The question is "Let $(x_i , y_i , z_i ), i = 1, 2, 3, 4, 5, 6, 7, 8, 9,$ be a set of nine distinct points with integer coordinates in $xyz$-space. Show that the midpoint of at least one pair of these points has integer coordinates."
Wouldn't the set ${(3,3,1),(2,2,1),(2,2,2),(2,2,3),(2,2,4),(2,2,5),(2,2,6),(2,2,7), (2,2,8)} $ be a counterexample to the theorem the exercise asks to find?
Thank you!
The midpoint between $(2,2,1)$ and $(2,2,3)$ has integer coordinates. Anyway, the parity - pigeonhole principle argument given earlier is correct.