We solve problems

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

In a room, there are 85 balloons – red and blue ones. It is known that: 1) at least one of the balls is red, 2) from each arbitrarily chosen pair of balls at least one is blue. How many red balls are there in the room?

Problem #PRU-103005

Problems Algebra Word Problems

10–11 2.0

Liz is 8 years older than Natasha. Two years ago Liz’s age was 3 times greater than Natasha’s. How old is Liz?

Problem #PRU-103008

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

10–11 2.0

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 Set theory and logic Algorithm Theory Algorithm Theory

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 Set theory and logic Algorithm Theory Algorithm Theory

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 Algebra Number theory. Divisibility Odd and even numbers

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 Set theory and logic Set theory Union, intersection, and set difference

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 Algebra Number systems Decimal number system

11–11 2.0

Find numbers equal to twice the sum of their digits.

Problem #PRU-103749

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

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

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?