I am not sure if this kind of "question" is welcome on MSE. Here is an olympiad-like problem that I would like to share with you:
Let $a,b,c$ be nonnegative integers. Prove that $$ \frac{(a+b)!^2}{(2a)!(2b)!} + \frac{(b+c)!^2}{(2b)!(2c)!} + \frac{(c+a)!^2}{(2c)!(2a)!} \leq 1 + 2\frac{(a+b)!(b+c)!(c+a)!}{(2a)!(2b)!(2c)!} $$ When does equality hold?
I can provide an answer later if anyone is interested.
Edit. I hoped there would be more interest in this. Do you prefer I give hints before the answer?