We solve problems

Problems Geometry Algebra and arithmetic Arithmetics. Mental maths Solid geometry Ratio of volumes Volume

When Gulliver came to Lilliput, he found that there all things were exactly 12 times shorter than in his homeland. Can you say how many Lilliputian matchboxes fit into one of Gulliver’s matchboxes?

Problem #PRU-103009

Problems Algorithm Theory Theory of algorithms (other) Discrete Mathematics

10–12 2.0

There are 6 locked suitcases and 6 keys to them. At the same time, it is not known to which suitcase each key fits. What is the smallest number of attempts you need to make in order to open all the suitcases for sure? And how many attempts will it take there are not 6 but 10 keys and suitcases?

Problem #PRU-103027

Problems Algorithm Theory Theory of algorithms (other) Discrete Mathematics

10–12 2.0

Decipher the following rebus (see the figure). Despite the fact that only two figures are known here, and all others are replaced by asterisks, the example can be restored.

Problem #PRU-103729

Problems Divisibility Odd and even numbers Number Theory

10–12 2.0

The parliament of a certain country has two houses with an equal number of members. In order to make a decision on an important issue all the members voted and there were no abstentions. When the chairman announced that the decision had been taken with a 23-vote advantage, the opposition leader declared that the results had been rigged. How did he know it?

Problem #PRU-103738

Problems Union, intersection, and set difference Discrete Mathematics Set theory and logic Set theory

11–12 2.0

Among some number of mathematicians, every seventh is a philosopher, and among some number of philosophers every ninth is a mathematician. Who are there more of: philosophers or mathematicians?

Problem #PRU-103744

Problems Numeral systems Decimal number system Number Theory

11–11 2.0

Find a nonzero number that is equal to twice the sum of its digits.

Problem #PRU-103749

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

10–12 2.0

Know-it-all came to visit the twin brothers Screw and Nut, knowing that one of them never speaks the truth, and asked one of them: “Are you Screw?”. “Yes,” he replied. When Know-it-all asked the second brother the same question, he received an equally clear answer and immediately determined who was who.

Who was called Screw?

Problem #PRU-103757

Problems Algebraic equations and systems of equations Algebra Higher order equations. Palindromic polynomial equations Equations of higher order (other) Real numbers Integer and fractional parts. Archimedean property Calculus

12–14 2.0

A resident of one foreign intelligence agency informed the centre about the forthcoming signing of a number of bilateral agreements between the fifteen former republics of the USSR. According to his report, each of them will conclude an agreement exactly with three others. Should this resident be trusted?

Problem #PRU-103764

Problems Order relations Discrete Mathematics Set theory and logic

11–11 2.0

Uncle Jack, the cat Whiskers, Spot and postman Pat are sitting on a bench. If Spot, sitting to the right of everyone, sits between Uncle Jack and the cat, then the cat will be at the extreme left. In what order do they sit?

Problem #PRU-103778

Problems Various dissection problems Discrete Mathematics Combinatorics Dissections, partitions, covers and tilings

11–11 2.0

Cut a square into three pieces, from which you can construct a triangle with three acute angles and three different sides.