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This might be silly, but here it goes.

Let $P,S>0$ be positive real numbers that satisfy $\frac{S}{n} \ge \sqrt[n]{P}$.

Does there exist a sequence of positive real numbers $a_1,\dots,a_n$ such that $S=\sum a_i,P=\prod a_i$?

Clearly, $\frac{S}{n} \ge \sqrt[n]{P}$ is a necessary condition, due to the AM-GM inequality. But is it sufficient?

For $n=2$, the answer is positive, as can be seen by analysing the discriminant of the associated quadratic equation. (In fact, the solvability criterion for the quadratic, namely- the non-negativity of the discriminant, is equivalent to the AM-GM inequality for the sum and the product).

What about $n>3$?

Answers