Kvant Math Problem 562

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Problem

On the segment $[0;1]$, a set $M$ is given, which is a union of several segments, such that the distance between any two points from $M$ is not equal to $0{,}1$. Prove that the sum of the lengths of the segments comprising $M$ is less than

  1. $0{,}55$;
  2. $0{,}5$.

A. Katz, 10th-grade student

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