RMO 2023 Questions, Solutions, Discussions
🔗
(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 \(36\).
🔗
Click here for the spoiler!
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)
Click on the icons below to download.
Topics Links Algebra Combinatorics Geometry Number Theory INMO Training Camp 2025, MP IMO Training Camp MOPSS Resources