2001 IMO Shortlist Problems/A1
Problem
Let denote the set of all ordered triples
of nonnegative integers. Find all functions
such that
![$f(p,q,r) = \begin{cases} 0 & \text{if} \; pqr = 0, \\ 1 + \tfrac{1}{6}\{f(p + 1,q - 1,r) + f(p - 1,q + 1,r) & \\ + f(p - 1,q,r + 1) + f(p + 1,q,r - 1) & \\ + f(p,q + 1,r - 1) + f(p,q - 1,r + 1)\} & \text{otherwise.} \end{cases}$](http://latex.artofproblemsolving.com/d/2/d/d2d4e8080418f5bb567971c0ee129fd02c7c7b13.png)
Solution
We can see that for
and
for
satisfies the equation. Suppose there exists another solution
. Let
. Plugging in
we see that
satisfies the relationship
, so that each value of
is equal to 6 points around it with an equal sum
. This implies that for fixed
,
is constant. Furthermore, some values of
are always zero; for example,
by the problem statement, and similarly,
, so
. Thus,
must be identically zero, so
is the only function satisfying this equation.