CONO-SUR 1991 Problem 6

Given a positive integrer number $n$ ( $n\ne 0$ ), let $f(n)$ be the average of all the positive divisors of $n$ . For example, $f(3)=\frac{1+3}{2}=2$ , and $f(12)=\frac{1+2+3+4+6+12}{6}=\frac{14}{3}$ .

a Prove that $\frac{n+1}{2} \ge f(n)\ge \sqrt{n}$ .

b Find all $n$ such that $f(n)=\frac{91}{9}$ .