HMMT 十一月 2013 · 冲刺赛 · 第 10 题

HMMT November 2013 — Guts Round — Problem 10

[ 8 ] How many pairs of real numbers ( x, y ) satisfy the equation 4 2 3 3 4 2 y − y = xy − xy = x y − xy = x − x = 0?

解析

Thus CE, ED are 2 , 8. Without loss of generality, assume CE = 8 and DE = 2. Assume our circle is centered at the origin, with points A = ( − 3 , 5), B = ( − 3 , − 5), C = (5 , − 3), 2 2 D = ( − 5 , − 3), and the equation of the circle is x + y = 34. Clearly the largest possible rectangle must lie in the first quadrant, and if we let ( x, y ) be the upper-right corner of the rectangle, then the √ √ 2 2 2 2 x + y x + y area of the rectangle is ( x + 3)( y + 3) = 9 + 6( x + y ) + xy ≤ 9 + 12 + = 26 + 6 17, where 2 2 √ equality holds if and only if x = y = 17.