CONO-SUR 2013 Problem 5

Let $d(k)$ be the number of positive divisors of integer $k$ . A number $n$ is called balanced if $d(n-1) \leq d(n) \leq d(n+1)$ or $d(n-1) \geq d(n) \geq d(n+1)$ . Show that there are infinitely many balanced numbers.