HMMT 十一月 2018 · 冲刺赛 · 第 32 题

HMMT November 2018 — Guts Round — Problem 32

[ 17 ] Over all real numbers x and y , find the minimum possible value of 2 2 2 ( xy ) + ( x + 7) + (2 y + 7) .

解析

[ 17 ] Over all real numbers x and y , find the minimum possible value of 2 2 2 ( xy ) + ( x + 7) + (2 y + 7) . Proposed by: James Lin Answer: 45 2 2 Solution 1: Rewrite the given expression as ( x + 4)(1 + y ) + 14( x + 2 y ) + 94. By Cauchy-Schwartz, 2 2 this is at least ( x + 2 y ) + 14( x + 2 y ) + 94 = ( x + 2 y + 7) + 45. The minimum is 45, attained when xy = 2 , x + 2 y = 7. Solution 2: Let z = 2 y, s = x + z, p = xz . We seek to minimize ⇣ ⌘ 2 2 xz p 2 2 2 2

( x + 7) + ( z + 7) = + ( x + z ) + 14( x + z ) + 98 2 4 2 p 2 = + s 2 p + 14 s + 98 4 ⇣ ⌘ 2 p 2 = 2 + ( s + 7) + 45 2 45 . 2 Equality holds when s = 7 , p = 4. Since s 4 p , this system has a real solution for x and z .