IMO Shortlist 1974 problem 4
Dodao/la:
arhiva2. travnja 2012. The sum of the squares of five real numbers
![a_1, a_2, a_3, a_4, a_5](/media/m/b/5/3/b53fb1ecc41d2df5081e34c6d34273b3.png)
equals
![1](/media/m/a/9/1/a913f49384c0227c8ea296a725bfc987.png)
. Prove that the least of the numbers
![(a_i - a_j)^2](/media/m/7/1/3/713e36b320f63c4f7f6a7419a786b2df.png)
, where
![i, j = 1, 2, 3, 4,5](/media/m/4/9/c/49c820e8a4ae27159d5d69452442b190.png)
and
![i \neq j](/media/m/a/5/f/a5f46b46ea53ae08f426e6adb8365875.png)
, does not exceed
%V0
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
Izvor: Međunarodna matematička olimpijada, shortlist 1974