Geometric-analytic problem
Let $f:\mathbb{R}^2\to\mathbb{R}$ be strictly positive Lipschitz function with constant $1/2$. Let $A$ be a nonempty subset of $\mathbb{R}^2$, such that if $x\in A$ and $y\in\mathbb{R}^2$ with $\|x-y\|=f(x)$, then $y\in A$. Prove that $A=\mathbb{R}^2$. The problem was proposed to Bulgarian TST, 2009. Do you know earlier source or some context of the problem?