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Strange power sum

Source: XIII Ibero American Olympiad For University Students

July 22, 2011
algebrapolynomialinductionalgebra proposed

Problem Statement

(a) Prove that, for any positive integers m≤ℓm\le \ell given, there is a positive integer nn and positive integers x1,⋯ ,xn,y1,⋯ ,ynx_1,\cdots,x_n,y_1,\cdots,y_n such that the equality ∑i=1nxik=∑i=1nyik \sum_{i=1}^nx_i^k=\sum_{i=1}^ny_i^k holds for every k=1,2,⋯ ,m−1,m+1,⋯ ,ℓk=1,2,\cdots,m-1,m+1,\cdots,\ell, but does not hold for k=mk=m.
(b) Prove that there is a solution of the problem, where all numbers x1,⋯ ,xn,y1,⋯ ,ynx_1,\cdots,x_n,y_1,\cdots,y_n are distinct.
Proposed by Ilya Bogdanov and Géza Kós.
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