Question 27
Let $a_{1}, a_{2}, \ldots, a_{9}$ be any non-negative numbers such that $a_{1}=a_{9}=0$ and at least one of the numbers is non-zero. Prove that for some $i$, $2 \leq i \leq 8$, the inequality $a_{i-1}+a_{i+1}\lt 2 a_{i}$ holds. Will the statement remain true if we change the number $2$ in the last inequality to $1.9$?