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HMMT 十一月 2021 · 冲刺赛 · 第 25 题

HMMT November 2021 — Guts Round — Problem 25

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [13] Let x, y, z be real numbers satisfying 2 x + y + 4 xy + 6 xz = − 6 y + 2 z + 2 xy + 6 yz = 4 x − z + 2 xz − 4 yz = − 3 2 2 2 Find x + y + z .
解析
  1. [13] Let x, y, z be real numbers satisfying 2 x + y + 4 xy + 6 xz = − 6 y + 2 z + 2 xy + 6 yz = 4 x − z + 2 xz − 4 yz = − 3 2 2 2 Find x + y + z . Proposed by: David Vulakh Answer: 29 1 1 Solution: We multiply the first, second, and third equations by , − , and − 1, respectively, then 2 2 add the three resulting equations. This gives xy + xz + yz = − 2. Doing the same with the coeffi- 2 2 2 2 cients − 1, 2, and 3 gives x + y + z = 5, from which ( x + y + z ) = 25. So x + y + z = 25 − 2 ·− 2 = 29.