HMMT 十一月 2020 · 冲刺赛 · 第 6 题
HMMT November 2020 — Guts Round — Problem 6
题目详情
- [6] If x, y, z are real numbers such that xy = 6, x − z = 2, and x + y + z = 9, compute − − . y x xy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMO 2020, November 14–21, 2020 — GUTS ROUND Organization Team Team ID#
解析
- [6] If x, y, z are real numbers such that xy = 6, x − z = 2, and x + y + z = 9, compute − − . y x xy Proposed by: Ragulan Sivakumar Answer: 2 2 2 2 x − yz − z x z z Solution: Let k = − − = . We have y x xy xy 2 2 2 2 x + xy − yz − z x − xz + xy − yz + zx − z ( x + y + z )( x − z ) 9 · 2 k + 1 = = = = = 3 , xy xy xy 6 so k = 2.