A [i]tile[/i] $T$ is a union of finitely many pairwise disjoint arcs of a unit circle $K$. The [i]size[/i] of $T$, denoted by $|T|$, is the sum of the lengths of the arcs $T$ consists of, divided by $2\pi$. A [i]copy[/i] of $T$ is a tile $T'$ obtained by rotating $T$ about the centre of $K$ through some angle. Given a positive real number $\varepsilon < 1$, does there exist an infinite sequence of tiles $T_1,T_2,\ldots,T_n,\ldots$ satisfying the following two conditions simultaneously: 1) $|T_n| > 1 - \varepsilon$ for all $n$; 2) The union of all $T_n'$ (as $n$ runs through the positive integers) is a proper subset of $K$ for any choice of the copies $T_1'$, $T_2'$, $\ldots$, $T_n', \ldots$? [hide=Note] In the extralist the problem statement had the clause "three conditions" rather than two, but only two are presented, the ones you see. I am quite confident this is a typo or that the problem might have been reformulated after submission.[/hide]