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?
Problem
Source: RMO 2018 P4
Tags: combinatorics
28.10.2018 15:14
Should All the red point pass by a white's circle ?
28.10.2018 15:46
Simply choose $10$ points such that no four points lie on a circle and colour them white. Now the number of red points is at most $\binom{10}{3} = 120>100$. If number of white points is less than $10$, then red points is less than $\binom{9}{3} = 84$. But $84+9<100$, the answer is $\boxed{10}$.
28.10.2018 16:05
@TDP I think this question is too easy because I know you are not a fool!
28.10.2018 19:52
Can we put all the red points in one circle? If it will be true we need only 3 white points to put all the 97 red points in one circle.
28.10.2018 20:13
A.Amoozadeh wrote: Can we put all the red points in one circle? If it will be true we need only 3 white points to put all the 97 red points in one circle. I guess you read the question incorrectly. It says that the red points are centers of the circle passing through three or more white points.
06.08.2020 06:04
As we have a unique circle passing through 3 points, we hence have a unique red point for each 3 unique combination of 3 white points. Hence, the no. of unique circles should be greater than the no of red points $\implies\binom{n}{3}\geq100-n$. We see that the smallest no. satisfying is $\boxed{10}$