AIME 2002 I · 第 14 题

AIME 2002 I — Problem 14

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

A set $\mathcal{S}$ of distinct positive integers has the following property: for every integer $x$ in $\mathcal{S},$ the arithmetic mean of the set of values obtained by deleting $x$ from $\mathcal{S}$ is an integer. Given that 1 belongs to $\mathcal{S}$ and that 2002 is the largest element of $\mathcal{S},$ what is the greatest number of elements that $\mathcal{S}$ can have?

解析

Solution

Let the sum of the integers in $\mathcal{S}$ be $N$ , and let the size of $|\mathcal{S}|$ be $n+1$ . After any element $x$ is removed, we are given that $n|N-x$ , so $x\equiv N\pmod{n}$ . Since $1\in\mathcal{S}$ , $N\equiv1\pmod{n}$ , and all elements are congruent to 1 mod $n$ . Since they are positive integers, the largest element is at least $n^2+1$ , the $(n+1)$ th positive integer congruent to 1 mod $n$ .

We are also given that this largest member is 2002, so $2002\equiv1\pmod{n}$ , and $n|2001=3\cdot23\cdot29$ . Also, we have $n^2+1\le2002$ , so $n \leq 44$ . The largest factor of 2001 less than 45 is 29, so $n=29$ and $n+1$ $\Rightarrow{\fbox{30}}$ is the largest possible. This can be achieved with $\mathcal{S}=\{1,30,59,88,\ldots,813,2002\}$ , for instance.