HMMT 十一月 2016 · THM 赛 · 第 8 题
HMMT November 2016 — THM Round — Problem 8
题目详情
英文原题
- Let P P . . . P be a convex octagon. An integer i is chosen uniformly at random from 1 to 7, inclusive.
1 2 8
For each vertex of the octagon, the line between that vertex and the vertex i vertices to the right ispainted red. What is the expected number times two red lines intersect at a point that is not one ofthe vertices, given that no three diagonals are concurrent?
解析
英文解析
- Let P P . . . P be a convex octagon. An integer i is chosen uniformly at random from 1 to 7, inclusive.
1 2 8
For each vertex of the octagon, the line between that vertex and the vertex i vertices to the right ispainted red. What is the expected number times two red lines intersect at a point that is not one ofthe vertices, given that no three diagonals are concurrent?
Proposed by: Eshaan Nichani
Answer:54
If i = 1 or i = 7, there are 0 intersections. If i = 2 or i = 6 there are 8. If i = 3 or i = 5 there are 7
16 intersections. When i = 4 there are 6 intersections (since the only lines drawn are the four longdiagonals).
8+16+6+16+854
Thus the final answer is =
77