23rd Eötvös Competition Problems 1916

1.  a, b are positive reals. Show that 1/x + 1/(x-a) + 1/(x+b) = 0 has two real roots one in [a/3, 2a/3] and the other in [-2b/3, -b/3]. 2.  ABC is a triangle. The bisector of ∠C meets AB at D. Show that CD2 < CA·CB.