, 2015

Show that for every positive real number $c>0$ there is a positive integer $n$ such that $\varphi\big(\sigma(n)\big)>cn$. (For an arbitrary postive integer $k$, $\varphi(k)$ denotes the number of positive integers not exceeding $k$ that are co-prime with $k$. $\sigma(k)$ is the sum of positive divisors of $k$.)
Proposed by: Barnabás Szabó, Budapest
(5 pont)