We solve problems

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

In the government of the planet of liars and truth tellers there are \(101\) ministers. In order to reduce the budget, it was decided to reduce the number of ministers by \(1.\) But each of the ministers said that if they were to be removed from the government, then the majority of the remaining ministers would be liars. How many truth tellers and how many liars are there in the government?

Problem #PRU-35512

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 3.0

Prove that in any group of friends there will be two people who have the same number of friends.

Problem #PRU-35534

Problems Methods Pigeonhole principle Pigeonhole principle (other)

12–14 2.0

In chess, ‘check’ is when the king is under threat of capture from another piece. What is the largest number of kings that it is possible to place on a standard \(8\times 8\) chess board so that no two check one another.

Problem #PRU-35541

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Solve the equation \(2x^x = \sqrt {2}\) for positive numbers.

Problem #PRU-35558

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

Let \(M\) be a finite set of numbers. It is known that among any three of its elements there are two, the sum of which belongs to \(M\).

What is the largest number of elements in \(M\)?

Problem #PRU-35563

Problems Discrete Mathematics Algorithm Theory Theory of algorithms (other)

11–14 2.0

Each of the three cutlets should be fried in a pan on both sides for five minutes each side. Only two cutlets can fit onto the frying pan. Is it possible to fry all three cutlets more quickly than in 20 minutes (if the time to turn over and transfer the cutlets is neglected)?

Problem #PRU-35566

Problems Geometry Solid geometry Lines and planes in space Methods Pigeonhole principle

14–15 2.0

An area of airspace contains clouds. It turns out that the area can be divided by 10 aeroplanes into regions such that each region contains no more than one cloud. What is the largest number of clouds an aircraft can fly through whilst holding a straight line course.

Problem #PRU-35573

Problems Calculus Functions of one variable. Continuity

15–16 4.0

Let \(f\) be a continuous function defined on the interval \([0; 1]\) such that \(f (0) = f (1) = 0\). Prove that on the segment \([0; 1]\) there are 2 points at a distance of 0.1 at which the function \(f 4(x)\) takes equal values.

Problem #PRU-35575

Problems Methods Methods of calculus Continuity considerations Geometry Plane geometry Convex and non-convex figures Convex polygons Calculus Functions of one variable. Continuity Intermediate value theorem. Connectedness.

14–16 5.0

A convex figure and point \(A\) inside it are given. Prove that there is a chord (that is, a segment joining two boundary points of a convex figure) passing through point \(A\) and dividing it in half at point \(A\).

Problem #PRU-35578

Problems Methods Extremal principle Extremal principle (other)

13–14 2.0

In how many ways can you rearrange the numbers from 1 to 100 so that the neighbouring numbers differ by no more than 1?