HMMT 二月 2018 · 冲刺赛 · 第 14 题

HMMT February 2018 — Guts Round — Problem 14

[ 9 ] Given that x is a positive real, find the maximum possible value of ( ( ) ( )) x x − 1 − 1 sin tan − tan . 9 16 2 2017

解析

[ 9 ] Given that x is a positive real, find the maximum possible value of ( ( ) ( )) x x − 1 − 1 sin tan − tan . 9 16 Proposed by: Yuan Yao 7 Answer: 25 Consider a right triangle AOC with right angle at O , AO = 16 and CO = x . Moreover, let B x x − 1 − 1 be on AO such that BO = 9. Then tan = ∠ CBO and tan = ∠ CAO , so their difference 9 16 is equal to ∠ ACB . Note that the locus of all possible points C given the value of ∠ ACB is part of a circle that passes through A and B , and if we want to maximize this angle then we need to make this circle as small as possible. This happens when OC is tangent to the circumcircle of ABC , so 4 2 2 OC = OA · OB = 144 = 12 , thus x = 12, and it suffices to compute sin( α − β ) where sin α = cos β = 5 3 4 3 7 2 2 and cos α = sin β = . By angle subtraction formula we get sin( α − β ) = ( ) − ( ) = . 5 5 5 25 2 2017