We solve problems

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-34904

Problems Fun Problems Word Problems Solving problems by inequalities. Going through the cases. Methods Pigeonhole principle Pigeonhole principle (other)

11–14 3.0

Is it possible to transport 50 stone blocks, whose masses are equal to \(370, 372,\dots, 468\) kg, from a quarry on seven 3-tonne trucks?

Problem #PRU-34905

Problems Geometry Plane geometry Area Area of a trapezium Area ratio Area ratio (other) Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

13–14 4.0

9 straight lines each divide a square into two quadrilaterals, with their areas having a ratio of \(2:3\). Prove that at least three of the nine lines pass through the same point.

Problem #PRU-34962

Problems Methods Pigeonhole principle Pigeonhole principle (other) Discrete Mathematics Combinatorics Systems of points and line segments Systems of points, lines, and line segments

12–14 2.0

Is it possible to arrange 1000 line segments in a plane so that both ends of each line segment rest strictly inside another line segment?

Problem #PRU-34970

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–14 3.0

Some open sectors – that is sectors of circles with infinite radii – completely cover a plane. Prove that the sum of the angles of these sectors is no less than \(360^\circ\).

Problem #PRU-34980

Problems Geometry Solid geometry Lines and planes in space Methods Pigeonhole principle

14–15 3.0

A spherical planet is surrounded by 25 point asteroids. Prove, that at any given moment there will be a point on the surface of the planet from which an astronomer will not be able to see more than 11 asteroids.

Problem #PRU-34991

Problems Methods Extremal principle Extremal principle (other) Pigeonhole principle Pigeonhole principle (other) Discrete Mathematics Combinatorics Systems of points and line segments Systems of points, lines, and line segments

13–15 4.0

On a line, there are 50 segments. Prove that either it is possible to find some 8 segments all of which have a shared intersection, or there can be found 8 segments, no two of which intersect.

Problem #PRU-35020

Problems Algebra and arithmetic Arithmetics. Mental maths Methods Pigeonhole principle Pigeonhole principle (other) Extremal principle Sorting in increasing (decreasing) order

11–13 3.0

10 people collected a total of 46 mushrooms in a forest. It is known that no two people collected the same number of mushrooms. How many mushrooms did each person collect?

Problem #PRU-35058

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Covers Methods Pigeonhole principle Pigeonhole principle (area and volume)

13–15 3.0

10 magazines lie on a coffee table, completely covering it. Prove that you can remove five of them so that the remaining magazines will cover at least half of the table.

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.