We solve problems

Problem #PRU-21982

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 (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 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-34875

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Geometry Plane geometry Polygons Hexagons Methods Pigeonhole principle Pigeonhole principle (angles and lengths) Polygons 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-58082

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Geometry Plane geometry Triangles Types of triangles Right-angled triangles Pythagorean theorem and its converse

12–14 4.0

A $3 \times 4$ rectangle contains 6 points. Prove that amongst them there will be two points, such that the distance between them is no greater than $\sqrt{5}$ .

Problem #PRU-65268

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A segment inside the triangle is smaller than the largest side ,Problems Probability and statistics Probability theory Discrete distribution

13–16 3.0

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?