HMMT 十一月 2020 · 团队赛 · 第 3 题

HMMT November 2020 — Team Round — Problem 3

[30] Let A be the area of the largest semicircle that can be inscribed in a quarter-circle of radius 1. 120 A Compute . π

解析

[30] Let A be the area of the largest semicircle that can be inscribed in a quarter-circle of radius 1. 120 A Compute . π Proposed by: Akash Das Answer: 20 Solution: X P M Y O N The optimal configuration is when the two ends X and Y of the semicircle lie on the arc of the quarter circle. Let O and P be the centers of the quarter circle and semicircle, respectively. Also, let M and N be the points where the semicircle is tangent to the radii of the quartercircle. √ Let r be the radius of the semicircle. Since P M = P N , P M ON is a square and OP = 2 r . By √ 2 2 the Pythagorean theorem on triangle OP X , 1 = 2 r + r , so r = 1 / 3. The area of the semicircle is π 1 π therefore = . 2 3 6