Question 49
Let $n \geq 3$ be an integer and let $1 \lt a_{1} \leq a_{2} \leq a_{3} \leq \cdots \leq a_{n}$ be $n$ real numbers such that $a_{1} + a_{2} + a_{3} + \cdots + a_{n} = 2n$. Prove that
$$
a_{1} a_{2} \cdots a_{n-1} + a_{1} a_{2} \cdots a_{n-2} + \cdots + a_{1} a_{2} + a_{1} + 2 \leq a_{1} a_{2} \cdots a_{n}
$$
$$
a_{1} a_{2} \cdots a_{n-1} + a_{1} a_{2} \cdots a_{n-2} + \cdots + a_{1} a_{2} + a_{1} + 2 \leq a_{1} a_{2} \cdots a_{n}
$$