HMMT 二月 2006 · 代数 · 第 6 题

HMMT February 2006 — Algebra — Problem 6

Let a, b, c be the roots of x − 9 x +11 x − 1 = 0 , and let s = a + b + c. Find s − 18 s − 8 s.

解析

Let a, b, c be the roots of x − 9 x + 11 x − 1 = 0 , and let s = a + b + c. Find 4 2 s − 18 s − 8 s. Answer: − 37 Solution: First of all, as the left side of the first given equation takes values − 1, 2, − 7, and 32 when x = 0, 1, 2, and 3, respectively, we know that a , b , and c are distinct √ √ √ positive reals. Let t = ab + bc + ca , and note that 2 s = a + b + c + 2 t = 9 + 2 t, √ 2 t = ab + bc + ca + 2 abcs = 11 + 2 s, 4 2 2 s = (9 + 2 t ) = 81 + 36 t + 4 t = 81 + 36 t + 44 + 8 s = 125 + 36 t + 8 s, 2 18 s = 162 + 36 t, 4 2 so that s − 18 s − 8 s = − 37 .