VMO 2016 Problem 7

a) Prove that if $n$ is an odd perfect number then $n$ has the following form $$n=p^sm^2$$ where $p$ is prime has form $4k+1$ , $s$ is positive integers has form $4h+1$ , and $m\in\mathbb{Z}^+$ , $m$ is not divisible by $p$ . b) Find all $n\in\mathbb{Z}^+$ , $n>1$ such that $n-1$ and $\frac{n(n+1)}{2}$ is perfect number