We solve problems

Problems Geometry Plane geometry Geometrical inequalities Triangle inequality

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

Problems Geometry Plane geometry Geometrical 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 Geometrical 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 Geometrical 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 Geometrical 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-116301

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Area

13–14 4.0

101 random points are chosen inside a unit square, including on the edges of the square, so that no three points lie on the same straight line. Prove that there exist some triangles with vertices on these points, whose area does not exceed 0.01.

Problem #PRU-21983

Problems Geometry Plane geometry Geometrical 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-32032

Problems Algebra Algebraic inequalities and systems of inequalities Linear inequalites and systems of inequalities Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Geometry Plane geometry Geometrical inequalities Triangle inequality Triangle inequality (other)

13–14 3.0

On a plane, there are 1983 points and a circle of unit radius. Prove that there is a point on the circle, from which the sum of the distances to these points is no less than 1983.

Problem #PRU-34875

Problems Geometry Plane geometry Geometrical 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-35149

Problems Geometry Plane geometry Geometrical inequalities Area inequalities Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (area and volume)

13–15 3.0

In a square which has sides of length 1 there are 100 figures, the total area of which sums to more than 99. Prove that in the square there is a point which belongs to all of these figures.