2023 IOQM/Problem 13

The ex-radii of a triangle are $10\frac{1}{2}$ , $12$ and $14$ . If the sides of the triangle are the roots of the cubic $x^3 - px^2 + qx - r$ , where $p$ , $q$ , $r$ are integers, find the integer nearest to $\sqrt{p+q+r}$