HMMT 十一月 2022 · GEN 赛 · 第 2 题
HMMT November 2022 — GEN Round — Problem 2
题目详情
- How many ways are there to arrange the numbers 1 , 2 , 3 , 4 , 5 , 6 on the vertices of a regular hexagon such that exactly 3 of the numbers are larger than both of their neighbors? Rotations and reflections are considered the same.
解析
- How many ways are there to arrange the numbers 1 , 2 , 3 , 4 , 5 , 6 on the vertices of a regular hexagon such that exactly 3 of the numbers are larger than both of their neighbors? Rotations and reflections are considered the same. Proposed by: Ankit Bisain Answer: 8 Solution: Label the vertices of the hexagon abcdef . The numbers that are larger than both of their neighbors can’t be adjacent, so assume (by rotation) that these numbers take up slots ace . We also have that 6 and 5 cannot be smaller than both of their neighbors, so assume (by rotation and reflection) that a = 6 and c = 5. Now, we need to insert 1 , 2 , 3 , 4 into b, d, e, f such that e is the largest among d, e, f . There are 4 ways to choose b , which uniquely determines e , and 2 ways to choose the ordering of d, f , giving 4 · 2 = 8 total ways.