ZA-SAMO 2014 Problem 6

Let $O$ be the centre of a two-dimensional coordinate system, and let $A_1, A_2, \ldots ,A_n$ be points in the first quadrant and $B_1, B_2, \ldots , B_m$ points in the second quadrant. We associate numbers $a_1, a_2, \ldots , a_n$ to the points $A_1, A_2, \ldots ,A_n$ and numbers $b_1, b_2, \ldots, b_m$ to the points $B_1, B_2, \ldots , B_m$ , respectively. It turns out that the area of triangle $OA_jB_k$ is always equal to the product $a_jb_k$ , for any $j$ and $k$ . Show that either all the $A_j$ or all the $B_k$ lie on a single line through $O$ .