We solve problems

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 \(\angle ABC < \angle BAC\) if and only if \(AC < BC\), that is, the larger side lies opposite the larger angle of the triangle, and opposite the larger side lies the larger angle.

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 \(\sqrt5\).

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-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.