HMMT 十一月 2011 · 团队赛 · 第 8 题
HMMT November 2011 — Team Round — Problem 8
题目详情
英文原题
- [ 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
英文解析
- [ 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
1B
2A
1G
2GG
3B
3A
5B
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
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