Pretty Prime Problem

This is one of the problems in our school math contest; I found it exotic, so I though I would share. Note, I have both the solution and the process by which one arrives to the solution, so this is not homework; I can send it to anyone who requests it. I'm not posting the process here (yet) because it's in Czech and I still have to translate it. I expect to be able to post it by the end of today. Here's the problem:

For a given prime number [math]p[/math], calculate the number of (all) ordered trios [math](a, b, c)[/math] from the set [math]\left\{1, 2, ..., 2p^2\right\}[/math] that satisfy the relation

[math] \frac{LCM(a, c) + LCM(b, c)}{a+b}=\frac{p^2+1}{p^2+2} \cdot c[/math]

where [math]LCM(x, y)[/math] is the Least Common Multiple of [math]x, y[/math].

The solution is [math]2p(p-1)[/math].

PS.: The process I'll post isn't a rigorous proof, it's more of a guide with explanations.