We solve problems

Problem #PRU-103869

Problems Discrete Mathematics Set theory and logic Order relations Methods Pigeonhole principle Pigeonhole principle (other)

11–13 2.0

Solve the puzzle: \(BAO \times BA \times B = 2002\).

Problem #PRU-103882

Problems Methods Algebraic methods Proof by exhaustion Discrete Mathematics Set theory and logic Mathematical logic Puzzles

11–12 2.0

Find the smallest four-digit number \(CEEM\) for which there exists a solution to the rebus \(MN + PORG = CEEM\). (The same letters correspond to the same numbers, different – different.)

Problem #PRU-103884

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Dissections with certain properties Methods Examples and counterexamples. Constructive proofs Algebra Fractions Simple fractions

11–12 2.0

A rectangle is cut into several smaller rectangles, the perimeter of each of which is an integer number of meters. Is it true that the perimeter of the original rectangle is also an integer number of meters?

Problem #PRU-104021

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Number Theory Divisibility Odd and even numbers Methods Pigeonhole principle Pigeonhole principle (other)

12–13 2.0

a) A 1 or a 0 is placed on each vertex of a cube. The sum of the 4 adjacent vertices is written on each face of the cube. Is it possible for each of the numbers written on the faces to be different?

b) The same question, but if 1 and \(-1\) are used instead.

Problem #PRU-105052

Problems Methods Pigeonhole principle Pigeonhole principle (other) Proof by contradiction Fun Problems Word Problems Tables and tournaments Tables and tournaments (other)

11–13 3.0

In a chess tournament, each participant played two games with each of the other participants: one with white pieces, the other with black. At the end of the tournament, it turned out that all of the participants scored the same number of points (1 point for a victory, \(\frac{1}{2}\) a point for a draw and 0 points for a loss). Prove that there are two participants who have won the same number of games using white pieces.

Problem #PRU-105208

Problems Fun Problems Word Problems Tables and tournaments Number tables and its properties Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–14 4.0

A teacher filled the squares of a chequered table with \(5\times5\) different integers and gave one copy of it to Janine and one to Zahara. Janine selects the largest number in the table, then she deletes the row and column containing this number, and then she selects the largest number of the remaining integers, then she deletes the row and column containing this number, etc. Zahara performs similar operations, each time choosing the smallest numbers. Can the teacher fill up the table in such a way that the sum of the five numbers chosen by Zahara is greater than the sum of the five numbers chosen by Janine?

Problem #PRU-107702

Problems Geometry Solid geometry Parallelepipeds Special cases of parallelepipeds Cube Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings Features of dissection pieces Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–16 3.0

The surface of a \(3\times 3\times 3\) Rubik’s Cube contains \(54\) squares. What is the maximum number of squares we can mark so that no marked squares share at least one vertex?

Make sure you show that both (a) you can achieve this maximum and (b) that you can’t do better than this maximum.

Problem #PRU-107703

Problems Discrete Mathematics Combinatorics Packing problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

11–13 2.0

Is it possible to place 12 identical coins along the edges of a square box so that touching each edge there were exactly: a) 2 coins, b) 3 coins, c) 4 coins, d) 5 coins, e) 6 coins, f) 7 coins.

You are allowed to place coins on top of one another. In the cases where it is possible, draw how this could be done. In the other cases, prove that doing so is impossible.

Problem #PRU-107802

Problems Methods Examples and counterexamples. Constructive proofs Fun Problems Word Problems Tables and tournaments Number tables and its properties Pigeonhole principle Pigeonhole principle (other)

12–14 4.0

Eight schoolchildren solved \(8\) tasks. It turned out that \(5\) schoolchildren solved each problem. Prove that there are two schoolchildren, who solved every problem at least once.
If each problem is solved by \(4\) pupils, prove that it is not necessary to have two schoolchildren who would solve each problem.

Problem #PRU-107810

Problems Discrete Mathematics Combinatorics Geometry on grid paper Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

13–15 3.0

The centres of all unit squares are marked in a \(10 \times 10\) chequered box (100 points in total). What is the smallest number of lines, that are not parallel to the sides of the square, that are needed to be drawn to erase all of the marked points?