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Algebra Difficulty 7.0 National Olympiad, round 2 Find the answer Hungary

An infinite set $S$ of positive numbers is called thick, if in every interval of the form $\big[1/(n+1), 1/n\big]$ (where $n$ is an arbitrary positive integer) there is a number which is the difference of two elements from $S$ . Does there exist a thick set such that the sum of its elements is finite?

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