We solve problems

Cut the trapezium \(ABCD\) into two parts which you can use to construct a triangle.

Problems Discrete Mathematics Algorithm Theory Theory of algorithms (other)

On the island of Contrast, both knights and liars live. Knights always tell the truth, liars always lie. Some residents said that the island has an even number of knights, and the rest said that the island has an odd number of liars. Can the number of inhabitants of the island be odd?

Problem #PRU-103843

Problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings

12–12 2.0

Cut the figure (on the boundaries of cells) into three equal parts (the same in shape and size).

Problem #PRU-103848

Problems Discrete Mathematics Set theory and logic Mathematical logic Puzzles

12–13 2.0

In the line of numbers and signs \({}* 1 * 2 * 4 * 8 * 16 * 32 * 64 = 27\) position the signs “\(+\)” or “\(-\)” instead of the signs “\(*\)”, so that the equality becomes true.

Problem #PRU-103850

Problems Discrete Mathematics Set theory and logic Mathematical logic Puzzles

12–13 2.0

The code of lock is a two-digit number. Ben forgot the code, but he remembers that the sum of the digits of this number, combined with their product, is equal to the number itself. Write all possible code options so that Ben could open the lock quickly.

Problem #PRU-103858

Problems Number Theory Divisibility The fundamental theorm of arithmetic. Prime factorisation.

12–13 2.0

Solve the rebus: \(AX \times UX = 2001\).

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-103883

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

11–12 2.0

A space traveller decided to visit the planet and upon arrival he met three inhabitants. The traveller asked each of the inhabitants: “How many truth tellers are there among your companions?”. The first one answered: “None.” The second one said: “One”. What did the third alien say?

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?