2002 AIME I Problems/Problem 14
Problem
A set of distinct positive integers has the following property: for every integer in the arithmetic mean of the set of values obtained by deleting from is an integer. Given that 1 belongs to and that 2002 is the largest element of what is the greatest number of elements that can have?
Solution
Let the sum of the integers in be , and let the size of be . After any element is removed, we are given that , so . Since , , and all elements are congruent to 1 mod . Since they are positive integers, the largest element is at least , the th positive integer congruent to 1 mod .
We are also given that this largest member is 2002, so , and . Also, we have , so . The largest factor of 2001 less than 45 is 29, so and is the largest possible. This can be achieved with , for instance.
See also
2002 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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