International Mathematical Olympiad 1961 Problem 4

Consider triangle $P_1P_2P_3$ and a point $P$ within the triangle. Lines $P_1P, P_2P, P_3P$ intersect the opposite sides in points $Q_1, Q_2, Q_3$ respectively. Prove that, of the numbers $$\frac{P_1P}{PQ_1}, \frac{P_2P}{PQ_2}, \frac{P_3P}{PQ_3}$$ at least one is $\leq 2$ and at least one is $\geq 2$.