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?

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