We call a place every point $(x, y)$ in the plane for which $x$ and $y$ are positive integers, each less than or equal to 20.

Initially, all 400 places are free. Anna and Balázs take turns placing tokens on the places, with Anna starting. On each of her turns, Anna places a new red token on a still free place in such a way that the distance between no two red tokens is equal to $\sqrt{5}$. On each of his turns, Balázs places a new blue token on a still free place. (The distance from a place occupied by a blue token to any other occupied place can be arbitrary.) The game ends when one of the players cannot make a move.

Determine the largest value of $K$ for which it is true that Anna can definitely place $K$ red tokens, no matter how Balázs plays.
Egri Máté's solution. We prove that $K=100$.

The distance between two points is $\sqrt{5}$ if they are "knight's move" apart, meaning we move two steps in one direction and one step in a perpendicular direction. Therefore, if we color the points in a "chessboard pattern," any two points with a distance of $\sqrt{5}$ will be of different colors. If Anna follows the strategy of placing only on black points (of which there are 200), she can place on at least half of them, i.e., 100 points. Thus, $K \geq 100$.

The $20 \times 20$ board can be divided into 25 $4 \times 4$ boards. We prove that on a $4 \times 4$ board, Balázs can ensure that Anna can place at most 4 times.

If we label the $4 \times 4$ board as follows and Balázs always places symmetrically to Anna's move with respect to the center of the board, then Anna cannot place more than once on the same letter, so she can indeed place at most four times. There are 25 $4 \times 4$ boards, so $K \leq 100$.

Since $K \geq 100$ and $K \leq 100$, it follows that $K=100$.

| $\mathrm{A}$ | $\mathrm{B}$ | $\mathrm{C}$ | $\mathrm{D}$ |
| :---: | :---: | :---: | :---: |
| $\mathrm{C}$ | $\mathrm{D}$ | $\mathrm{A}$ | $\mathrm{B}$ |
| $\mathrm{B}$ | $\mathrm{A}$ | $\mathrm{D}$ | $\mathrm{C}$ |
| $\mathrm{D}$ | $\mathrm{C}$ | $\mathrm{B}$ | $\mathrm{A}$ |