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2015 HMIC #1: Multiplicative number theory

Source:

May 11, 2015
HMICnumber theory

Problem Statement

Let SS be the set of positive integers nn such that the inequality ϕ(n)⋅τ(n)≥n33 \phi(n) \cdot \tau(n) \geq \sqrt{\frac{n^3}{3}} holds, where ϕ(n)\phi(n) is the number of positive integers k≤nk \le n that are relatively prime to nn, and τ(n)\tau(n) is the number of positive divisors of nn. Prove that SS is finite.
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