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#### Covers , Inscribed and circumscribed polygons , Pigeonhole principle (finite number of poits, lines etc.)

A unit square contains 51 points. Prove that it is always possible to cover three of them with a circle of radius $\frac{1}{7}$.

#### Countable and uncountable subsets , Covers

Prove that rational numbers from [0; 1] can be covered by a system of intervals of total length no greater than 1/1000.

#### Covers , Pigeonhole principle (area and volume)

10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.

#### Arithmetic progression , Covers , Pigeonhole principle (angles and lengths)

Identical to problem 34834.

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