Question 10
Let $r \gt 0$ be a real number. All the interior points of the disc $D(r)$ of radius $r$ are colored with one of two colors, red or blue.
1. If $r \gt \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that the distance $AB = \pi$ and $A$ and $B$ have the same color.
2. Does the conclusion in (a) hold if $r \gt \frac{\pi}{2}$?
1. If $r \gt \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that the distance $AB = \pi$ and $A$ and $B$ have the same color.
2. Does the conclusion in (a) hold if $r \gt \frac{\pi}{2}$?