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HMMT 二月 2003 · 冲刺赛 · 第 19 题

HMMT February 2003 — Guts Round — Problem 19

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  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

英文解析

  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 ?
    Solution: b − 2 ac 2
    3 2
    We have ( x − r )( x − s )( x − t ) = x + ax + bx + c , soa = − ( 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.