ILL 1978 Problem 35

A sequence $(a_n)_0^N$ of real numbers is called concave if $2a_n\ge a_{n-1} + a_{n+1}$ for all integers $n, 1 \le n \le N - 1$ . $(a)$ Prove that there exists a constant $C >0$ such that $$\left( \sum_{n=0}^{N}a_n\right)^2\ge C(N - 1) \sum_{n=0}^{N}a_n^2\:\:\:\:\:(1)$$ for all concave positive sequences $(a_n)^N_0$ $(b)$ Prove that $(1)$ holds with $C = \frac{3}{4}$ and that this constant is best possible.