Problems – We Solve Problems
Filter Problems
Showing 1 to 20 of 46 entries

#### Central angle. Arc length and circumference , Chords and secants (other) , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

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.

#### 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)

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

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

2022 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?

#### 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 $\frac{1}{7}$.

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

There are 25 points on a plane, and among any three of them there can be found two points with a distance between them of less than 1. Prove that there is a circle of radius 1 containing at least 13 of these points.

#### 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.

#### Lines, segments, rays, and angles (other) , Pigeonhole principle (finite number of poits, lines etc.) , Triangles (other)

Prove that no straight line can cross all three sides of a triangle, at points away from the vertices.

#### Incirlce and circumcircle of a triangle , Pentagons , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

All the points on the edge of a circle are coloured in two different colours at random. Prove that there will be an equilateral triangle with vertices of the same colour inside the circle – the vertices are points on the circumference of the circle.

#### Partitions into pairs and groups bijections , Pigeonhole principle (finite number of poits, lines etc.) , Regular polygons

2001 vertices of a regular 5000-gon are painted. Prove that there are three coloured vertices lying on the vertices of an isosceles triangle.

My Problem Set reset
No Problems selected