Question 2
(BUL 3) Let $a_0,a_1,\ldots,a_n,a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
$$a_0=a_{n+1}=0,\qquad |a_{k-1}-2a_k+a_{k+1}|\le1\quad(k=1,2,\ldots,n).$$
Prove that $|a_k|\le\frac{k(n+1-k)}2\quad(k=0,1,\ldots,n+1)$.
$$a_0=a_{n+1}=0,\qquad |a_{k-1}-2a_k+a_{k+1}|\le1\quad(k=1,2,\ldots,n).$$
Prove that $|a_k|\le\frac{k(n+1-k)}2\quad(k=0,1,\ldots,n+1)$.