PUMaC 2012 · 数论（A 组） · 第 4 题

PUMaC 2012 — Number Theory (Division A) — Problem 4

[ 4 ] Find the sum of all possible sums a + b where a and b are nonnegative integers such that a b 4 + 2 + 5 is a perfect square.

解析

The only residues of squares modulo 4 are 0 and 1. If all of the squares have residues of 1 modulo 4, then they are all odd and we consider the problem modulo 8. The only residues of squares modulo 8 are 0, 1, and 4, and because 2012 2 ≡ 0 (mod 8), we see that the squares cannot all be odd, so they must all be even. If all of the squares are even, then we divide both sides by 4 and repeat the process. We see that the only solution is 1005 a = b = c = d = 2 , so there is only 1 solution. Problem contributed by Wesley Cao. x x k k