Problem

Source: Kazakhstan National Olympiad 2024 (9 grade), P2

Tags: combinatorics



Given an integer $n>1$. The board $n\times n$ is colored white and black in a chess-like manner. We call any non-empty set of different cells of the board as a figure. We call figures $F_1$ and $F_2$ similar, if $F_1$ can be obtained from $F_2$ by a rotation with respect to the center of the board by an angle multiple of $90^\circ$ and a parallel transfer. (Any figure is similar to itself.) We call a figure $F$ connected if for any cells $a,b\in F$ there is a sequence of cells $c_1,\ldots,c_m\in F$ such that $c_1 = a$, $c_m = b$, and also $c_i$ and $c_{i+1}$ have a common side for each $1\le i\le m - 1$. Find the largest possible value of $k$ such that for any connected figure $F$ consisting of $k$ cells, there are figures $F_1,F_2$ similar to $F$ such that $F_1$ has more white cells than black cells and $F_2$ has more black cells than white cells in it.