There are really lot of problems on AM-GM inequality because of its elementary nature and powerful applications.
What I want is a collection of questions/problems which look very complex but get solved swiftly and powerfully through use of AM-GM Inequality.
Edit:I see that 'interesting' is subjective hence to make this question more specific please provide examples which seem interesting or remarkable to you and don't think about what's interesting to me.
One cute application is to the well-known lemma:
Lemma: Let $\{x_n\}$ be a sequence of positive real numbers. Then $$\prod_{n=1}^{\infty} (1+x_n)$$ converges if and only if $$\sum_{n=1}^{\infty} x_n$$ converges.
Proof: One direction is clear, as $\prod_{n=1}^{N} (1+x_n) > \sum_{n=1}^{N} x_n$. For the other direction, use the AM-GM inequality. $$\sqrt[N]{\prod_{n=1}^{N} (1+x_n)}\le \frac{\sum_{n=1}^{N} (1+x_n)}{N} \implies \prod_{n=1}^{N} (1+x_n) \le \left(1 + \frac{x_1 + \cdots + x_N}{N} \right)^N$$ where the right hand side converges to $\exp{\sum_{n=1}^{\infty} x_n}$.
The usual way of proving this lemma is using $1+x\le e^x$ to get that upper bound. For some reason, I find it easy to forget this inequality, so AM-GM is a good fallback.