Question

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

Answer 1

red point are 40 and white 60

Answer 2

Answer:

Least possible number of white points = 9

Step-by-step explanation:

Let x be the number of red points

Then no. of white points = 100-x

No. of red points in 1 circle = 1

No. of white points in each circle = at least 3

Since there can be more than one circle passing through to points, therefore

no. of circles passing through at least 3 white points = $\binom{100-x}{3}$

No. circles = no. of red points = x

so, the equation is:  $\binom{100-x}{3}$ ≤ x

By solving, we get x = 91 (approx)

Therefore, no. of white points = 100-x = 100-91 =9

Hence, least possible number of white points = 9