IMO Shortlist 1975 problem 4
Dodao/la:
arhiva2. travnja 2012. Let
![a_1, a_2, \ldots , a_n, \ldots](/media/m/a/f/3/af350fcd61b80024d0cacf174687babe.png)
be a sequence of real numbers such that
![0 \leq a_n \leq 1](/media/m/1/a/0/1a013de2c4c91b7b02e3c8c60e1d2ee4.png)
and
![a_n - 2a_{n+1} + a_{n+2} \geq 0](/media/m/2/6/7/26727d1f0d3465457708585a557635e8.png)
for
![n = 1, 2, 3, \ldots](/media/m/d/5/9/d59e410f8c37653f899a4e74178138f0.png)
. Prove that
%V0
Let $a_1, a_2, \ldots , a_n, \ldots$ be a sequence of real numbers such that $0 \leq a_n \leq 1$ and $a_n - 2a_{n+1} + a_{n+2} \geq 0$ for $n = 1, 2, 3, \ldots$. Prove that
$$0 \leq (n + 1)(a_n - a_{n+1}) \leq 2 \qquad \text{ for } n = 1, 2, 3, \ldots$$
Izvor: Međunarodna matematička olimpijada, shortlist 1975