ELMO SL 2014 N9

Let $d$ be a positive integer and let $\varepsilon$ be any positive real. Prove that for all sufficiently large primes $p$ with $\gcd(p-1,d) \neq 1$ , there exists an positive integer less than $p^r$ which is not a $d$ th power modulo $p$ , where $r$ is defined by $$\log r = \varepsilon - \frac{1}{\gcd(d,p-1)}.$$Proposed by Shashwat Kishore