VN-TST 2012 Problem 5

Prove that $c=10\sqrt{24}$ is the largest constant such that if there exist positive numbers $a_1,a_2,\ldots ,a_{17}$ satisfying: $$\sum_{i=1}^{17}a_i^2=24,\ \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i<c$$ then for every $i,j,k$ such that $1\le 1<j<k\le 17$ , we have that $x_i,x_j,x_k$ are sides of a triangle.