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#### Tilings with ordinary and domino tiles

Fill an ordinary chessboard with the tiles shown in the figure.

#### Partition rearrangement

Cut an arbitrary triangle into 3 parts and out of these pieces construct a rectangle.

#### 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.

#### Convex polygons , Pentagons , Pigeonhole principle (finite number of poits, lines etc.) , Plane dissected by lines

Two points are placed inside a convex pentagon. Prove that it is always possible to choose a quadrilateral that shares four of the five vertices on the pentagon, such that both of the points lie inside the quadrilateral.

#### 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?

#### Odd and even numbers , Tilings with ordinary and domino tiles

Is it possible to fill a $5 \times 5$ board with $1 \times 2$ dominoes?

#### Pigeonhole principle (finite number of poits, lines etc.) , Tilings with ordinary and domino tiles

What is the smallest number of ‘L’ shaped ‘corners’ out of 3 squares that can be marked on an 8×8 square grid, so that no more ‘corners’ would fit?

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

Identical to problem 34834.

#### Cube , Features of dissection pieces , Pigeonhole principle (finite number of poits, lines etc.)

The surface of a 3x3x3 Rubik’s Cube contains 54 squares. What is the maximum number of squares we can mark, so that no marked squares share a vertex or are directly adjacent to another marked square?

#### Dissections with certain properties , Examples and counterexamples. Constructive proofs , Simple fractions

A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?

#### Dissections, partitions, covers and tilings

Cut the figure $($on the boundaries of cells$)$ into three equal parts $($the same in shape and size$)$.

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