# Win Rate评估报告

## 基本信息

- **评估日期**: 2025-10-11T12:43:22.568017
- **评委模型**: gpt-4o
- **对比次数**: 20 次

## 评估结果

### 胜率统计

| 指标 | 数值 | 百分比 |
|------|------|--------|
| 生成数据胜出 | 5 次 | 25.00% |
| 参考数据胜出 | 13 次 | 65.00% |
| 平局 | 2 次 | 10.00% |

### 结果分析

**Win Rate**: 25.00%


❌ **需改进**: 生成数据质量明显低于参考数据。建议检查生成Pipeline并进行优化。


## 详细对比结果


### 对比 1

- **生成题目**: 25
- **参考题目**: gen_aime_26
- **胜者**: 🤝 Problem A
- **理由**: Upon evaluation, Problem A and Problem B are both solid in terms of mathematical correctness and clarity. However, Problem A exhibits a higher level of problem quality and complexity, which aligns more closely with the AIME standards. Problem A involves an intricate combinatorial setup, requiring the solver to understand properties of a regular polygon and engage in non-trivial counting arguments to find the number of ways to pair vertices, which is a challenging task suitable for AIME. In contrast, Problem B, while correctly formulated and solvable, relies on a straightforward application of the properties of a regular hexagon inscribed in a circle. The solution to Problem B is more direct and less complex, as it quickly reduces to finding the radius of the circle, a relatively simple task. Therefore, considering the dimensions of difficulty appropriateness and problem quality, Problem A is the winner due to its higher level of complexity and challenge appropriate for AIME.

### 对比 2

- **生成题目**: gen_aime_19
- **参考题目**: 10
- **胜者**: 🤝 Tie
- **理由**: Failed to parse response

### 对比 3

- **生成题目**: 4
- **参考题目**: gen_aime_28
- **胜者**: 🤝 Problem A
- **理由**: Upon evaluating both problems, Problem A is superior in terms of mathematical correctness, clarity, and difficulty appropriateness. Problem A is a combinatorial problem that requires the solver to understand permutations and divisibility rules, which are common themes in AIME-style problems. The problem is clearly stated, and the answer provided is consistent with the problem requirements. In contrast, Problem B has several issues. The solution provided for Problem B incorrectly concludes that the area of one of the smaller triangles is 1, while logically, given the problem statement, the area should be less than 1 due to the division of the area among the smaller triangles. Additionally, the solution lacks clarity, especially in the justification of the final step where it abruptly concludes the area as 1 without proper reasoning, which detracts from its mathematical correctness and clarity. Therefore, Problem A is the better-quality problem.

### 对比 4

- **生成题目**: gen_aime_7
- **参考题目**: 27
- **胜者**: 🤝 Problem B
- **理由**: Problem B has a clear and unambiguous statement with a correct and straightforward answer. The mathematical correctness of the sequence operations in Problem B is sound and leads to a rational number, where the task is to find the remainder when the sum of the numerator and denominator is divided by 1000. This is a typical AIME-style problem that involves sequences and modular arithmetic, which are common themes in such competitions. Problem A, on the other hand, suffers from a solution that is both unnecessarily complicated and possibly incorrect in its simplification process. The inclusion-exclusion principle is correctly applied, but the final simplification error suggests a lack of clarity or correctness in the solution verification, which affects the overall quality. While both problems have a certain degree of complexity, Problem B's approach to sequences provides a more elegant and appropriate challenge for the AIME context.

### 对比 5

- **生成题目**: 29
- **参考题目**: gen_aime_6
- **胜者**: 🤝 Problem A
- **理由**: Problem A is mathematically correct and concise. It clearly defines a polynomial function and asks for the sum of specific values of k, which is a well-posed question. The problem is appropriate for AIME standards as it involves understanding polynomial behavior and requires analytical thinking. On the other hand, Problem B has multiple issues. The solution provided in Problem B is incorrect, as it contains errors in calculating the maximum value of f(p). The problem statement in Problem B is somewhat misleading because it suggests maximizing the sum of permutations without a clear strategy, leading to confusion. Additionally, the solution attempts to use permutations without clearly explaining the rationale or strategy behind the selections. While Problem B has potential, the errors in its solution detract from its overall quality. Therefore, Problem A is the winner due to its correctness, clarity, and appropriate difficulty level.

### 对比 6

- **生成题目**: 20
- **参考题目**: gen_aime_29
- **胜者**: 🤝 Tie
- **理由**: Both problems have their unique attributes that make them suitable for AIME standards, and neither problem stands out significantly over the other across the evaluation dimensions. 

1. Mathematical Correctness: Both Problem A and Problem B are mathematically correct. The answers provided are consistent with the problem statements, and the reasoning in Problem B's solution verifies the correctness.

2. Clarity: Problem B is clearer in terms of presentation and understanding. Problem A, while clear, involves a complex geometric setup that might take longer for students to parse compared to the straightforward modular arithmetic problem in Problem B.

3. Difficulty Appropriateness: Problem A involves geometry and requires understanding of several geometric properties and relationships, which aligns well with AIME's difficulty level. Problem B, on the other hand, involves modular arithmetic and application of Fermat's Little Theorem, which are also suitable for AIME.

