ILL 1990 Problem 92

Let $n$ be a positive integer and $m = \frac{(n+1)(n+2)}{2}$ . In coordinate plane, there are $n$ distinct lines $L_1, L_2, \ldots, L_n$ and $m$ distinct points $A_1, A_2, \ldots, A_m$ , satisfying the following conditions:i) Any two lines are non-parallel.ii) Any three lines are non-concurrent.iii) Only $A_1$ does not lies on any line $L_k$ , and there are exactly $k + 1$ points $A_j$ 's that lie on line $L_k$ $(k = 1, 2, \ldots, n).$ Prove that there exist a unique polynomial $p(x, y)$ with degree $n$ satisfying $p(A_1) = 1$ and $p(A_j) = 0$ for $j = 2, 3, \ldots, m.$