HMMT 二月 2014 · 代数 · 第 5 题
HMMT February 2014 — Algebra — Problem 5
题目详情
- Find the sum of all real numbers x such that 5 x 10 x + 10 x 5 x 11 = 0. 2 2
解析
- Find the sum of all real numbers x such that 5 x − 10 x + 10 x − 5 x − 11 = 0. 5 5 Answer: 1 Rearrange the equation to x + (1 − x ) − 12 = 0. It’s easy to see this has two real roots, and that r is a root if and only if 1 − r is a root, so the answer must be 1. 4 3 2 2 Alternate solution: Note that 5 x − 10 x + 10 x − 5 x − 11 = 5 x ( x − 1)( x − x + 1) − 11 = 5 u ( u + 1) − 11, 2 2 1 where u = x − x . Of course, x − x = u has real roots if and only if u ≥ − , and distinct real roots 4 √ √ 2 − 5 ± 5 +4(5)(11) 1 − 5 ± 7 5 if and only if u > − . But the roots of 5 u ( u + 1) − 11 are = , one of which is 4 2 · 5 10 1 1 2 greater than − and the other less than − . For the larger root, x − x = u has exactly two distinct 4 4 real roots, which sum up to 1 by Vieta’s. 2 2