For what value(s) of $a$ does the inequality $\prod_{i=0}^{a}(n-i) \geq a^{a+1}$ hold?

Question

For what value(s) of $a$ does the inequality $\displaystyle \prod_{i=0}^{a}(n-i) \geq a^{a+1}$ hold? ($n,a \in \mathbb{N}, n > a$)

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Supposedly for $a=\dfrac{n}{2}$:

$n \cdot (n-1) \cdot (n-2) \dots (n-\frac{n}{2}) \geq \left( \dfrac{n}{2}\right)^{\frac{n}{2}+1}$

But I've not thought of a way to show this.

Answer 1

For $a-\frac{n}{2}$ you have $$n \geq \frac{n}{2} \\ n-1 \geq \frac{n}{2}\\ ....\\ n-\frac{n}{2} \geq \frac{n}{2}$$

Multiply them.

In general, the relation can only hold if there is a relation between $n$ and $a$.