We solve problems

Problem #PRU-57304

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle Inequalities in a triangle involving medians

13–13 2.0

Prove that $(a + b - c) / 2 < m_{c} < (a + b) / 2$ , where $a$ , $b$ and $c$ are the lengths of the sides of an arbitrary triangle and $m_{c}$ is the median to side $c$ .

Problem #PRU-57416

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle Inequalities in a triangle involving heights

13–14 2.0

Prove that in any triangle the sum of the lengths of the heights is less than the perimeter.

Problem #PRU-57470

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle The longest side of the triangle lies opposite the largest angle

14–14 2.0

Prove that $∠ A B C < ∠ B A C$ if and only if $A C < B C$ , that is, the larger side lies opposite the larger angle of the triangle, and opposite the larger side lies the larger angle.

Problem #PRU-57480-3

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle A perpendicular distance is the shortest. Inequalities for right triangles

13–13 2.0

$A B C$ is a right angled triangle with a right angle at $C$ . Prove that $c^{n} > a^{n} + b^{n}$ for $n > 2$ .

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 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?

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 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-54025

Problems Geometry Plane geometry Geometrical inequalities Inequalities in a triangle The longest side of the triangle lies opposite the largest angle

13–14 2.0

Prove that the segment connecting the vertex of an isosceles triangle to a point lying on the base is no greater than the lateral side of the triangle.