IMO Shortlist 1988 problem 24
Dodao/la:
arhiva2. travnja 2012. Let
![\{a_k\}^{\infty}_1](/media/m/3/2/a/32acce62e0b242e570639eb24c6593ac.png)
be a sequence of non-negative real numbers such that:
and
![\sum^k_{j = 1} a_j \leq 1](/media/m/9/c/8/9c89386bc42cd104c417df19d9b8c14a.png)
for all
![k = 1,2, \ldots](/media/m/9/8/8/9885a0f638c3fad65715ec70dbb021d3.png)
. Prove that:
for all
![k = 1,2, \ldots](/media/m/9/8/8/9885a0f638c3fad65715ec70dbb021d3.png)
.
%V0
Let $\{a_k\}^{\infty}_1$ be a sequence of non-negative real numbers such that:
$$a_k - 2 \cdot a_{k + 1} + a_{k + 2} \geq 0$$
and $\sum^k_{j = 1} a_j \leq 1$ for all $k = 1,2, \ldots$. Prove that:
$$0 \leq (a_{k} - a_{k + 1}) < \frac {2}{k^2}$$
for all $k = 1,2, \ldots$.
Izvor: Međunarodna matematička olimpijada, shortlist 1988