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#### Auxiliary simialr triangles , Extremal principle (other) , Pigeonhole principle (area and volume)

A unit square is divided into $n$ triangles. Prove that one of the triangles can be used to completely cover a square with side length $\frac{1}{n}$.

#### Area of a circle, sector, segment , Area of curvilinear shapes , Pigeonhole principle (area and volume) , Rectangles and squares. Criteria and properties

A square of side 15 contains 20 non-overlapping unit squares. Prove that it is possible to place a circle of radius 1 inside the large square, so that it does not overlap with any of the unit squares.

#### Isometry helps solve the problem , Pigeonhole principle (area and volume) , Point symmetry

On the planet Tau Ceti, the landmass takes up more than half the surface area. Prove that the Tau Cetians can drill a hole through the centre of their planet that connects land to land.

#### Area inequalities , Partitions into pairs and groups bijections , Pigeonhole principle (area and volume)

In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.

#### 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.

#### Geometry on grid paper , Pigeonhole principle (area and volume) , Various dissection problems

One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?

#### Packing problems , Pigeonhole principle (area and volume) , Systems of points

Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.

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

Leo’s grandma placed five empty plates on a square 1 metre $\times$ 1 metre table for dinner. Show that some two of these plates were less than $75$ cm apart.

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