HMMT 十一月 2023 · THM 赛 · 第 4 题

HMMT November 2023 — THM Round — Problem 4

Let LOV ER be a convex pentagon such that LOV E is a rectangle. Given that OV = 20 and LO = V E = RE = RL = 23 , compute the radius of the circle passing through R , O , and V . 3 n − 1989

解析

Let LOV ER be a convex pentagon such that LOV E is a rectangle. Given that OV = 20 and LO = V E = RE = RL = 23, compute the radius of the circle passing through R , O , and V . Proposed by: Pitchayut Saengrungkongka Answer: 23 Solution: R L E X O V Let X be the point such that RXOL is a rhombus. Note that line RX defines a line of symmetry on the pentagon LOV ER . Then by symmetry RXV E is also a rhombus, so RX = OX = V X = 23. This makes X the center of the circle, and the radius is 23. 3 n − 1989