HMMT 二月 2024 · 团队赛 · 第 5 题

HMMT February 2024 — Team Round — Problem 5

[40] Determine, with proof, whether there exist positive integers x and y such that x + y , x + y , and 3 3 x + y are all perfect squares.

解析

[40] Determine, with proof, whether there exist positive integers x and y such that x + y , x + y , and 3 3 x + y are all perfect squares. Proposed by: Rishabh Das Answer: Yes 2 2 2 2 3 3 2 Solution: Take ( x, y ) = (184 , 345) . Then x + y = 23 , x + y = 391 , and x + y = 6877 . 2 2 2 2 Remark. We need x + y, x + y , x − xy + y to be perfect squares. We will find a, b such that 2 2 2 2 a + b , a − ab + b are perfect squares, and then let x = a ( a + b ) and y = b ( a + b ) . Experimenting with small Pythagorean triples gives a = 8 , b = 15 as a solution. 2 2 Remark. The smallest solution we know of not of the form (184 k , 345 k ) is (147 916 017 521 041 , 184 783 370 001 360) .