HMMT 二月 2004 · 冲刺赛 · 第 21 题
HMMT February 2004 — Guts Round — Problem 21
题目详情
- [8] Find all ordered pairs of integers ( x, y ) such that 3 4 = 2 + 2 . 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND
解析
- Find all ordered pairs of integers ( x, y ) such that 3 4 = 2 + 2 . Solution: (0 , 1) , (1 , 1) , (2 , 2) 5 x + y x + y − 1 The right side is 2 (1 + 2 ). If the second factor is odd, it needs to be a power of 3, so the only options are x + y = 2 and x + y = 4. This leads to two solutions, namely (1,1) and (2,2). The second factor can also be even, if x + y − 1 = 0. Then x y x + y = 1 and 3 4 = 2 + 2, giving (0 , 1) as the only other solution.