We solve problems

Problems Geometry Plane geometry Geometric inequalities Triangle inequality

A point \(P\) is somewhere inside the triangle \(\triangle ABC\). Show that \(AP + BP < AC + BC\).

Problems Geometry Plane geometry Geometric inequalities Triangle inequality

The distance between London and Warsaw equals \(1450\) km, between Warsaw and Kyiv is \(680\) km. The distance from London to New Delhi, is \(6700\) km and the distance from Kyiv to New Delhi is \(4570\) km. What is the distance from London to Kyiv?

Problem #PRU-100679

Problems Geometry Plane geometry Geometric inequalities Triangle inequality

Show that if all sides of a triangle have integer lengths and one of them is equal to \(1\), then the other two have lengths equal to each other.

Problem #PRU-100682

Problems Geometry Plane geometry Geometric inequalities Triangle inequality

A billiard ball lies on a table in the shape of an acute angle. How should you hit the ball so that it returns to its starting location after hitting each of the two banks once? Is it always possible to do so?
(When the ball hits the bank, it bounces. The way it bounces is determined by the shortest path rule – if it begins at some point \(D\) and ends at some point \(D'\) after bouncing, the path it takes is the shortest possible path that includes the bounce.)

Problem #PRU-110207

Problems Methods Pigeonhole principle Pigeonhole principle (other) Geometry Plane geometry Geometric inequalities Triangle inequality Triangle inequality algebra problems

12–14 3.0

Natural numbers from 1 to 200 are divided into 50 sets. Prove that in one of the sets there are three numbers that are the lengths of the sides of a triangle.

Problem #PRU-21982

Problems Geometry Plane geometry Geometric inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

11–13 3.0

Prove that it is not possible to completely cover an equilateral triangle with two smaller equilateral triangles.

Problem #PRU-21983

Problems Geometry Plane geometry Geometric inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

51 points were thrown into a square of side 1 m. Prove that it is possible to cover some set of 3 points with a square of side 20 cm.

Problem #PRU-34875

Problems Geometry Plane geometry Geometric inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Polygons Hexagons Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Regular polygons

12–14 2.0

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.

Problem #PRU-35234

Problems Geometry Plane geometry Geometric inequalities Polygons (inequalities)

12–14 2.0

A pentagon is inscribed in a circle of radius 1. Prove that the sum of the lengths of its sides and diagonals is less than 17.

Problem #PRU-57326

Problems Geometry Plane geometry Geometric inequalities Triangle inequality Triangle inequality (other)

11–13 2.0

In a triangle, the lengths of two of the sides are 3.14 and 0.67. Find the length of the third side if it is known that it is an integer.