Let $(n_1,m_1),(n_2,m_2),. . .,(n_9,m_9)$ be integer lattice points in the plane (ie. $n_i$ and $m_i$ are integers). Show that the midpoint of the line joining some pair of points is also an integer lattice point.
I think that I need to use the pigeonhole principle but I'm not sure how to get to that point.
Consider the pigeonholes $(\text{even}, \text{even}), (\text{even}, \text{odd}), (\text{odd}, \text{odd}), (\text{odd}, \text{even})$, where a point is in a pigeonhole depending of the parity of each coordinate. By pigeonhole principle, we have $2$ points with the same parity of x-coordinate and y-coordinate. (Actually, $5$ points in total is enough.)
When $2$ points have the same parity of x-coordinate and y-coordinate, we can get that their midpoint is a lattice point.