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#### Geometry on grid paper , Painting problems , Pigeonhole principle (finite number of poits, lines etc.)

A ream of squared paper is shaded in in two colours. Prove that there are two horizontal and two vertical lines, the points of intersection of which are shaded in the same colour.

#### 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?

#### Geometry on grid paper , Integer lattices (other) , Painting problems , Pigeonhole principle (finite number of poits, lines etc.)

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

#### Geometry on grid paper

Cut the shape $($see the figure$)$ into two identical pieces $($coinciding when placed on top of one another$)$.

#### Geometry on grid paper

How many squares are shown in the picture?

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