After login you will be able to create your own lists of problems.

Found: 10

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?

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.

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.)

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.

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

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.

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.

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.

Prove that \(\angle ABC > 90^{\circ}\) if and only if the point \(B\) lies inside a circle with diameter \(AC\).

There are 25 children in a class. At random, two are chosen. The probability that both children will be boys is \(3/25\). How many girls are in the class?