MathDB

Let

a

be a fixed positive integer. For a given positive integer

n

, consider the following assertion.
In an infinite two-dimensional grid of squares,

n

different cells are colored black. Let

K

denote the number of

a

a

squares in the grid containing exactly

a

black cells. Then over all possible choices of the

n

black cells, the maximum possible

K

a(n+1-a)

.
Prove that there exists a positive integer

N

such that for all

n\ge N

, this assertion is true.
(link is http://www.imojp.org/challenge/old/jmo25mq.html for anyone who wants to correct my translation)

Problem Statement