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HMMT 十一月 2016 · 冲刺赛 · 第 27 题

HMMT November 2016 — Guts Round — Problem 27

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 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#

英文原题

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

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

英文解析

  1. [ 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 .