We solve problems

Problems Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

Prove that any convex polygon contains not more than $35$ vertices with an angle of less than $170^{\circ}$ .

Problems Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (area and volume) Systems of points and line segments Systems of points

13–15 3.0

Prove that, in a circle of radius 10, you cannot place 400 points so that the distance between each two points is greater than 1.

Problem #PRU-109018

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

12–14 3.0

A circle is covered with several arcs. These arcs can overlap one another, but none of them cover the entire circumference. Prove that it is always possible to select several of these arcs so that together they cover the entire circumference and add up to no more than $720^{\circ}$ .

Problem #PRU-109441

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

We are given a $100 \times 100$ square grid and $N$ counters. All of the possible arrangements of the counters on the grid which follow the following rule are considered: no two counters lie in adjacent squares.

What is the largest value of $N$ for which, in every single possible arrangement of counters following this rule, it is possible to find at least one counter such that moving it to an adjacent square does not break the rule. Squares are considered adjacent if they share a side.

Problem #PRU-109527

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Algebra and arithmetic Word problems Tables and tournaments Tables and tournaments (other)

11–13 3.0

What is the largest number of counters that can be put on the cells of a chessboard so that on each horizontal, vertical and diagonal (not only on the main ones) there is an even number of counters?

Problem #PRU-109574

Problems Methods Mathematical induction Mathematical induction (other) Pigeonhole principle Pigeonhole principle (other)

13–16 4.0

On a particular day it turned out that every person living in a particular city made no more than one phone call. Prove that it is possible to divide the population of this city into no more than three groups, so that within each group no person spoke to any other by telephone.

Problem #PRU-109748

Problems Geometry Plane geometry Convex and non-convex figures Convex polygons Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Polygons Polygons (other)

13–15 4.0

We are given a convex 200-sided polygon in which no three diagonals intersect at the same point. Each of the diagonals is coloured in one of 999 colours. Prove that there is some triangle inside the polygon whose sides lie some of the diagonals, so that all 3 sides are the same colour. The vertices of the triangle do not necessarily have to be the vertices of the polygon.

Problem #PRU-109766

Problems Algebra and arithmetic Word problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Is it possible to arrange natural numbers from 1 to $2002^{2}$ in the cells of a $2002 \times 2002$ table so that for each cell of this table one could choose a triplet of numbers, from a row or column, where one of the numbers is equal to the product of the other two?

Problem #PRU-109808

Problems Combinatorics Geometry on grid paper Integer lattices Integer lattices (other) Painting problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 3.0

All of the points with whole number co-ordinates in a plane are plotted in one of three colours; all three colours are present. Prove that there will always be possible to form a right-angle triangle from these points so that its vertices are of three different colours.

Problem #PRU-109877

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

11–13 3.0

A regular hexagon with sides of length $5$ is divided by straight lines, that are parallel to its sides, to form regular triangles with sides of length 1. We call the vertices of all such triangles nodes. It is known that more than half of the nodes are marked. Prove that there are five marked nodes lying on one circle.