IMO Shortlist 2007 problem A6
Dodao/la:
arhiva2. travnja 2012. Let
![a_1, a_2, \ldots, a_{100}](/media/m/e/4/6/e46a230d6980f33034650417473efaa1.png)
be nonnegative real numbers such that
![a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.](/media/m/b/1/9/b1907b7026e4eec1a08a996a483a3ae8.png)
Prove that
Author: Marcin Kuzma, Poland
%V0
Let $a_1, a_2, \ldots, a_{100}$ be nonnegative real numbers such that $a^2_1 + a^2_2 + \ldots + a^2_{100} = 1.$ Prove that
$$a^2_1 \cdot a_2 + a^2_2 \cdot a_3 + \ldots + a^2_{100} \cdot a_1 < \frac {12}{25}.$$
Author: Marcin Kuzma, Poland
Izvor: Međunarodna matematička olimpijada, shortlist 2007