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HMMT 二月 2007 · 代数 · 第 5 题

HMMT February 2007 — Algebra — Problem 5

专题
Contest Math / 竞赛数学
难度
L3
来源
HMMT

题目详情

英文原题

  1. [ 5 ] A convex quadrilateral is determined by the points of intersection of the curves x + y = 100 andxy = 4; determine its area.
    3 2
解析

英文解析

  1. [ 5 ] A convex quadrilateral is determined by the points of intersection of the curves x + y = 100 andxy = 4; determine its area.
    √
    Answer: 4 17 . By symmetry, the quadrilateral is a rectangle having x = y and x = − y as axes ofsymmetry. Let ( a, b ) with a > b > 0 be one of the vertices. Then the desired area is
    ( ) ( )
    √ √
    √ √ √
    2 2
    4 2 2 4 2
    2( a − b ) · 2( a + b ) = 2( a − b ) = 2 a − 2 a b + b = 2 100 − 2 · 4 = 4 17 .
    3 2