Find the sum of all positive integers $n,$ where the inequality $\sqrt{a + \sqrt{b + \sqrt{c}}} \ge \sqrt[n]{abc}$ holds for all nonnegative real numbers $a,$ $b,$ and $c.$
I tried squaring both sides but I'm not sure how to continue and apply the am-gm inequality.
Let $$(a,b,c)=(t,12t^2,576t^4)$$ $$\implies abc=6912t^7$$
The given inequality has to hold in this case as well.
$$\sqrt{a+\sqrt{b+\sqrt{c}}}=\sqrt{7t} \ge \sqrt[n]{6912t^7}$$ $$\Longleftrightarrow t^{\frac{1}{2}-\frac{7}{n}}\ge \frac{\sqrt[n]{6912}}{\sqrt{7}}$$ Regardless of $n$, we can set $t$ to an arbitrarily large or small positive real number so that this won't hold.
Therefore, no such $n$ exists.
