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In Research Problems in Discrete Geometry by Peter Brass, William O. J. Moser, János Pach, in section 2.1 page 75, "Decomposition of multiple packings and coverings", problems with a similar flavour are listed, some with solutions, about covering of the plane with discs, or the space with spheres. Typically, for spheres, for some $a$ and $b$, $\{a\le F_x\le b, \forall x\}$ entails that there exists a partition $F=\tilde F\cup \hat F$ with $\{\tilde F_x\ge 1, \forall x\}$ and the same for $\hat F$. A probabilistic proof uses the local Lovasz lemma.