HMMT 十一月 2024 · 冲刺赛 · 第 7 题

HMMT November 2024 — Guts Round — Problem 7

[7] Let P be a regular 10 -gon in the coordinate plane. Mark computes the number of distinct x -coordinates that vertices of P take. Across all possible placements of P in the plane, compute the sum of all possible answers Mark could get.

解析

[7] Let P be a regular 10-gon in the coordinate plane. Mark computes the number of distinct x - coordinates that vertices of P take. Across all possible placements of P in the plane, compute the sum of all possible answers Mark could get. Proposed by: Srinivas Arun Answer: 21 Solution: 10 distinct coordinates 5 distinct coordinates 6 distinct coordinates Let P have vertices P P . . . P . If no two vertices have the same x -coordinate, then Mark gets 10 . 1 2 10 Otherwise, two vertices P and P have the same x -coordinate. Then P and P also have the i j k i + j − k same x -coordinate (indices taken modulo 10), as P P ∥ P P . i j k i + j − k If i + j is odd, the ten vertices of P pair up into 5 pairs of the form ( P , P ), so Mark gets 5. If k i + j − k i + j i + j i + j is even, then the vertices P and P do not pair up, and the remaining 8 vertices form 4 +5 2 2 pairs, so Mark gets 6. Thus, the answer is 10 + 5 + 6 = 21 .