1976 IMO 第 2 题

(BUL 3) Let $a_{0}, a_{1}, \ldots, a_{n}, a_{n+1}$ be a sequence of real numbers satisfying the following conditions: $$\begin{aligned} a_{0} & =a_{n+1}=0 \\ \left|a_{k-1}-2 a_{k}+a_{k+1}\right| & \leq 1 \quad(k=1,2, \ldots, n) \end{aligned}$$ Prove that $\left|a_{k}\right| \leq \frac{k(n+1-k)}{2}(k=0,1, \ldots, n+1)$.

(BUL 3) 设 $a_{0}, a_{1}, \ldots, a_{n}, a_{n+1}$ 为满足以下条件的实数序列： \begin{aligned} a_{0} & =a_{n+1}=0 \\ \left|a_{k-1}-2 a_{k}+a_{k+1}\right| & \leq 1 \quad(k=1,2, \ldots, n) \end{对齐}证明$\left|a_{k}\right| \leq \frac{k(n+1-k)}{2}(k=0,1, \ldots, n+1)$。