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#### Central angle. Arc length and circumference , Chords and secants (other) , Isosceles, inscribed, and circumscribed trapeziums , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons , Two tangent lines to a circle, intersecting at a particular point

In a regular shape with 25 vertices, all the diagonals are drawn.
Prove that there are no nine diagonals passing through one interior point of the shape.

#### Examples and counterexamples. Constructive proofs , Pigeonhole principle (finite number of poits, lines etc.)

The smell of a flowering lavender plant diffuses through a radius of 20m around it. How many lavender plants must be planted along a straight 400m path so that the smell of the lavender reaches every point on the path.

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

15 points are placed inside a $4 \times 4$ square. Prove that it is possible to cut a unit square out of the $4 \times 4$ square that does not contain any points.

#### Convex polygons , Pigeonhole principle (finite number of poits, lines etc.) , Tilings with ordinary and domino tiles , Various dissection problems

A square piece of paper is cut into 6 pieces, each of which is a convex polygon. 5 of the pieces are lost, leaving only one piece in the form of a regular octagon $($see the drawing$)$. Is it possible to reconstruct the original square using just this information?

#### Equilateral triangle , Painting problems , Pigeonhole principle (finite number of poits, lines etc.)

A white plane is arbitrarily sprinkled with black ink. Prove that for any positive l there exists a line segment of length l with both ends of the same colour.

#### Pigeonhole principle (finite number of poits, lines etc.)

A square $ABCD$ contains 5 points. Prove that the distance between some pair of these points does not exceed $\frac{1}{2} AC$

#### Discrete geometry (other) , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons , Trapeziums (other)

A regular 1981-gon has 64 vertices. Prove that there exists a trapezium with vertices at the marked points.

#### Circle, sector, segment, etc , Covers , Pigeonhole principle (angles and lengths) , Pigeonhole principle (finite number of poits, lines etc.)

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?

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

1956 points are selected from a cube, whose edge is equal to 13 units. Is it possible to place a cube with edge of 1 unit in this cube so that there is not one selected point inside it?

#### Chessboard colouring , Examples and counterexamples. Constructive proofs , Pigeonhole principle (finite number of poits, lines etc.)

What is the minimum number of 1×1 squares that need to be drawn in order to get an image of a 25×25 square divided into 625 smaller 1×1 squares?

#### Convex polygons , Pigeonhole principle (finite number of poits, lines etc.)

What is the minimum number of points necessary to mark inside a convex $n$-sided polygon, so that at least one marked point always lies inside any triangle whose vertices are shared with those of the polygon?

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