Problem
COM-B1-M05-P015 Integer Midpoint
#15
★★★☆☆ Level 3 of 5
Five points with integer coordinates are chosen in the plane. Prove that the midpoint of some segment between two chosen points also has integer coordinates.
There are \(4\) parity classes of coordinates.
Each point has parity type \((x\bmod2,y\bmod2)\), only \(4\) types. Among \(5\) points, two have the same type. Then the sums of their \(x\)-coordinates and \(y\)-coordinates are even, so the midpoint has integer coordinates.
Coordinate pigeonhole.