Problem

Source: Iran TST 2005

Tags: function, induction, modular arithmetic, analytic geometry, combinatorics proposed, combinatorics



Suppose $S= \{1,2,\dots,n\}$ and $n \geq 3$. There is $f:S^k \longmapsto S$ that if $a,b \in S^k$ and $a$ and $b$ differ in all of elements then $f(a) \neq f(b)$. Prove that $f$ is a function of one of its elements.