We solve problems

Problem #PRU-103825

Problems Algebra Arithmetics. Mental maths Methods Examples and counterexamples. Constructive proofs Set theory and logic Algorithm Theory Algorithm Theory

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 Set theory and logic Algorithm Theory Algorithm Theory

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 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 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 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 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 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-103885

Problems Combinatorics Dissections, partitions, covers and tilings Tilings with ordinary and domino tiles

11–12 2.0

At the disposal of a tile layer there are 10 identical tiles, each of which consists of 4 squares and has the shape of the letter L (all tiles are oriented the same way). Can he make a rectangle with a size of \(5 \times 8\)? (The tiles can be rotated, but you cannot turn them over). For example, the figure shows the wrong solution: the shaded tile is incorrectly oriented.

Problem #PRU-103888

Problems Methods Algebraic methods Proof by exhaustion Geometry Solid geometry Visual geometry in space

11–12 2.0

A square napkin was folded in half, the resulting rectangle was then folded in half again (see the figure). The resulting square was then cut with scissors (in a straight line). Could the napkin have been broken up a) into 2 parts? b) into 3 parts? c) into 4 parts? d) into 5 parts? If yes – illustrate such a cut, if not – write the word “no”.