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Fifth largest numbers in a partition of {1,2,...,900}

Source: XIII Rioplatense Mathematical Olympiad (2004), Level 3

July 26, 2011
combinatorics unsolvedcombinatorics

Problem Statement

Consider a partition of {1,2,…,900}\{1,2,\ldots,900\} into 3030 subsets S1,S2,…,S30S_1,S_2,\ldots,S_{30} each with 3030 elements. In each SkS_k, we paint the fifth largest number blue. Is it possible that, for k=1,2,…,30k=1,2,\ldots,30, the sum of the elements of SkS_k exceeds the sum of the blue numbers?
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