IMO 1972 SL 12

Origin: USS

Problem

A set of 10 positive integers is given such that the decimal expansion of each of them has two digits. Prove that there are two disjoint subsets of the set with equal sums of their elements.

Solution

First we observe that it is not essential to require the subsets to be disjoint (if they aren’t, one simply excludes their intersection). There are 210−1 = 1023 different subsets and at most 990 different sums. By the pigeonhole principle there are two different subsets with equal sums.