IMO Shortlist 1972 problem 3
Dodao/la:
arhiva2. travnja 2012. The least number is
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and the greatest number is
![M](/media/m/f/7/f/f7f312cf6ba459a332de8db3b8f906c4.png)
among
![a_1 ,a_2 ,\ldots,a_n](/media/m/a/3/3/a33ac2ef57e5b8d3204a50dea9622f59.png)
satisfying
![a_1 +a_2 +...+a_n =0](/media/m/5/6/0/5606eec07dbe89280d34393d6950dd14.png)
. Prove that
%V0
The least number is $m$ and the greatest number is $M$ among $a_1 ,a_2 ,\ldots,a_n$ satisfying $a_1 +a_2 +...+a_n =0$. Prove that
$$a_1^2 +\cdots +a_n^2 \le-nmM$$
Izvor: Međunarodna matematička olimpijada, shortlist 1972