Bogdanov I.I.

The distance between two cells on an infinite chessboard is defined as the minimum number of moves in the path of a king between these cells. On the board, three cells are marked, the pairwise distances between which are 100. How many cells exist such that the distances from them to all three marked cells are 50?
Consider two arbitrary cells $A$ and $B$. Let the difference in the abscissas of their centers be $x \geq 0$, and the difference in the ordinates be $y \geq 0$. Then the distance $\rho(A, B)$ between these cells is $\max \{x, y\}$.

Let cells $A, B, C$ be marked. Then for each pair of cells, there exists a coordinate in which they differ by exactly 100. For two pairs of cells, this will be the same coordinate; for definiteness, let these be the pairs $(A, B)$ and $(A, C)$, differing horizontally. Then the abscissas of points $B$ and $C$ either differ by 200 or coincide. The first case is impossible, since $\rho(B, C)=100$. Therefore, their abscissas coincide, and their ordinates differ by 100. We can assume that the cells have coordinates $B(0,0), C(0,100), A(100, x)(0 \leq x \leq 100)$. Consider a point $X$ that is 50 units away from points $A, B$, and $C$. Its abscissa must be 50, otherwise $\rho(X, B)>50$ or $\rho(X, A)>50$. Similarly, the ordinate of $X$ is 50. Therefore, the coordinates of $X$ are $(50,50)$, and this cell fits. Thus, the sought cell is exactly one.

## Answer

One cell