What is the greatest number of points that can be placed inside a circle? Singapore Math Competition Question I'm just doing some practice papers when this question came. 
What is the greatest number of points that can be placed inside a circle such that the distance between any two points is greater than the radius of a circle?
I have a catch that the answer should be 5 or 6. However, I do not know which one should be correct. 
Anyone help?
 A: Let's examine placing a particular point $P$ in the circle $C$. Then a circular region of radius $r$ (the region in which no other points can be placed) around $P$ and inside $C$ includes $\textit{all}$ of the circular region of radius $r$ around $P'$ and inside $C$, where $P'$ is the closest point to $P$ on the perimeter of the circle.

Note that if $P$ starts at the center of $C$, then the whole of $C$ is excluded, so we can do better by choosing $P'$ anywhere else.
Hence, a maximum arrangement of points greater than $r$ away from each other can be obtained when all points are on the perimeter (note that there can still be maximum arrangements with every point inside the circle, we have just proved that if there is one of those, then we could also draw a maximum arrangement on the perimeter).
The minimum distance two points need to be away from each other is $r$, so every two points must be greater than $\frac{1}{3}\pi r$ away from each other along the perimeter of the circle (we can see this by noting that when two points on the perimeter are exactly $r$ distant from each other, then the angle between them measured from the center of the circle is $\frac{1}{3}\pi$). However, the total length of the perimeter is $2\pi r$, so by the pigeon hole principle, if we have $6$ points arranged on the circle, then the counterclockwise distance between some two neighbors must be $\leq\frac{2\pi r}{6}=\frac{1}{3}\pi r$. But this contradicts the fact that the distance between any two neighbors must be strictly greater than $\frac{1}{3}\pi r$. Hence there are no satisfying arrangements of $6$ points.
Finally, noting that an arrangement of a regular pentagon of sidelength $r$ fits inside our circle of radius $r$, we can conclude that the maximum number of points we can place in accordance with the given rule is $5$.
