HMMT 十一月 2016 · THM 赛 · 第 8 题

HMMT November 2016 — THM Round — Problem 8

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 is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?

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 is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent? Proposed by: Eshaan Nichani 54 Answer: 7 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 16 intersections. When i = 4 there are 6 intersections (since the only lines drawn are the four long diagonals). 54 8+16+6+16+8 Thus the final answer is = 7 7