We solve problems

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-35561

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

11–12 2.0

Someone arranged a 10-volume collection of works in an arbitrary order. We call a “disturbance” a situation where there are two volumes for which a volume with a large number is located to the left. For this volume arrangement, we call the number \(S\) the number of all of the disturbances. What values can \(S\) take?

Problem #PRU-35563

Problems Set theory and logic Algorithm Theory Algorithm Theory

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-35571

Problems Methods Pigeonhole principle Pigeonhole principle (finite number of poits, lines etc.)

12–13 2.0

A standard chessboard has more than a quarter of its squares filled with chess pieces. Prove that at least two adjacent squares, either horizontally, vertically, or diagonally, are occupied somewhere on the board.

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?

Problem #PRU-35584

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

11–12 2.0

There are 18 sweets in one piles, and 23 in another. Two play a game: in one go one can eat one pile of sweets, and the other can be divided into two piles. The loser is one who cannot make a move, i.e. before this player’s turn there are two piles of sweets with one sweet in each. Who wins with a regular game?

Problem #PRU-35592

Problems Set theory and logic Algorithm Theory Algorithm Theory

13–14 2.0

Your task is to find out a five-digit phone number, asking questions that can be answered with either “yes” or “no.” What is the smallest number of questions for which this can be guaranteed (provided that the questions are answered correctly)?