HMMT 二月 2004 · 代数 · 第 7 题
HMMT February 2004 — Algebra — Problem 7
题目详情
- If x, y, k are positive reals such that ( ) ( ) 2 2 x y x y 2 3 = k + + k + , 2 2 y x y x find the maximum possible value of k . 3 7 2
解析
- If x, y, k are positive reals such that ( ) ( ) 2 2 x y x y 2 3 = k + + k + , 2 2 y x y x find the maximum possible value of k . √ Solution: ( − 1 + 7) / 2 2 2 2 2 2 2 2 We have 3 = k ( x /y + y /x ) + k ( x/y + y/x ) ≥ 2 k + 2 k , hence 7 ≥ 4 k + 4 k + 1 = √ 2 (2 k +1) , hence k ≤ ( 7 − 1) / 2. Obviously k can assume this value, if we let x = y = 1. 3 7 2