返回题库

HMMT 二月 2003 · 冲刺赛 · 第 19 题

HMMT February 2003 — Guts Round — Problem 19

专题
Discrete Math / 离散数学
难度
L3
来源
HMMT

题目详情

  1. [8] Let r, s, t be the solutions to the equation x + ax + bx + c = 0. What is the value 2 2 2 of ( rs ) + ( st ) + ( rt ) in terms of a, b, and c ?
解析
  1. Let r, s, t be the solutions to the equation x + ax + bx + c = 0. What is the value of 2 2 2 ( rs ) + ( st ) + ( rt ) in terms of a, b, and c ? 2 Solution: b − 2 ac 3 2 We have ( x − r )( x − s )( x − t ) = x + ax + bx + c , so a = − ( r + s + t ) , b = rs + st + rt, c = − rst. So we have 2 2 2 2 2 ( rs ) + ( st ) + ( rt ) = ( rs + st + rt ) − 2 rst ( r + s + t ) = b − 2 ac.