HMMT 二月 2004 · 冲刺赛 · 第 12 题

HMMT February 2004 — Guts Round — Problem 12

[7] A convex quadrilateral is drawn in the coordinate plane such that each of its 2 2 vertices ( x, y ) satisfies the equations x + y = 73 and xy = 24. What is the area of this quadrilateral? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HARVARD-MIT MATHEMATICS TOURNAMENT, FEBRUARY 28, 2004 — GUTS ROUND

解析

A convex quadrilateral is drawn in the coordinate plane such that each of its vertices 2 2 ( x, y ) satisfies the equations x + y = 73 and xy = 24. What is the area of this quadrilateral? Solution: 110 2 2 2 The vertices all satisfy ( x + y ) = x + y + 2 xy = 73 + 2 · 24 = 121, so x + y = ± 11. 2 2 2 Similarly, ( x − y ) = x + y − 2 xy = 73 − 2 · 24 = 25, so x − y = ± 5. Thus, there are four solutions: ( x, y ) = (8 , 3) , (3 , 8) , ( − 3 , − 8) , ( − 8 , − 3). All four of these solutions satisfy the original equations. The quadrilateral is therefore a rectangle with side lengths of √ √ 5 2 and 11 2, so its area is 110.