IR-TST 2023 Problem 12

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