IMO Shortlist 1976 problem 2
Dodao/la:
arhiva2. travnja 2012. Let
![a_0, a_1, \ldots, a_n, a_{n+1}](/media/m/3/3/7/3371f6efba3bf3a2d970b2f8f9a258ba.png)
be a sequence of real numbers satisfying the following conditions:
![|a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).](/media/m/4/8/7/4875cacb0603ca31de069bd48759a01b.png)
Prove that
%V0
Let $a_0, a_1, \ldots, a_n, a_{n+1}$ be a sequence of real numbers satisfying the following conditions:
$$a_0 = a_{n+1 }= 0,$$ $$|a_{k-1} - 2a_k + a_{k+1}| \leq 1 \quad (k = 1, 2,\ldots , n).$$
Prove that $|a_k| \leq \frac{k(n+1-k)}{2} \quad (k = 0, 1,\ldots ,n + 1).$
Izvor: Međunarodna matematička olimpijada, shortlist 1976