2001 IMO Shortlist Problems/A1
Let denote the set of all ordered triples of nonnegative integers. Find all functions such that
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.