4. Problem Quality: Problem A is well-designed with a complex geometric configuration that tests a variety of skills. Problem B is simpler in terms of setup but still requires deep understanding of number theory concepts.

Overall, both problems meet the AIME standard comprehensively, and therefore neither problem significantly outshines the other across all dimensions.

### 对比 7

- **生成题目**: gen_aime_22
- **参考题目**: 8
- **胜者**: 🤝 Problem A
- **理由**: Both Problem A and Problem B are mathematically interesting and invite the solver to apply a range of mathematical techniques. However, Problem A stands out for a few reasons: 

1. **Mathematical Correctness**: Both problems are mathematically sound and have correct answers. 

2. **Clarity**: Problem A provides a clear condition involving congruences that is easy to understand and follow. Problem B, while clear, involves a transformation that might be less intuitive for some solvers in terms of visualizing the rotation and intersection. 

3. **Difficulty Appropriateness**: Both problems are challenging and appropriate for AIME, but Problem A involves a more direct application of modular arithmetic and factorization, which aligns well with typical AIME problem-solving strategies. Problem B involves additional geometric visualization and understanding of transformations, making it potentially more complex than typical AIME problems which focus more on algebra and number theory.

4. **Problem Quality**: Problem A is well-designed, with a clear path to the solution that involves logical steps and common number theory techniques. Problem B is also well-designed but may require more abstract thinking due to the geometric aspect. 

Overall, Problem A provides a clearer and more direct problem-solving experience, making it slightly superior in this evaluation.

### 对比 8

- **生成题目**: 22
- **参考题目**: gen_aime_8
- **胜者**: 🤝 Problem A
- **理由**: Both problems are mathematically correct and have clear problem statements, but Problem A stands out in terms of complexity and originality. Here is a breakdown of the evaluation dimensions:

1. Mathematical Correctness: Both problems are mathematically sound. Problem A deals with the application of the greedy algorithm to determine when it provides an optimal solution, while Problem B involves probability and divisor counting, both of which are correctly handled.

2. Clarity: Both problem statements are clear and unambiguous, providing sufficient information for the solver to understand what is required without confusion.

3. Difficulty Appropriateness: Both problems are suitable for AIME standards. Problem A's exploration of the greedy algorithm's success across a range of values introduces a unique challenge that requires deeper analysis beyond straightforward calculation. Problem B, while also challenging, involves more standard divisor and probability calculations.

4. Problem Quality: Problem A exhibits higher quality due to its innovative approach in examining the conditions under which a common algorithm succeeds. It encourages deeper thinking about algorithmic efficiency, which adds to the problem's educational value. Problem B, although well-crafted, follows a more conventional path of evaluating divisors and probability.

Overall, Problem A provides a more engaging and thought-provoking challenge, thus making it the higher quality problem.

### 对比 9

- **生成题目**: gen_aime_13
- **参考题目**: 22
- **胜者**: 🤝 Problem B
- **理由**: Both problems are challenging and suitable for the AIME level, but Problem B is the winner based on its higher quality in several evaluation dimensions. Firstly, both problems are mathematically sound, but Problem B is more clearly presented and free from errors or ambiguities in its description. Problem A has some inconsistencies in the solution, particularly in the calculation and explanation of the last three digits of m+n, which is confusing and incorrect as the correct last three digits should simply be the sum 10, not 375. Problem B, on the other hand, is clear and well-structured throughout with a correct answer. Regarding difficulty, both problems are appropriate for AIME, but Problem B offers a more interesting and less straightforward combinatorial challenge, requiring understanding of algorithms and optimization, which adds to its complexity. Overall, Problem B is better designed with a more consistent problem statement and solution, making it the superior problem.

### 对比 10

- **生成题目**: 4
- **参考题目**: gen_aime_10
- **胜者**: 🤝 Problem B
- **理由**: Both problems present interesting challenges, but Problem B stands out in several key aspects. Firstly, the mathematical correctness of Problem B is clearly demonstrated with a detailed solution that shows a complete and accurate derivation of the problem's parameters. The problem involves well-known geometric principles such as the properties of a triangle inscribed in a circle and the Pythagorean theorem, which are executed correctly and contribute to a well-rounded mathematical experience. In terms of clarity, Problem B is succinct in its presentation of the geometric scenario, making it straightforward to understand the setup without ambiguity. Problem A, while interesting, lacks a complete solution, which makes it difficult to fully evaluate its correctness and reasoning. The difficulty of Problem B is appropriate for an AIME-style question, as it involves multiple steps and requires a solid understanding of geometry and algebra, making it challenging yet solvable. Finally, Problem B is better designed as it naturally engages with mathematical concepts and has a logical progression, whereas Problem A seems more mechanical and lacks a creative spark in problem design. Hence, Problem B is judged to be of higher quality.

*（仅显示前10次对比的详细结果，完整结果请查看JSON文件）*

## 结论

基于Win Rate评估，生成数据集的质量**仍有提升空间**（Win Rate = 25.00%）。

建议：
1. 优化题目生成的提示词
2. 增加质量过滤步骤
3. 使用更强的生成模型
4. 增加人工审核环节


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*报告生成时间: 2025-10-11 12:43:22*
