
4. Suppose 100 points in the plane are coloured using two colours, red and white, such that each red point is the centre of a circle passing through at least three white points. What is the least possible number of white points?


Solution: Let $n$ be the number of white points. Then we can draw at most $\binom{n}{3}$ circles. Thus the number of white and red points together is at most $n+\binom{n}{3}$. We observe that

$$
9+\binom{9}{3}=93, \quad 10+\binom{10}{3}=130
$$

Thus $n \geq 10$. We show that $n=10$ works.

Take any 10 points such that no three are collinear and no four are concyclic. Then we get $\binom{10}{3}=120$ points as centre of distinct circles. Among these, there may be some points from the original 10 points. Even if we leave out these 10 points, we have at least 110 points which are centres of circles formed by the ten points we have chosen. Choose any 90 points from them and colour them red and colour the original 10 points white. We get 100 points of which 10 are white and remaining 90 are red. Each of these 90 red points is the centre of a circle passing through some three white points.
