HMMT 十一月 2022 · GEN 赛 · 第 3 题

HMMT November 2022 — GEN Round — Problem 3

Let ABCD be a rectangle with AB = 8 and AD = 20. Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to AB and AD , and the other passing through both C and D . What is the area inside the rectangle and outside of both circles? 2 3

解析

Let ABCD be a rectangle with AB = 8 and AD = 20. Two circles of radius 5 are drawn with centers in the interior of the rectangle - one tangent to AB and AD , and the other passing through both C and D . What is the area inside the rectangle and outside of both circles? Proposed by: Ankit Bisain Answer: 112 − 25 π Solution: Let O and O be the centers of the circles, and let M be the midpoint of CD . We can see that 1 2 ′ △ O M C and △ O M D are both 3-4-5 right triangles. Now let C be the intersection of circle O and 2 2 2 ′ BC (that isn’t C ), and let D be the intersection of circle O and AD (that isn’t D ). We know that 2 ′ ′ ′ AD = BC = 14 because BC = 2 O M = 6. 2 ′ ′ ′ ′ All of the area of ABCD that lies outside circle O must lie within rectangle ABC D because C CDD 2 ′ ′ is completely covered by circle O . Now, notice that the area of circle O that lies inside ABC D is 2 2 ′ ′ ′ ′ the same as the area of circle O that lies outside ABC D . Thus, the total area of ABC D that is 1 covered by either of the two circles is exactly the area of one of the circles, 25 π . The remaining area is 8 · 14 − 25 π , which is our answer. 2 3