1972 IMO Problems/Problem 1
Prove that from a set of ten distinct two-digit numbers (in the decimal system), it is possible to select two disjoint subsets whose members have the same sum.
There are distinct subsets of our set of 10 two-digit numbers. The sum of the elements of any subset of our set of 10 two-digit numbers must be between and . (There are fewer attainable sums.) As , the Pigeonhole Principle implies there are two distinct subsets whose members have the same sum. Let these sets be and . Now, let and . Notice and are disjoint. They are also nonempty because if or , then one of and is a subset of the other, so they are either not distinct or have different sums. Therefore and are disjoint subsets our set of 10 distinct two-digit numbers, which proves the claim.
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