Suppose $100$ points in the plane are coloured using two colours, red and white such that each red point is the centre of circle passing through at least three white points. What is the least possible number of white points?
I couldn't reach anywhere in this question. I tried making different images and patterns of circles but couldn't find a one where the red circle had only 3 white points on it's circumference.
Help.
The red points only have to be the centre of a circle through some three white points – there are no restrictions on the location or radius of the circles, nor the white points. Hence, given $n$ white points in general position, we may maximise the number of red points by considering all size-$3$ combinations of white points and constructing the centres of all $\binom n3$ circles formed from those combinations.
Thus the problem is equivalent to finding the smallest $n$ such that $n+\binom n3\ge100$. By trial and error, we see that $9+\binom93=93$ but $10+\binom{10}3=130$, so at least $10$ white points are required.
Let the $w$ white points be in general position, then every set of three of them defines a circle and thus a possible red point at its center. Then solve $w+\binom{w}3\ge100$. The weakened condition $(w-1)^3\ge600\implies w\ge 10$ provides an upper bound. Computing $$ 9+\frac{9\cdot 8\cdot 7}6=9+84=93<100 $$ shows that $w=10$ is also the minimal solution for this construction, giving $120$ center positions to select the 90 locations of the red points from.
