This problem was question A1 on the 2013 Putnam contest. Is there a better way to solve this problem than just using pigeonhole principle? Specifically, is there a group theoretic way to interpret this problem? The problem goes:
"Recall that a regular icosohedron is a convex polyhedron having 12 vertices and 20 faces; the faces are congruent equilateral triangles. On each face of a regular icoshedron is written a nonnegative integer such that the sum of all 20 integers is 39. Show that there are two faces that share a vertex and have the same integer written on them."
Every face touches $3$ vertices, thus every number can appear at most $4$ times (if you don't want vertices with two faces sharing the same number). Now consider the minimal filling of number which would be possible t do this way, using only $0,1,2,3$ and $4$, each of them appearing exactly $4$ times: $$4\cdot(0+1+2+3+4)=40$$ This implies that at least one of the numbers from $0$ to $3$ must appear at least twice.