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HMMT 二月 2006 · TEAM2 赛 · 第 12 题

HMMT February 2006 — TEAM2 Round — Problem 12

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  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
  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, 2 27 which implies x = y = z = = 9. 3