$ a_{1},a_{2},....,a_{n}(n\geq 3) $ are positive numbers that : $(a_{1}+a_{2}+....+a_{n})^2 >\frac{3n-1}{3}(a_{1}^2+a_{2}^2+....+a_{n}^2)$
Prove that for any triple $a_{i},a_{j},a_{k} $ are three edge lengths of some triangle, where natural numbers $ i,j,k $ satifying $ 0< i< j< k\leq n $
It seems a Interesting problem , i tried to prove it in case $n=3$ starting an induction on but i failed at that level
please help me. Thank
Maybe can use Induction