CZ-SK-MO 1999 Problem 3

Show that there exists a triangle $ABC$ such that $a \ne b$ and $a+t_a = b+t_b$ , where $t_a,t_b$ are the medians corresponding to $a,b$ , respectively. Also prove that there exists a number $k$ such that every such triangle satisfies $a+t_a = b+t_b = k(a+b)$ . Finally, find all possible ratios $a : b$ in such triangles.