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Probablity for a modular equation

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September 1, 2010
probabilitymodular arithmeticfunctionalgebracombinatoricsIMO ShortlistIMO Longlist

Problem Statement

Suppose that n numbers x1,x2,...,xnx_1, x_2, . . . , x_n are chosen randomly from the set {1,2,3,4,5}\{1, 2, 3, 4, 5\}. Prove that the probability that x12+x22+⋯+xn2≡0(mod5)x_1^2+ x_2^2 +\cdots+ x_n^2 \equiv 0 \pmod 5 is at least 15.\frac 15.
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