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 $n<45$ . The largest factor of 2001 less than 45 is 29, so $n=29$ and $n+1=\fbox{30}$ is the largest possible. This can be achieved with $\mathcal{S}=\{1,30,59,88,\ldots,813,2002\}$ , for instance.

2002 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2002 AIME I Problems/Problem 14

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