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2n subset from {0,1,2,...,2n-1}

Source: Iran TST 2023 ; Exam 2 Problem 6

March 16, 2023
Subsetscombinatorics

Problem Statement

Suppose that we have 2n2n non-empty subset of {0,1,2,...,2n−1} \big\{0,1,2,...,2n-1\big\} that sum of the elements of these subsets is (2n+12) \binom{2n+1}{2} . Prove that we can choose one element from every subset that some of them is (2n2) \binom{2n}{2}
Proposed by Morteza Saghafian and Afrouz Jabalameli
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