Answer is $(a, b, c) = (20, 3, 1)$?

The only solution is $\boxed{(20,3,1)}$, which obviously works. Now we show it's the only solution. First note that the LHS is less than $a + 1$ (and greater than $a$), so $a = 20$. This implies that $b + \frac 1c = 4$. Since $b,4$ are integers, $\frac 1c$ is also an integer, so $c = 1$ and $b = 3$.