HMMT 十一月 2016 · 冲刺赛 · 第 27 题

HMMT November 2016 — Guts Round — Problem 27

专题: Discrete Math / 离散数学
难度: L3
来源: HMMT

题目详情

[ 13 ] Let r , r , r , r be the four roots of the polynomial x − 4 x + 8 x − 7 x + 3. Find the value of 1 2 3 4 2 2 2 2 r r r r 1 2 3 4

2 2 2 2 2 2 2 2 2 2 2 2 r + r + r r + r + r r + r + r r + r + r 2 3 4 1 3 4 1 2 4 1 2 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT November 2016, November 12, 2016 — GUTS ROUND Organization Team Team ID#

解析

[ 13 ] Let r , r , r , r be the four roots of the polynomial x − 4 x + 8 x − 7 x + 3. Find the value of 1 2 3 4 2 2 2 2 r r r r 1 2 3 4

2 2 2 2 2 2 2 2 2 2 2 2 r + r + r r + r + r r + r + r r + r + r 2 3 4 1 3 4 1 2 4 1 2 3 Proposed by: Henrik Boecken Answer: − 4 Add 1 to each fraction to get 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 r + r + r + r r + r + r + r r + r + r + r r + r + r + r 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4

2 2 2 2 2 2 2 2 2 2 2 2 r + r + r r + r + r r + r + r r + r + r 2 3 4 1 3 4 1 2 4 1 2 3 This seems like a difficult problem until one realizes that 2 2 2 2 2 2 r + r + r + r = ( r + r + r + r ) − 2( r r + r r + r r + r r + r r + r r ) = 4 − 2 · 8 = 0 1 2 3 4 1 2 1 3 1 4 2 3 2 4 3 4 1 2 3 4 Thus, our current expression is 0. Noting that we added 4, the original value had to be − 4 .