We solve problems

Problem #PRU-32046

Problems Methods Algebraic methods Counting in two ways Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

11–13 3.0

You are given 25 numbers. The sum of any 4 of these numbers is positive. Prove that the sum of all 25 numbers is also positive.

Problem #PRU-32052

Problems Methods Algebraic methods Processes and operations

12–15 3.0

There were seven boxes. In some of them, seven more boxes were placed inside (not nested in each other), etc. As a result, there are 10 non-empty boxes. How many boxes are there now in total?

Problem #PRU-32057

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

A class contains 25 pupils. It is known that within any group of 3 pupils there are two friends. Prove that there is a pupil who has no fewer than 12 friends.

Problem #PRU-32068

Problems Discrete Mathematics Combinatorics Painting problems Methods Pigeonhole principle Pigeonhole principle (area and volume)

11–13 3.0

A square area of size \(100\times 100\) is covered in tiles of size \(1\times 1\) in 4 different colours – white, red, black, and grey. No two tiles of the same colour touch one another, that is share a side or a corner. How many red tiles can there be?

Problem #PRU-32124

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Geometry Solid geometry Tetrahedron and pyramid Particular cases of tetrahedra Regular tetrahedron

13–15 3.0

A scone contains raisins and sultanas. Prove that inside the scone there will always be two points 1cm apart such that either both lie inside raisins, both inside sultanas, or both lie outside of either raisins or sultanas.

Problem #PRU-32135

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) Discrete Mathematics Combinatorics Systems of points and line segments Systems of points and line segments (other)

12–14 3.0

101 points are marked on a plane; not all of the points lie on the same straight line. A red pencil is used to draw a straight line passing through each possible pair of points. Prove that there will always be a marked point on the plane through which at least 11 red lines pass.

Problem #PRU-32783

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

33 representatives of four different races – humans, elves, gnomes, and goblins – sit around a round table.

It is known that humans do not sit next to goblins, and that elves do not sit next to gnomes. Prove that some two representatives of the same peoples must be sitting next to one another.

Problem #PRU-32787

Problems Number Theory Numeral systems Decimal number system Methods Algebraic methods Partitions into pairs and groups; bijections Pigeonhole principle Pigeonhole principle (other)

12–13 3.0

Is it possible to find 57 different two digit numbers, such that no sum of any two of them was equal to 100?

Problem #PRU-32789

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

A number is written on each edge of a cube. The sum of the 4 numbers on the adjacent edges is written on each face. Place the numbers \(1\) and \(-1\) on the edges so that the numbers written on the faces are all different.

Problem #PRU-34834

Problems Algebra Sequences Progressions Arithmetic progression Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

12–14 3.0

A road of length 1 km is lit with streetlights. Each streetlight illuminates a stretch of road of length 1 m. What is the maximum number of streetlights that there could be along the road, if it is known that when any single streetlight is extinguished the street will no longer be fully illuminated?