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ab+bc+ca

Source: 2012cmo,problem5

January 12, 2012
modular arithmeticinequalitiesnumber theory proposednumber theory

Problem Statement

Consider a square-free even integer nn and a prime pp, such that 1) (n,p)=1(n,p)=1; 2) p≤2np\le 2\sqrt{n}; 3) There exists an integer kk such that p∣n+k2p|n+k^2. Prove that there exists pairwise distinct positive integers a,b,ca,b,c such that n=ab+bc+can=ab+bc+ca.
Proposed by Hongbing Yu
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