This is again some homework translated (hopefully not too badly) from my book
The graph $K_{n}$ is colored using $n$ different colors, in a way that each color is used at least once. Prove that there exists a triangle with its edges colored in different colors.
On
Consider the following simpler case (and use it as a lemma for your problem). Suppose you have a cycle $(v_1,\ldots,v_k)$ where each edge is coloured differently (so with $k$ colors out of the $n$). Now draw "chords" from $v_1$ to $v_3,\ldots,v_{k-1}$, and color these edges using any of the $n$ colors. This gives you $n-2$ triangles. Show that one of these triangles must have all edges colored differently.
Find the smallest cycle $C$ in your graph with all edges distinct colors. We know $C$ must exist, because there are $n$ colors (I'll let you think about why). If $C \neq C_3$, consider a chord of $C$. The chord divides $C$ into two smaller cycles. One of these cycles should provide a contradiction to the minimality of $C$.