HMMT 十一月 2011 · 团队赛 · 第 8 题

HMMT November 2011 — Team Round — Problem 8

[ 4 ] Let G, A , A , A , A , B , B , B , B , B be ten points on a circle such that GA A A A is a regular 1 2 3 4 1 2 3 4 5 1 2 3 4 pentagon and GB B B B B is a regular hexagon, and B lies on minor arc GA . Let B B intersect 1 2 3 4 5 1 1 5 3 B A at G , and let B A intersect GB at G . Determine the degree measure of ∠ GG G . 1 2 1 5 3 3 2 2 1 ′

解析

[ 4 ] Let G, A , A , A , A , B , B , B , B , B be ten points on a circle such that GA A A A is a regular 1 2 3 4 1 2 3 4 5 1 2 3 4 pentagon and GB B B B B is a regular hexagon, and B lies on minor arc GA . Let B B intersect 1 2 3 4 5 1 1 5 3 B A at G , and let B A intersect GB at G . Determine the degree measure of ∠ GG G . 1 2 1 5 3 3 2 2 1 ◦ Answer: 12 B 1 A 2 G 1 G G 2 B 3 A 3 B 5 Note that GB is a diameter of the circle. As a result, A , A are symmetric with respect to GB , as 3 2 3 3 are B , B . Therefore, B A and B A intersect along line GB , so in fact, B , A , G , G are collinear. 1 5 1 2 5 3 3 1 2 1 2 We now have ◦ ◦ ̂ ̂ GB − B A 60 − 36 1 3 2 ◦ ∠ GG G = ∠ GG B = = = 12 . 2 1 2 1 2 2 Team Round ′