HMMT 二月 2005 · 代数 · 第 7 题

英文解析

7. Let x be a positive real number. Find the maximum possible value of
   √
   x + 2 − x + 442
   .
   2 x
   √
   Solution: 2 2 − 2
   Rationalizing the numerator, we get
   √ √
   2 2 2 2 4
   4 4
   x + 2 − x + 4 x + 2 + x + 4 ( x + 2) − ( x + 4)
   · √ = √
   2 4 2 4
   x + 2 + x + 4 x ( x + 2 + x + 4)x
   4 x 2
   √
   2 4=
   x ( x + 2 + x + 4)
   √4
   1=
   2 4
   ( x + 2 + x + 4)
   4 x
   √
   = .
   2 4
   x + + x +2
   x x 2
   Since we wish to maximize this quantity, we wish to minimize the denominator. By
   √ √
   2 4
   AM-GM, x + ≥ 2 2 and x + ≥ 4, so that the denominator is at least 2 2 + 2.2
   x x 2
   Therefore,
   √
   √42
   x + 2 − x + 4 4
   √
   ≤ = 2 2 − 2 ,
   2 2 + 2 x
   √
   with equality when x = 2.