Prove there are 3 points on the circle having same colour 
All the points of a circle are randomly coloured red or blue. Prove there
  are 3 points on the circle having same colour, representing an
  isosceles triangle.

 A: In the wonderfully clever solution given, starting with that pentagon, the two possible monochromatic triangles are not congruent. It may be interesting to note that this is of necessity.
Say an $\alpha$-triangle is an isoceles triangle in which the two sides of equal length meet at an angle $\alpha$.
Fix an angle $\alpha$. There exists a coloring of the points of the cirle with two colors, such that no $\alpha$-triangle is monochromatic.
Say the circle is the unit circle $S$ in the complex plane. Given $\alpha$, there exists a complex number $\beta$ with $|\beta|=1$ such that the vertices of any $\alpha$-triangle are $z,\beta z,\beta^2z$ for some $z$. For $z\in S$ define $$C_z=\{\beta^nz:n\in\Bbb Z\}.$$Then the $C_z$ form a partition of $S$; choose a coloring so that $\beta^nz$ and $\beta^{n+1}z$ always have different colors.
Oops Unless the sequence $\beta^n z$ is periodic with period $k$, for $k$ odd. In that case color $z,\dots,\beta^{k-1}z$ with alternating colors. (So now $\beta^{k-1}z$ and $\beta^{k}z$ have the same color, but it doesn't matter, there are no three consecutive $n$ for which $\beta^nz$ all have the same color.) Thanks to stewbasic for noticing the gap.
A: Hint:
Take regular pentagon. The three points are the same color.

