symmetric 2 variable function

Suppose $S = \{1,2,\dots,n\}$ and $f: S \times S \to R$ satisfies the following: $$f(i,j) + f(j,i) = 0 \forall i,j \in S$$ Now for any two number $i,j$, we say that $i$ is superior to $j$ if there exists a $k$ (not necessarily distinct from $i,j$) such that $f(i,k) + f(k,j) \geq 0$. Show that there exists a number $x \in S$ such that $x$ is superior to all elements of $S$.

Graph with degree 3

A graph where each vertex has degree 3 has all of its edges colored with red, green, or blue. How many colorings are there such that every 3 edges that meet in a vertex are either of the same color or have 3 different colors?

Gas stations on a circular track

On a circular track, there are several gas stations. The total amount of gas in all these stations are just enough for a car to complete one lap. Prove that, starting with an empty car, one can choose an initial position such that he can complete the lap by subsequently filling gas in these stations.