We solve problems

Problem #PRU-32068

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

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-32075

Problems Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (area and volume) Dissections, partitions, covers and tilings Various dissection problems

12–14 4.0

One corner square was cut from a chessboard. What is the smallest number of equal triangles that can be cut into this shape?

Problem #PRU-32092

Problems Methods Algebraic methods Counting in two ways Algebra Word Problems Tables and tournaments Number tables and its properties

10–12 2.0

In each square of a rectangular table of size \(M \times K\), a number is written. The sum of the numbers in each row and in each column, is 1. Prove that \(M = K\).

Problem #PRU-32095

Problems Geometry Plane geometry Visual geometry

10–13 2.0

Is it possible to draw this picture (see the figure), without taking your pencil off the paper and going along each line only once?

Problem #PRU-32104

Problems Set theory and logic Mathematical logic Mathematical logic (other)

10–13 2.0

One of five brothers baked a cake for their Mum. Alex said: “This was Vernon or Tom.” Vernon said: “It was not I and not Will who did it.” Tom said: “You’re both lying.” David said: “No, one of them told the truth, and the other was lying.” Will said: “No David, you’re wrong.” Mum knows that three of her sons always tell the truth. Who made the cake?

Problem #PRU-32110

Problems Algebra Number systems Decimal number system Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Reception pupil Peter knows only the number 1. Prove that he can write a number divisible by 1989.

Problem #PRU-32135

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.) 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-32784

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

What is the maximum number of rooks – also known as castles – you could place on an 8 by 8 chess board such that no two could take one another? Rooks can attack any number of squares horizontally and vertically, but not diagonally.

Problem #PRU-32785

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

Lessons at the Evening Mathematical School take place in nine auditoriums. Amongst the class were 19 students from the same school.

a) Prove that no matter how these students are arranged at least one auditorium will contain no fewer than 3 of these students.

b) Is it true that one of the auditoriums must contain exactly 3 of these students?