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Four lamps need to be hung over a square ice-rink so that they fully illuminate it. What is the minimum height needed at which to hang the lamps if each lamp illuminates a circle of radius equal to the height at which it hangs?

Several pieces of carpet are laid along a corridor. Pieces cover the entire corridor from end to end without omissions and even overlap one another, so that over some parts of the floor lie several layers of carpet. Prove that you can remove a few pieces, perhaps by taking them out from under others and leaving the rest exactly in the same places they used to be, so that the corridor will still be completely covered and the total length of the pieces left will be less than twice the length corridor.

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

A straight corridor of length 100m is covered with 20 rugs that have a total length of 1km. The width of each rug is equal to the width of the corridor. What is the longest possible total length of corridor that is not covered by a rug?

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

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.

Some open sectors – that is sectors of circles with infinite radii – completely cover a plane. Prove that the sum of the angles of these sectors is no less than $360^\circ$.

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than 720$^{\circ}$.