HMIC 2015 Problem 1

Let $S$ be the set of positive integers $n$ such that the inequality

$$\phi(n) \cdot \tau(n) \geq \sqrt{\frac{n^{3}}{3}}$$

holds, where $\phi(n)$ is the number of positive integers $k \leq n$ that are relatively prime to $n$, and $\tau(n)$ is the number of positive divisors of $n$. Prove that $S$ is finite.