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 greatet number of elements that
can have?
Solution
Let the sum of the integers in be
. We are given that
and
are integers. Thus
is a multiple of
. Now
, so either
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
, and there aren't enough numbers like that. So
is the maximum.
See also
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 | ||
All AIME Problems and Solutions |