Question 171
Fix a natural number $n \geqslant 2$. Determine all real numbers $x \geqslant -1$ such that for all real numbers $a_{1}, \ldots, a_{n} \geqslant 1$ we have
$$
\frac{a_{1}+x}{2} \times \cdots \times \frac{a_{n}+x}{2} \leqslant \frac{a_{1} a_{2} \cdots a_{n}+x}{2} .
$$
$$
\frac{a_{1}+x}{2} \times \cdots \times \frac{a_{n}+x}{2} \leqslant \frac{a_{1} a_{2} \cdots a_{n}+x}{2} .
$$