MathDB

For a given positive integer

n

, let

A_n

be the set of all points

(x,y)

in the coordinate plane with

x,y \in \{0,1,...,n\}

. A point

(i, j)

is called internal if

0 < i, j < n

. A real function

f

, defined on

A_n

, is called good if it has the following property: For every internal point

x

, the value of

f(x)

is the arithmetic mean of its values on the four neighboring points (i.e. the points at the distance

1

from

x

). Prove that if

f

and

g

are good functions that coincide at the non-internal points of

A_n

, then

f \equiv g

Problem Statement