HMMT 二月 2006 · 代数 · 第 6 题

英文解析

6. 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 thats = a + b + c + 2 t = 9 + 2 t,2
   √
   t = ab + bc + ca + 2 abcs = 11 + 2 s,2
   4 2 2
   s = (9 + 2 t ) = 81 + 36 t + 4 t = 81 + 36 t + 44 + 8 s = 125 + 36 t + 8 s,
   18 s = 162 + 36 t,2
   4 2
   so that s − 18 s − 8 s = − 37 .