RMO 2023 Questions, Solutions, Discussions
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(RMO 2023a P2, AoPS) Given a prime number \(p\) such that \(2p\) is equal to the sum of the squares of some four consecutive positive integers. Prove that \(p-7\) is divisible by \(36\).
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Show that the sum of four consecutive squares is congruent to \(6\) modulo \(8\), and conclude that \(p \equiv 3 \bmod 4\). Considering congruence conditions modulo \(3\), prove that the smallest of the four consecutive numbers is a multiple of \(3\). Deduce that the sum of the four consecutive squares is \(5\) modulo \(9\).
Lecture notes for Math Olympiad (IOQM, RMO, INMO)
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