We solve problems

Problem #PRU-103818

Problems Geometry Plane geometry Quadrilateral Parallelogram Special cases Rectangles and squares. Criteria and properties

11–12 2.0

Which rectangles with whole sides are there more of: with perimeter of 1996 or with perimeter of 1998? (The rectangles \(a \times b\) and \(b \times a\) are assumed to be the same).

Problem #PRU-103824

Problems Geometry Solid geometry Spheres Circle on a sphere Visual geometry in space

11–13 2.0

A globe has 17 parallels and 24 meridians. How many parts is the globe’s surface divided into? The meridian is an arc connecting the North Pole with the South Pole. A parallel is a circle parallel to the equator (the equator is also a parallel).

Problem #PRU-103825

Problems Algebra and arithmetic Arithmetics. Mental maths Methods Examples and counterexamples. Constructive proofs Discrete Mathematics Algorithm Theory Theory of algorithms (other)

10–12 2.0

Three hedgehogs divided three pieces of cheese of mass of 5g, 8g and 11g. The fox began to help them. It can cut off and eat 1 gram of cheese from any two pieces at the same time. Can the fox leave the hedgehogs equal pieces of cheese?

Problem #PRU-103827

Problems Dissections

11–11 2.0

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

Problem #PRU-103833

Problems Discrete Mathematics Algorithm Theory Theory of algorithms (other)

10–12 2.0

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-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-103873

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

11–12 3.0

Jessica, Nicole and Alex received 6 coins between them: 3 gold coins and 3 silver coins. Each of them received 2 coins. Jessica doesn’t know which coins the others received but only which coins she has. Think of a question which Jessica can answer with either “yes”, “no” or “I don’t know” such that from the answer you can know which coins Jessica has.

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?