HMMT 二月 2011 · 冲刺赛 · 第 6 题

HMMT February 2011 — Guts Round — Problem 6

[ 5 ] Square ABCD is inscribed in circle ω with radius 10. Four additional squares are drawn inside ω but outside ABCD such that the lengths of their diagonals are as large as possible. A sixth square is drawn by connecting the centers of the four aforementioned small squares. Find the area of the sixth square. √

解析

[ 5 ] Square ABCD is inscribed in circle ω with radius 10. Four additional squares are drawn inside ω but outside ABCD such that the lengths of their diagonals are as large as possible. A sixth square is drawn by connecting the centers of the four aforementioned small squares. Find the area of the sixth square. Answer: 144 Let DEGF denote the small square that shares a side with AB , where D and E lie on AB . Let O denote the center of ω , K denote the midpoint of F G , and H denote the center of DEGF . The area 2 of the sixth square is 2 · OH . √ 2 2 2 2 2 2 Let KF = x . Since KF + OK = OF , we have x + (2 x + 5 2) = 10 . Solving for x , we get √ √ 2 x = 2. Thus, we have OH = 6 2 and 2 · OH = 144. Guts Round √