HMMT 十一月 2018 · 团队赛 · 第 4 题

HMMT November 2018 — Team Round — Problem 4

[ 30 ] Let a and b be real numbers greater than 1 such that ab = 100. The maximum possible value of 2 (log b ) x 10 a can be written in the form 10 for some real number x . Find x .

解析

[ 30 ] Let a and b be real numbers greater than 1 such that ab = 100. The maximum possible value of 2 (log b ) x 10 a can be written in the form 10 for some real number x . Find x . Proposed by: James Lin 32 Answer: 27 Let p = log a, q = log b . Since a, b > 1, p and q are positive. The condition ab = 100 translates to 10 10 p + q = 2. We wish to maximize 2 (log b ) 2 2 10 x = log a = (log a )(log b ) = pq . 10 10 10 By AM-GM, ( ) 3 27 q q 2 pq ≤ p + + = 8 . 4 2 2 32 2 4 2 Hence pq ≤ with equality when p = , q = . 27 3 3