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Problems Methods Examples and counterexamples. Constructive proofs Set theory and logic Algorithm Theory

On a table, there are five coins lying in a row: the middle one lies with a head facing upwards, and the rest lie with the tails side up. It is allowed to simultaneously flip three adjacent coins. Is it possible to make all five coins positioned with the heads side facing upwards with the help of several such overturns?

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-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-32038

Problems Methods Examples and counterexamples. Constructive proofs Geometry Plane geometry Lines, segments, rays, and angles Measurement of segments and angles. Adjacent angles.

11–13 2.0

Is it possible to draw five lines from one point on a plane so that there are exactly four acute angles among the angles formed by them? Angles between not only neighboring rays, but between any two rays, can be considered.

Problem #PRU-32861

Problems Algebra Number theory. Divisibility Divisibility of a number. General properties Methods Examples and counterexamples. Constructive proofs

12–12 2.0

A country is called a Fiver if, in it, each city is connected by airlines with exactly with five other cities (there are no international flights).

a) Draw a scheme of airlines for a country that is made up of 10 cities.

b) How many airlines are there in a country of 50 cities?

c) Can there be a Fiver country, in which there are exactly 46 airlines?

Problem #PRU-35662

Problems Set theory and logic Algorithm Theory Balance puzzles Methods Examples and counterexamples. Constructive proofs

12–13 2.0

What weights can three weights have so that they can weigh any integer number of kilograms from 1 to 10 on weighing scales (weights can be put on both cups)? Give an example.

Problem #PRU-89945

Problems Methods Examples and counterexamples. Constructive proofs

10–12 2.0

Five oaks are planted along two linear park alleys in such a way that there are three oaks along each alley, see picture. Where should we plant the sixth oak so that it will be possible to lay two more linear alleys, along each of which there would also be three oak trees growing?

Problem #PRU-98648

Problems Methods Examples and counterexamples. Constructive proofs Set theory and logic Mathematical logic Mathematical logic (other)

11–12 2.0

Elephants, rhinoceroses, giraffes. In all zoos where there are elephants and rhinoceroses, there are no giraffes. In all zoos where there are rhinoceroses and there are no giraffes, there are elephants. Finally, in all zoos where there are elephants and giraffes, there are also rhinoceroses. Could there be a zoo in which there are elephants, but there are no giraffes and no rhinoceroses?

Problem #PRU-98670

Problems Methods Examples and counterexamples. Constructive proofs

11–12 2.0

Among 4 people there are no three with the same name, the same middle name and the same surname, but any two people have either the same first name, middle name or surname. Can this be so?