MathDB

Let

m

n

be positive integers. On an

m\times{n}

checkerboard, divided into

1\times1

squares, we consider all paths that go from upper right vertex to the lower left vertex, travelling exclusively on the grid lines by going down or to the left. We define the area of a path as the number of squares on the checkerboard that are below this path. Let

p

be a prime such that

r_{p}(m)+r_{p}(n)\geq{p}

, where

r_{p}(m)

denotes the remainder when

m

is divided by

p

and

r_{p}(n)

denotes the remainder when

n

is divided by

p

. How many paths have an area that is a multiple of

p

Problem Statement