We solve problems

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

A traveller who came to the planet hired a local as a guide. They went for a walk and saw another alien. The traveller sent the guide to find out to whether this native is a liar or truth teller. The guide returned and said: “The native says that they are a truth teller.” Who was the guide: a liar or a truth teller?

Problem #PRU-35344

Problems Number Theory Divisibility Division with remainders. Arithmetic of remainders Division with remainder Methods Pigeonhole principle Pigeonhole principle (other)

13–14 3.0

Prove that if $a, b, c$ are odd numbers, then at least one of the numbers $a b - 1$ , $b c - 1$ , $c a - 1$ is divisible by 4.

Problem #PRU-35347

Problems Discrete Mathematics Algorithm Theory Balance puzzles

12–13 2.0

In a basket there are 13 apples. There are scales, with which you can find out the total weight of any two apples. Think of a way to find out from 8 weighings the total weight of all the apples.

Problem #PRU-35352

Problems Probability and statistics Probability theory Probability theory (other)

14–16 3.0

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-35362

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

A raisin bag contains 2001 raisins with a total weight of 1001 g, and no raisin weighs more than 1.002 g.

Prove that all the raisins can be divided onto two scales so that they show a difference in weight not exceeding 1 g.

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 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-35410

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

10 numbers are written around the circle, the sum of which is equal to 100. It is known that the sum of every three numbers standing side by side is not less than 29.

Specify the smallest number $A$ such that in any such set of numbers each of the numbers does not exceed $A$ .

Problem #PRU-35415

Problems 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-35427

Problems Discrete Mathematics Algorithm Theory Game theory Symmetric strategies

12–14 2.0

A daisy has a) 12 petals; b) 11 petals. Consider the game with two players where: in one turn a player is allowed to remove either exactly one petal or two petals which are next to each other. The loser is the one who cannot make a turn. How should the second player act, in cases a) and b), in order to win the game regardless of the moves of the first player?