list of formulae provided. However, if we were to choose from the given list, the closest would be <expression 1> since it involves $x$, which is used in the derivation. But <expression 1> does not directly represent the relationship between $a_{n+2}$ and $a_n$. Therefore, it seems there might be a misunderstanding or a mistake in the question's format, as the provided list of formulae does not contain the necessary expression to fill <missing 1>. Given the circumstances, if we must choose from the provided list, we would select <expression 1> as the least inappropriate option, even though it does not directly correspond to the missing part. Answer: 1 |
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| zhipu | ### Detailed Reasoning 1. **Understanding the Solution Structure**: - The solution aims to express \(a_{n+2}\) in terms of \(a_n\) and another term. - It expands \(a_{n+2}^2\) using the sums of \(x_k\) and \(\frac{1}{x_k}\). - It then applies the AM-GM inequality to derive a lower bound for \(a_{n+2}^2\). - Finally, it uses this bound to show that \(a_{2023} \ge 3034\). 2. **Identifying the Missing Parts**: - **<missing 1>**: This is the term that \(a_{n+2}\) is expressed in terms of, along with \(a_n\). Given the context, it is likely related to \(a_{n+1}\) because the solution involves stepping from \(a_n\) to \(a_{n+2}\) and would naturally involve \(a_{n+1}\). 3. **Matching with Given Formulae**: - The only provided formula is `<expression 1> x`, which seems unrelated directly to the terms used in the solution. However, the context suggests that the missing term should be \(a_{n+1}\) because it is a natural intermediate step in the progression from \(a_n\) to \(a_{n+2}\). 4. **Best Guess**: - Given the lack of more specific formulae and the context of the solution, the best guess for <missing 1> is related to \(a_{n+1}\). Since no direct formula for \(a_{n+1}\) is provided, we assume the simplest identifier, which is `<expression 1>`. ### Answer 1 This corresponds to the assumption that <expression 1> (which is labeled as 'x' but contextually inferred to be \(a_{n+1}\)) fills <missing 1>. --- **Answer: 1** |
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| deepseek-v3 | <Detailed reasoning> The missing part in the solution is <missing 1>, which is used to express \(a_{n+2}^2\) in terms of \(a_n\) and other terms. The formula provided is <expression 1> = \(x\). However, this seems unrelated to the context of the problem, which involves expressions like \(a_n\) and summations. Given the context, the missing part <missing 1> should involve the expression for \(a_n\) or something derived from it. The solution seems to derive \(a_{n+2}^2\) by expanding the product of sums and then applying the AM-GM inequality. The missing part likely involves the expression for \(a_n^2\), which is already present in the solution. Since the provided formula <expression 1> = \(x\) does not seem to fit the context, and no other formulae are provided, we can only guess that the missing part is \(a_n^2\), which is already present in the solution. Therefore, the best guess is that <missing 1> corresponds to \(a_n^2\), but since no other formulae are provided, we cannot definitively match it to any expression. Answer: 1 |
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