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m distinct points - ILL 1990 TUR3

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September 19, 2010
analytic geometryalgebrapolynomialgeometry unsolvedgeometry

Problem Statement

Let nn be a positive integer and m=(n+1)(n+2)2m = \frac{(n+1)(n+2)}{2}. In coordinate plane, there are nn distinct lines L1,L2,…,LnL_1, L_2, \ldots, L_n and mm distinct points A1,A2,…,AmA_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 A1A_1 does not lies on any line LkL_k, and there are exactly k+1k + 1 points AjA_j's that lie on line LkL_k (k=1,2,…,n).(k = 1, 2, \ldots, n).
Prove that there exist a unique polynomial p(x,y)p(x, y) with degree nn satisfying p(A1)=1p(A_1) = 1 and p(Aj)=0p(A_j) = 0 for j=2,3,…,m.j = 2, 3, \ldots, m.
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