Question 45
There is a frog jumping on a $2k \times 2k$ chessboard, composed of unit squares. The frog's jumps are of length $\sqrt{1+k^{2}}$ and they carry the frog from the center of a square to the center of another square. Some $m$ squares of the board are marked with an $x$, and all the squares into which the frog can jump from an $x$'d square (whether they carry an $x$ or not) are marked with an $\circ$. There are $n$ $\circ$'d squares. Prove that $n \geqslant m$.