IMO Shortlist 2012 problem A3
Dodao/la:
arhiva3. studenoga 2013. Let
![n\ge 3](/media/m/f/2/6/f26f60f0cef07be122e851956e32cd1d.png)
be an integer, and let
![a_2,a_3,\ldots ,a_n](/media/m/d/1/a/d1ad2917c896049545c602335deded7c.png)
be positive real numbers such that
![a_{2}a_{3}\cdots a_{n}=1](/media/m/d/5/4/d548ca041cfe5eac7ec9c0c8488f503c.png)
. Prove that
![(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.](/media/m/8/7/f/87f5d69561fae0dea52365a062a29aec.png)
Proposed by Angelo Di Pasquale, Australia
%V0
Let $n\ge 3$ be an integer, and let $a_2,a_3,\ldots ,a_n$ be positive real numbers such that $a_{2}a_{3}\cdots a_{n}=1$. Prove that
$$(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.$$
Proposed by Angelo Di Pasquale, Australia
Izvor: Međunarodna matematička olimpijada, shortlist 2012