HMMT 十一月 2011 · THM 赛 · 第 10 题

HMMT November 2011 — THM Round — Problem 10

[ 7 ] Let Ω be a circle of radius 8 centered at point O , and let M be a point on Ω. Let S be the set of points P such that P is contained within Ω, or such that there exists some rectangle ABCD containing P whose center is on Ω with AB = 4, BC = 5, and BC ‖ OM . Find the area of S .

解析

[ 7 ] Let Ω be a circle of radius 8 centered at point O , and let M be a point on Ω. Let S be the set of points P such that P is contained within Ω, or such that there exists some rectangle ABCD containing P whose center is on Ω with AB = 4, BC = 5, and BC ‖ OM . Find the area of S . Theme Round Answer: 164 + 64 π We wish to consider the union of all rectangles ABCD with AB = 4, BC = 5, and BC ‖ OM , with center X on Ω. Consider translating rectangle ABCD along the radius XO to a ′ ′ ′ ′ rectangle A B C D now centered at O . It is now clear that that every point inside ABCD is a translate ′ ′ ′ ′ of a point in A B C D , and furthermore, any rectangle ABCD translates along the appropriate radius ′ ′ ′ ′ to the same rectangle A B C D . We see that the boundary of this region can be constructed by constructing a quarter-circle at each vertex, then connecting these quarter-circles with tangents to form four rectangular regions. Now, splitting our region in to four quarter circles and five rectangles, we compute the desired area to be 1 2 4 · (8) π + 2(4 · 8) + 2(5 · 8) + (4 · 5) = 164 + 64 π. 4 O M Theme Round