返回题库

HMMT 二月 2004 · 冲刺赛 · 第 21 题

HMMT February 2004 — Guts Round — Problem 21

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [8] Find all ordered pairs of integers ( x, y ) such that 3 4 = 2 + 2 .
    HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND

英文原题

[8] Find all ordered pairs of integers ( x, y ) such that 3 x 4 y = 2 x + y + 2 2( x + y ) − 1 .
3
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND

解析

英文解析

  1. Find all ordered pairs of integers ( x, y ) such that 3 4 = 2 + 2 .
    Solution: (0 , 1) , (1 , 1) , (2 , 2)
    x + y x + y − 15
    The right side is 2 (1 + 2 ). If the second factor is odd, it needs to be a powerof 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. Thenx yx + y = 1 and 3 4 = 2 + 2, giving (0 , 1) as the only other solution.