Ball Game

$N$ balls are placed at **distinct** coordinates on the $X-$axis. The initial coordinate of the $i^{th}$ ball is given as $A_i \ (A_i \gt 0)$. A hole is present at the origin. The $i^{th}$ ball moves towards the hole with a speed of $B_i$ units per second. If multiple balls collide **before** reaching the hole, they merge into one ball, taking the maximum speed of the initial balls. Determine