5. On the coordinate plane, squares are considered, all vertices of which have natural coordinates, and the center is located at the point $(55 ; 25)$. Find the number of such squares.
Answer: 600.

Solution. Draw through the given point $(55 ; 25)$ vertical and horizontal lines $(x=55$ and $y=25)$. There are two possible cases.

a) The vertices of the square lie on these lines (and its diagonals are parallel to the coordinate axes). Then the "lower" vertex of the square can be located in 24 ways: $(55 ; 1),(55 ; 2), \ldots,(55 ; 24)$ (the position of the other vertices is uniquely determined in this case).

b) The vertices of the square do not lie on the specified lines. This means that the vertices lie one in each of the four regions into which the lines $x=55$ and $y=25$ divide the plane. Consider the "lower left" vertex (its location uniquely determines the other vertices). For the coordinates of all vertices of the square to be non-negative, it is necessary and sufficient for this vertex to fall into the square $31 \leqslant x \leqslant 54,1 \leqslant y \leqslant 24$. We get $24^{2}$ ways.

The total number of ways is $24^{2}+24=24 \cdot 25=600$.