Problems – We Solve Problem
Filter Problems
Showing 1 to 7 of 7 entries

#### 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.

#### 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.

#### 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.

#### A segment inside the triangle is smaller than the largest side , Hexagons , Pigeonhole principle (angles and lengths) , Regular polygons

There are 7 points placed inside a regular hexagon of side length 1 unit. Prove that among the points there are two which are less than 1 unit apart.

My Problem Set reset
No Problems selected