HMMT 二月 2011 · 冲刺赛 · 第 6 题
HMMT February 2011 — Guts Round — Problem 6
题目详情
英文原题
- [ 5 ] Square ABCD is inscribed in circle ω with radius 10. Four additional squares are drawn inside ωbut outside ABCD such that the lengths of their diagonals are as large as possible. A sixth square isdrawn by connecting the centers of the four aforementioned small squares. Find the area of the sixthsquare.
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解析
英文解析
- [ 5 ] Square ABCD is inscribed in circle ω with radius 10. Four additional squares are drawn inside ωbut outside ABCD such that the lengths of their diagonals are as large as possible. A sixth square isdrawn by connecting the centers of the four aforementioned small squares. Find the area of the sixthsquare.
Answer: 144
Let DEGF denote the small square that shares a side with AB , where D and E lie on AB . Let Odenote the center of ω , K denote the midpoint of F G , and H denote the center of DEGF . The areaof the sixth square is 2 · OH .2
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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
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x = 2. Thus, we have OH = 6 2 and 2 · OH = 144.2
Guts Round
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