MathDB

Let

m

and

n

be positive integers. Fuming Zeng gives James a rectangle, such that

m-1

lines are drawn parallel to one pair of sides and

n-1

lines are drawn parallel to the other pair of sides (with each line distinct and intersecting the interior of the rectangle), thus dividing the rectangle into an

m\times n

grid of smaller rectangles. Fuming Zeng chooses

m+n-1

of the

mn

smaller rectangles and then tells James the area of each of the smaller rectangles. Of the

\dbinom{mn}{m+n-1}

possible combinations of rectangles and their areas Fuming Zeng could have given, let

C_{m,n}

be the number of combinations which would allow James to determine the area of the whole rectangle. Given that

A=\sum_{m=1}^\infty \sum_{n=1}^\infty \frac{C_{m,n}\binom{m+n}{m}}{(m+n)^{m+n}},

then find the greatest integer less than

1000A

.
Proposed by James Lin

Problem Statement