Finding the couples $(n,m)$ which make this inequality true

Question

Let $\Delta \subset \mathbb{R}^3$ be a convex polyhedron with $V$ vertices, $E$ edged and $F$ faces. Assume that each face of $\Delta$ is an $n$-gon (where $n \ge 3$) and at each vertex of $\Delta$ meet $m$ edges (where $m \ge 3$). Prove that:

1. $2E/V = m$ and $2E/F = n$;
2. $V/2E −1/2 + F/2E = 1/E > 0$.

Find all couples $(n,m)$ of integers such that $n \ge 3$, $m \ge 3$ and $1/m + 1/ n −1/2 > 0$.

So I’ve done parts 1 and 2, but I’m having a little trouble with part 3 where I’m assuming we need to use parts 1 and 2 to solve this inequality. I tried doing $2E/V > 3$ and $2E/ F > 3$ and putting them in (2) but I’m not getting any concrete answers. Not sure what I’m doing wrong?

Answer 1

I do not think you need part I and II.

If $x > y > 0$ then $\frac{1}{x}< \frac{1}{y}$.

So you can see that only (trial and error) $(3,3), (3,4), (3,5)$ and $(4,3), (5,3)$ work. If $m$ or $n$ is bigger, then $\frac{1}{n}+\frac{1}{m}-\frac{1}{2} \leq 0$.