MathDB

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.

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