# 2002 AIME I Problems/Problem 14

## 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?

## 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.

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