Do there exist a circle and an infinite set of points on it such that the distance between any two of the points is rational?

A plane figure of area $A > n$ is given, where $n$ is a positive integer. Prove that this figure can be placed onto a Cartesian plane so that it covers at least $n+1$ points with integer coordinates.

Let $a,b,c\not= 0$ and $x,y,z\in\mathbb{R}^+$ such that $x+y+z=3$. Prove that \[\frac{3}{2}\sqrt{\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}}\geq\frac{x}{1+a^2}+\frac{y}{1+b^2}+\frac{z}{1+c^2}\]

In triangle $\triangle ABC$ we have $BC=a,CA=b,AB=c$ and $\angle B=4\angle A$ Show that \[ab^2c^3=(b^2-a^2-ac)((a^2-b^2)^2-a^2c^2)\]