PUMaC 2021 · 代数（B 组） · 第 3 题
PUMaC 2021 — Algebra (Division B) — Problem 3
题目详情
- Compute the sum of all real numbers x which satisfy the following equation x x 8 − 19 · 4 = 2 . x 16 − 25 · 2
解析
- Compute the sum of all real numbers x which satisfy the following equation x x 8 − 19 · 4 = 2 . x 16 − 25 · 2 Proposed by: Nancy Xu Answer: 5 3 2 y − 19 y x 3 2 Let y = 2 . Then the equation becomes = 2, which gives us y − 19 y + 50 y − 32 = 0. 16 − 25 y All roots of this polynomial must divide 32, so by testing the divisors of 32 we find that 2 2 y − 19 y + 50 y − 32 = ( y − 1)( y − 2)( y − 16), so that x = 0 , 1 , or 4. Thus the desired sum is 0 + 1 + 4 = 5.