HMMT 二月 2004 · 冲刺赛 · 第 12 题

英文解析

12. 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 thisquadrilateral?
    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 foursolutions: ( x, y ) = (8 , 3) , (3 , 8) , ( − 3 , − 8) , ( − 8 , − 3). All four of these solutions satisfythe original equations. The quadrilateral is therefore a rectangle with side lengths of
    √ √
    5 2 and 11 2, so its area is 110.