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HMMT 十一月 2013 · 冲刺赛 · 第 12 题

HMMT November 2013 — Guts Round — Problem 12

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 8 ] Given that 62 + 122 = 18728, find positive integers ( n, m ) such that n + m = 9364.
    HMMT NOVEMBER 2013, 9 NOVEMBER 2013 — GUTS ROUND

英文原题

[ 8 ] Given that 62 2 + 122 2 = 18728, find positive integers ( n, m ) such that n 2 + m 2 = 9364.
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HMMT NOVEMBER 2013, 9 NOVEMBER 2013 — GUTS ROUND

解析

英文解析

  1. [ 8 ] Given that 62 + 122 = 18728, find positive integers ( n, m ) such that n + m = 9364.
    2 2 2 2
    a + b a − b 2 a +2 b a + b
    2 2 2 2
    Answer: (30 , 92) OR (92 , 30) If a + b = 2 c , then ( ) + ( ) = = = c . Thus,
    2 2 4 2
    62+122 122 − 62
    n = = 92 and m = = 30 works.
    2 2