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HMMT 十一月 2008 · 冲刺赛 · 第 12 题

HMMT November 2008 — Guts Round — Problem 12

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

  1. [ 8 ] Solve the following system of equations for w .
    2 w + x + y + z = 1
    w + 2 x + y + z = 2
    w + x + 2 y + z = 2
    w + x + y + 2 z = 1 .
    1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUNDst

英文原题

[ 8 ] Solve the following system of equations for w .
2 w + x + y + z = 1
w + 2 x + y + z = 2
w + x + 2 y + z = 2
w + x + y + 2 z = 1 .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1 st HARVARD-MIT NOVEMBER TOURNAMENT, 8 NOVEMBER 2008 — GUTS ROUND

解析

英文解析

  1. [ 8 ] Solve the following system of equations for w .
    2 w + x + y + z = 1
    w + 2 x + y + z = 2
    w + x + 2 y + z = 2
    w + x + y + 2 z = 1 .
    − 1 6
    Answer: Add all the equations together to find that 5 x + 5 y + 5 z + 5 w = 6, or x + y + z + w = .
    5 5 − 1
    We can now subtract this equation from the first equation to see that w = .
    1 HARVARD-MIT NOVEMBER TOURNAMENT, 8 SATURDAY 2008 — GUTS ROUNDst 2