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A zero-sum problem

Source: Kürschák 2000, problem 3

July 13, 2014
pigeonhole principlemodular arithmeticnumber theory unsolvednumber theory

Problem Statement

Let k≥0k\ge 0 be an integer and suppose the integers a1,a2,…,ana_1,a_2,\dots,a_n give at least 2k2k different residues upon division by (n+k)(n+k). Show that there are some aia_i whose sum is divisible by n+kn+k.
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