Putnam 1996 B6

Let $(a_1, b_1), (a_2, b_2), \ldots, (a_n, b_n)$ be the vertices of a convex polygon which contains the origin in its interior. Prove that there exist positive real numbers $x$ and $y$ such that

$$\begin{gathered}(a_1, b_1)x^{a_1} y^{b_1} + (a_2, b_2)x^{a_2}y^{b_2} + \cdots \\ + (a_n, b_n)x^{a_n}y^{b_n} = (0,0).\end{gathered}$$