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Bosnia and Herzegovina TST 2012 Problem 3

Source:

May 19, 2012
calculusintegrationmodular arithmeticquadraticsnumber theoryprime numbersnumber theory proposed

Problem Statement

Prove that for all odd prime numbers pp there exist a natural number m<pm<p and integers x1,x2,x3x_1, x_2, x_3 such that:
mp=x12+x22+x32.mp=x_1^2+x_2^2+x_3^2.
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