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Problem 5 of Third round - Divisibility of a^n+b^n+c^n+d^n

Source: VI International Festival of Young Mathematicians Sozopol, Theme for 10-12 grade

December 24, 2019
number theoryDivisibilityprimes

Problem Statement

Let p>3p>3 be a prime number. The natural numbers a,b,c,da,b,c, d are such that a+b+c+da+b+c+d and a3+b3+c3+d3a^3+b^3+c^3+d^3 are divisible by pp. Prove that for all odd nn, an+bn+cn+dna^n+b^n+c^n+d^n is divisible by pp.
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