HMMT 十一月 2020 · 冲刺赛 · 第 7 题
HMMT November 2020 — Guts Round — Problem 7
题目详情
- [7] Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism.
解析
- [7] Compute the maximum number of sides of a polygon that is the cross-section of a regular hexagonal prism. Proposed by: Hahn Lheem Answer: 8 Solution: Note that since there are 8 faces to a regular hexagonal prism and a cross-section may only intersect a face once, the upper bound for our answer is 8. ′ ′ ′ ′ ′ ′ Indeed, we can construct a cross-section of the prism with 8 sides. Let ABCDEF and A B C D E F ′ be the two bases of the prism, with A being directly over A . Choose points P and Q on line segments ′ ′ ′ ′ ′ ′ AB and BC , respectively, and choose points P and Q on segments D E and E F , respectively, such ′ ′ ′ that P Q ‖ P Q . Then, the cross-section of the prism from the plane that goes through P , Q , P , and ′ Q forms a polygon with 8 sides.