Question

一场音乐会中，成年人占观众人数的 $\frac{5}{12}$ 。后来一辆载有 $50$ 人的公共汽车到达，此时成年人占音乐会现场人数的 $\frac{11}{25}$ 。求公共汽车到达后，现场成年人可能的最少人数。

Adults made up $\frac{5}{12}$ of the crowd of people at a concert. After a bus carrying $50$ more people arrived, adults made up $\frac{11}{25}$ of the people at the concert. Find the minimum number of adults who could have been at the concert after the bus arrived.

答案

答案：154

知识点：分数整除性最优化

难度评级：1890

解答：

设原来有 $12k$ 名观众，其中 $5k$ 人是成年人。公共汽车到达后共有 $12k + 50$ 人，成年人数为 $\frac{11}{25}(12k + 50)$ 。要使其为整数， $25$ 必须整除 $12k + 50$ ，所以 $25 \mid 12k$ ，因为 $\gcd(12, 25) = 1$ ，所以 $k$ 是 $25$ 的倍数。

成年人数 $\frac{11}{25}(12k + 50)$ 随 $k$ 增大而增大，因此最小值在 $k = 25$ 时取得：新总人数为 $350$ ，成年人数为 $\frac{11}{25} \cdot 350 = 154$ 。这是可以实现的，例如公共汽车上有 $29$ 名成年人和 $21$ 名非成年人，所以答案是 $154$ 。

Let the original crowd have $12k$ people, of whom $5k$ are adults. After the bus arrives there are $12k + 50$ people, and the number of adults is $\frac{11}{25}(12k + 50).$ For this to be an integer, $25$ must divide $12k + 50,$ so $25 \mid 12k,$ and since $\gcd(12, 25) = 1$ this means $k$ is a multiple of $25.$

The adult count $\frac{11}{25}(12k + 50)$ increases with $k,$ so the minimum occurs at $k = 25:$ the new total is $350$ and the number of adults is $\frac{11}{25} \cdot 350 = 154.$ This is achievable, for example if the bus carries $29$ adults and $21$ non-adults, so the answer is $154.$

2022 AIME II 第 1 题

答案：154

难度评级：1890

解答：

其他年份的第 1 题