Let G be a connected 5-regular embedded planar graph, in which every face has the same degree. Find the number of faces of G.
So let there be $e$ edges, $f$ faces with each degree $d$ and $v$ vertices each of degree $5$ so using the handshaking lemma for vertices and faces we get equations:
$f \cdot d = 2e = 5v$
Euler's formula states $v + f - e = 2$
$$\implies fd/5 + f - fd/2 = 2$$
$$\implies 10f - 3df = 20$$
Also euler's $e \le 3v - 6$
But now I am stuck, any ideas?