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 $S$ . We are given that $\dfrac{S-1}{num(\mathcal{S})-1}$ and $\dfrac{S-2002}{num(\mathcal{S})-1}$ are integers. Thus $2001$ is a multiple of $num(\mathcal{S})-1$ . Now $2001=3*667$ , so either $num(\mathcal{S})$ is 2002, 668, 4, or 2. 2 is guaranteed possible, 2002 is not. 4 is: 1, 4, 7, 2002. For 668, all 668 numbers must be congruent mod $667$ , and there aren't enough numbers like that. So $004$ is the maximum.

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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