HMMT 十一月 2017 · GEN 赛 · 第 2 题
HMMT November 2017 — GEN Round — Problem 2
题目详情
- Determine the sum of all distinct real values of x such that ||| · · · || x | + x | · · · | + x | + x | = 1 , where there are 2017 x ’s in the equation.
解析
- Determine the sum of all distinct real values of x such that ||| · · · || x | + x | · · · | + x | + x | = 1 , where there are 2017 x ’s in the equation. Proposed by: Yuan Yao 2016 Answer: − 2017 Note that | x + | x || = 2 x when x is nonnegative, and is equal to 0 otherwise. Thus, when there are 2017 x ’s, the expression equals 2017 x when x ≥ 0 and − x otherwise, so the two solutions to the equation 1 2016 are x = − 1 and , and their sum is − . 2017 2017