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 greatet 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 $n$ . We are given that $\dfrac{N-1}{n-1}$ and $\dfrac{N-2002}{n-1}$ are integers. Thus $2001$ is a multiple of $n-1$ . Now $2001= 3 \times 23 \times 29$ , Template:Incomplete

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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