Question 59
Prove that $c=10\sqrt{24}$ is the greatest positive real number with the following property: if 17 positive real numbers $a_1,\ldots,a_{17}$ satisfy both conditions
$$\sum_{i=1}^{17}a_i^2=24,\qquad
\sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i\lt c,$$
then for every $1\le i\lt j\lt k\le17$, the numbers $a_i,a_j,a_k$ are the side lengths of a triangle.
$$\sum_{i=1}^{17}a_i^2=24,\qquad
\sum_{i=1}^{17}a_i^3+\sum_{i=1}^{17}a_i\lt c,$$
then for every $1\le i\lt j\lt k\le17$, the numbers $a_i,a_j,a_k$ are the side lengths of a triangle.