Question 16
Given an odd integer $n \ge 3$ and a regular $n$-gon with vertex set $V$, let $\mathcal{P}$ denote the set of all regular polygons with vertices in $V$. For example, if $n = 15$, then $\mathcal{P}$ includes 1 regular 15-gon, 3 regular pentagons, and 5 equilateral triangles.
Two players, Alice and Bob, play the following game. Initially, all points in $V$ are uncolored. Alice moves first, and they take turns coloring an uncolored point in $V$: Alice colors red, and Bob colors blue. The game ends when all points are colored. A polygon in $\mathcal{P}$ is called **red-dominated** if it has more red vertices than blue ones.
Find the largest integer $k$ such that, no matter how Bob plays, Alice can guarantee at least $k$ red-dominated polygons in $\mathcal{P}$ at the end.
Two players, Alice and Bob, play the following game. Initially, all points in $V$ are uncolored. Alice moves first, and they take turns coloring an uncolored point in $V$: Alice colors red, and Bob colors blue. The game ends when all points are colored. A polygon in $\mathcal{P}$ is called **red-dominated** if it has more red vertices than blue ones.
Find the largest integer $k$ such that, no matter how Bob plays, Alice can guarantee at least $k$ red-dominated polygons in $\mathcal{P}$ at the end.