AI 中文总结
受数学谜题启发，研究自我指涉多项选择题测试，以‘有多少正确答案选项’为问题，答案为正整数。定义可解和\(k\)可解测试，推导出生成函数枚举相关情况，还研究了极值问题，如最小成本和有效值得最大数量。
AI 中文摘要
我们研究自我指涉的多项选择题测试，问题为：“有多少个正确答案选项？”答案选项为正整数。若值\(a\)在答案选项中恰好出现\(a\)次，则称\(a\)为“有效”。测试的“成本”是所有答案选项之和，这将问题与整数划分联系起来。我们定义了可解和\(k\)可解测试，并推导出生成函数，通过成本、不同有效值的数量和选项数量对其进行枚举。我们还研究了极值问题，包括最小成本和有效值得最大可能数量。本文受《数学谜题与奇趣》中的一个谜题启发。
英文摘要
We study self-referential multiple-choice tests with the question: \emph{How many correct answer choices are there?} The answer choices are positive integers. A value $a$ is called \emph{valid} if it occurs exactly $a$ times among answer choices. The \emph{cost} of a test is the sum of all answer choices, linking the problem to integer partitions. Using this framework, we define solvable and $k$-solvable tests and derive generating functions that enumerate them by cost, number of distinct valid values, and number of options. We also investigate extremal questions, including minimum costs and the maximum possible number of valid values. The paper was inspired by a puzzle from \emph{Mathematical Puzzles and Curiosities}.
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