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Proportional areas mean that points lie on a line

Source: South African MO 2014 Q6

October 25, 2014
geometryanalytic geometrygeometry unsolved

Problem Statement

Let OO be the centre of a two-dimensional coordinate system, and let A1,A2,…,AnA_1, A_2, \ldots ,A_n be points in the first quadrant and B1,B2,…,BmB_1, B_2, \ldots , B_m points in the second quadrant. We associate numbers a1,a2,…,ana_1, a_2, \ldots , a_n to the points A1,A2,…,AnA_1, A_2, \ldots ,A_n and numbers b1,b2,…,bmb_1, b_2, \ldots, b_m to the points B1,B2,…,BmB_1, B_2, \ldots , B_m, respectively. It turns out that the area of triangle OAjBkOA_jB_k is always equal to the product ajbka_jb_k, for any jj and kk. Show that either all the AjA_j or all the BkB_k lie on a single line through OO.
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