HMMT 二月 2004 · 代数 · 第 5 题

HMMT February 2004 — Algebra — Problem 5

There exists a positive real number x such that cos(tan ( x )) = x . Find the value of 2 x . 4 4

解析

There exists a positive real number x such that cos(tan ( x )) = x . Find the value of 2 x . √ Solution: ( − 1 + 5) / 2 − 1 Draw a right triangle with legs 1 , x ; then the angle θ opposite x is tan x , and we can √ √ 2 2 compute cos( θ ) = 1 / x + 1. Thus, we only need to solve x = 1 / x + 1. This is √ 4 2 4 2 2 equivalent to x x + 1 = 1. Square both sides to get x + x = 1 ⇒ x + x − 1 = 0. √ 2 2 Use the quadratic formula to get the solution x = ( − 1 + 5) / 2 (unique since x must be positive). 4 4