MathDB

Let

m

and

n

be positive integers. Let

S

be the set of integer points

(x,y)

with

1\leq x\leq 2m

and

1\leq y\leq 2n

. A configuration of

mn

rectangles is called happy if each point in

S

is a vertex of exactly one rectangle, and all rectangles have sides parallel to the coordinate axes. Prove that the number of happy configurations is odd.
Proposed by Serena An and Claire Zhang

Problem Statement