MEMO 2010 ekipno problem 2
Dodao/la:
arhiva28. travnja 2012. For each integer
![n\geqslant2](/media/m/7/d/e/7de025e723264e181387bc1b65daa65b.png)
, determine the largest real constant
![C_n](/media/m/c/4/7/c4741f92cb04e74aca4942700ae6f7bf.png)
such that for all positive real numbers
![a_1, \ldots, a_n](/media/m/4/6/2/4620bab4413c05b6365d3f7e207d102a.png)
we have
%V0
For each integer $n\geqslant2$, determine the largest real constant $C_n$ such that for all positive real numbers $a_1, \ldots, a_n$ we have $$\frac{a_1^2+\ldots+a_n^2}{n}\geqslant\left(\frac{a_1+\ldots+a_n}{n}\right)^2+C_n\cdot(a_1-a_n)^2\mbox{.}$$
Izvor: Srednjoeuropska matematička olimpijada 2010, ekipno natjecanje, problem 2