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HMMT 二月 2006 · TEAM2 赛 · 第 12 题

HMMT February 2006 — TEAM2 Round — Problem 12

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [15] Find all ordered triples ( x, y, z ) of positive reals such that x + y + z = 27 and x + y + z − xy −
    yz − zx = 0. Prove that your answer is correct.
解析

英文解析

  1. [15] Find all ordered triples ( x, y, z ) of positive reals such that x + y + z = 27 and
    2 2 2
    x + y + z − xy − yz − zx = 0. Prove that your answer is correct.
    Answer: (9 , 9 , 9)
    2 2 2
    ( x − y ) + ( y − z ) + ( z − x )
    2 2 2
    Solution: We have x + y + z − xy − yz − zx = = 0,
    which implies x = y = z = = 9.272 3