We solve problems

Problems Probability and statistics Probability theory Probability theory (other)

When boarding a plane, a line of $n$ passengers was formed, each of whom has a ticket for one of the $n$ places. The first in the line is a crazy old man. He runs onto the plane and sits down in a random place (perhaps, his own). Then passengers take turns to take their seats, and in the case that their place is already occupied, they sit randomly on one of the vacant seats. What is the probability that the last passenger will take his assigned seat?

Problem #PRU-35371

Problems Methods Pigeonhole principle Pigeonhole principle (other)

13–14 3.0

We are given 101 natural numbers whose sum is equal to 200. Prove that we can always pick some of these numbers so that the sum of the picked numbers is 100.

Problem #PRU-35379

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra and arithmetic Algebraic equations and systems of equations Systems of linear equations

14–15 3.0

Find all functions $f (x)$ defined for all real values of $x$ and satisfying the equation $2 f (x) + f (1 - x) = x^{2}$ .

Problem #PRU-35415

Problems Algebra and arithmetic Number theory. Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

13–15 4.0

10 natural numbers are written on a blackboard. Prove that it is always possible to choose some of these numbers and write “ $+$ ” or “ $-$ ” between them so that the resulting algebraic sum is divisible by 1001.

Problem #PRU-35428

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–14 2.0

On the board the number 1 is written. Two players in turn add any number from 1 to 5 to the number on the board and write down the total instead. The player who first makes the number thirty on the board wins. Specify a winning strategy for the second player.

Problem #PRU-35429

Problems Set theory and logic Theory of algotithms Game theory Symmetric strategies

12–14 2.0

There are two stacks of coins on a table: in one of them there are 30 coins, and in the other – 20. You can take any number of coins from one stack per move. The player who cannot make a move is the one that loses. Which player wins with the correct strategy?

Problem #PRU-35476

Problems Calculus Functions of one variable. Continuity Methods Algebraic methods Iterations

15–16 3.0

Let $f (x)$ be a polynomial about which it is known that the equation $f (x) = x$ has no roots. Prove that then the equation $f (f (x)) = x$ does not have any roots.

Problem #PRU-35483

Problems Methods Pigeonhole principle Pigeonhole principle (other)

14–16 4.0

Prove that there exist numbers, that can be presented in no fewer than 100 ways in the form of a summation of 20001 terms, each of which is the 2000th power of a whole number.

Problem #PRU-35501

Problems Geometry Plane geometry Polygons Polygons (other)

12–14 2.0

Arrows are placed on the sides of a polygon. Prove that the number of vertices in which two arrows converge is equal to the number of vertices from which two arrows emerge.

Problem #PRU-35503

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

12–14 2.0

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?