HMMT 十一月 2022 · THM 赛 · 第 6 题

HMMT November 2022 — THM Round — Problem 6

A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends in a draw. Given that both players play optimally, find all possible areas of the convex polygon formed by Alice’s points at the end of the game.

解析

A regular octagon is inscribed in a circle of radius 2. Alice and Bob play a game in which they take turns claiming vertices of the octagon, with Alice going first. A player wins as soon as they have selected three points that form a right angle. If all points are selected without either player winning, the game ends in a draw. Given that both players play optimally, find all possible areas of the convex polygon formed by Alice’s points at the end of the game. Proposed by: Rishabh Das √ √ Answer: 2 2 , 4 + 2 2 Solution: A player ends up with a right angle iff they own two diametrically opposed vertices. Under optimal play, the game ends in a draw: on each of Bob’s turns he is forced to choose the diametrically opposed vertex of Alice’s most recent choice, making it impossible for either player to win. At the end, the two possibilities are Alice’s points forming the figure in red or the figure in blue (and rotations of √ these shapes). The area of the red quadrilateral is 3[ △ OAB ] − [ △ OAD ] = 2 2 (this can be computed 1 using the ab sin θ formula for the area of a triangle). The area of the blue quadrilateral can be 2 calculated similarly by decomposing it into four triangles sharing O as a vertex, giving an area of √ 4 + 2 2.