HMMT 二月 2014 · 代数 · 第 5 题

英文解析

5. 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 realroots, 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
   √
   − 5 ± 5 +4(5)(11)2
   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
   greater than − and the other less than − . For the larger root, x − x = u has exactly two distinct 2
   4 4
   real roots, which sum up to 1 by Vieta’s.
   2 2