VMO 1977 Problem 5

The real numbers $a_0, a_1, ... , a_{n+1}$ satisfy $a_0 = a_{n+1} = 0$ and $|a_{k-1} - 2a_k + a_{k+1}| \le 1$ for $k = 1, 2, ... , n$ . Show that $|a_k| \le \frac{ k(n + 1 - k)}{2}$ for all $k$ .