Question 5
Prove that $C=10\sqrt{24}$ is the largest constant with the following property: if 17 positive real numbers $a_1,\ldots,a_{17}$ satisfy
$$\sum_{i=1}^{17}a_i^2=24,\qquad \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i\lt C,$$
then $a_i,a_j,a_k$ are the side lengths of a triangle for every $1\le i\lt j\lt k\le17$.
$$\sum_{i=1}^{17}a_i^2=24,\qquad \sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i\lt C,$$
then $a_i,a_j,a_k$ are the side lengths of a triangle for every $1\le i\lt j\lt k\le17$.