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a_1+2^ka_2+3^ka_3+...+n^ka_n is divisible by k!

Source: Polish second round 1999 p6

January 19, 2020
factorialnumber theorySum

Problem Statement

Suppose that a1,a2,...,ana_1,a_2,...,a_n are integers such that a1+2ia2+3ia3+...+nian=0a_1 +2^ia_2 +3^ia_3 +...+n^ia_n = 0 for i=1,2,...,k−1i = 1,2,...,k -1, where k≥2k \ge 2 is a given integer. Prove that a1+2ka2+3ka3+...+nkana_1+2^ka_2+3^ka_3+...+n^ka_n is divisible by k!k!.
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