2001 IMO Shortlist Problems/A1
Revision as of 22:12, 17 July 2009 by Brut3Forc3 (talk | contribs)
Problem
Let denote the set of all ordered triples
of nonnegative integers. Find all functions
such that

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.