HMMT February

Answer: 36 Solution: Consider such a polyhedron with $V$ vertices, $E$ edges, and $F=n$ faces. By Euler's formula we have $V+F=E+2$. Next, note that the number of pairs of incident faces and edges is both $2E$ and $3F$, so $2E=3F$. Now, since our polyhedron is convex, the sum of the degree measures at each vertex is strictly less than $360=36 \cdot 10$. As all angle measures of the faces of our polyhedron are divisible by 36, the maximum degree measure at a given vertex is $36 \cdot 9=324$. On the other hand, the total degree measure at all vertices is the total degree measure over all faces, which is $180F$. Thus we have $180F \leq 324V$, or $10F \leq 18V$. Putting our three conditions together, we have $$10F \leq 18V=18(E+2-F)=9(2E)+36-18F=9(3F)+36-18F=9F+36$$ Thus $F \leq 36$. $F=36$ is attainable by taking a 9-gon antiprism with a 9-gon pyramid attached on the top and the bottom. Thus the answer is 36.