HMMT 二月 2014 · 冲刺赛 · 第 16 题

英文解析

16. [ 8 ] Suppose that x and y are positive real numbers such that x − xy + 2 y = 8. Find the maximum
    2 2
    possible value of x + xy + 2 y .
    √
    √
    72+32 2 2 2 u
    Answer: Let u = x + 2 y . By AM-GM, u ≥ 8 xy , so xy ≤ √ . If we let xy = kuwhere k ≤ √ , then we have 187
    u (1 − k ) = 88
    2 2
    u (1 + k ) = x + xy + 2 y
    1 + kthat is, u (1 + k ) = 8 · . It is not hard to see that the maximum value of this expression occurs at
    1 − k
    √
    √1
    1 +
    1 72 + 32 2
    k = √ , so the maximum value is 8 · = .8
    √1
    1 −7
    88