IMO Shortlist 2002 problem A2
Dodao/la:
arhiva2. travnja 2012. Let
![a_1,a_2,\ldots](/media/m/d/b/c/dbce63436fd54e80a8e6c1712c9f50ca.png)
be an infinite sequence of real numbers, for which there exists a real number
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
with
![0\leq a_i\leq c](/media/m/e/a/e/eaea8f82f461bf9a53322d73174eebf1.png)
for all
![i](/media/m/3/2/d/32d270270062c6863fe475c6a99da9fc.png)
, such that
Prove that
![c\geq1](/media/m/3/c/0/3c0c0937666857a7c9fc9e7e3c2da363.png)
.
%V0
Let $a_1,a_2,\ldots$ be an infinite sequence of real numbers, for which there exists a real number $c$ with $0\leq a_i\leq c$ for all $i$, such that
$$\left|\,a_i-a_j\,\right|\geq{1\over i+j}{\rm \forall}i,j \quad \textnormal{with} \quad i\ne j.$$
Prove that $c\geq1$.
Izvor: Međunarodna matematička olimpijada, shortlist 2002