We solve problems

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

Peter thought of a number between 1 to 200. What is the fewest number of questions for which you can guess the number if Peter answers

a) “yes ” or “no”;

b) “yes”, “no” or “I do not know”

for every question?

Problem #PRU-60921

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

11–12 2.0

There are 4 coins. Of the four coins, one is fake (it differs in weight from the real ones, but it is not known if it is heavier or lighter). Find the fake coin using two weighings on scales without weights.

Problem #PRU-61215

Problems Calculus Functions of one variable. Continuity Periodicity and aperiodicity Algebra and arithmetic Trigonometry Trigonometry (other)

14–16 2.0

Prove that the function $\cos \sqrt{x}$ is not periodic.

Problem #PRU-61421

Problems Algebra and arithmetic 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

Draw all of the stairs made from four bricks in descending order, starting with the steepest $(4, 0, 0, 0)$ and ending with the shallowest $(1, 1, 1, 1)$ .

Problem #PRU-61473

Problems Algebra and arithmetic 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 frog jumps over the vertices of the triangle $A B C$ , moving each time to one of the neighbouring vertices.

How many ways can it get from $A$ to $A$ in $n$ jumps?

Problem #PRU-61486

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Let $(1 + \sqrt{2} + \sqrt{3})^{n} = p_{n} + q_{n} \sqrt{2} + r_{n} \sqrt{3} + s_{n} \sqrt{6}$ for $n \geq 0$ . Find:

a) $lim_{n \to \infty} \frac{p_{n}}{q_{n}}$ ; b) $lim_{n \to \infty} \frac{p_{n}}{r_{n}}$ ; c) $lim_{n \to \infty} \frac{p_{n}}{s_{n}}$ ;

Problem #PRU-61507

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

Find the generating functions of the sequences of Chebyshev polynomials of the first and second kind: $F_{T} (x, z) = \sum_{n = 0}^{\infty} T_{n} (x) z^{n}; F_{U} (x, z) = \sum_{n = 0}^{\infty} U_{n} (X) z^{n} .$

Definitions of Chebyshev polynomials can be found in the handbook.

Problem #PRU-61525

Problems Geometry Solid geometry Visual geometry in space

11–13 2.0

We denote by $P_{k, l} (n)$ the number of partitions of the number $n$ into at most $k$ terms, each of which does not exceed $l$ . Prove the equalities:

a) $P_{k, l} (n) - P_{k, l - 1} (n) = P_{k - 1, l} (n - l)$ ;

b) $P_{k, l} (n) - P_{k - 1, l} (n) = P_{k, l - 1} (n - k)$ ;

c) $P_{k, l} (n) = P_{l, k} (n)$ ;

d) $P_{k, l} (n) = P_{k, l} (k l - n)$ .

Problem #PRU-64305

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

11–12 2.0

Author: D.E. Shnol

On the island of Truthland, all of the inhabitants may be mistaken, but the younger ones never contradict the elders, and when the older ones contradict the younger ones, they (the elders) are not mistaken. Between the residents A, B and C there was such a conversation:

A: B is the tallest.

B: A is the tallest.

C: I’m taller than B.

Does it follow from this conversation that the younger the person, the taller he or she is (for the three people having this conversation)?

Problem #PRU-64315

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

11–12 2.0

Author: I.V. Izmestyev

Postman Pat did not want to give away the parcel. So, Matt suggested that he play the following game: every move, Pat writes in a line from left to right the letters M and P, randomly alternating them, until he has a line made up of 11 letters. Matt, after each of Pat’s moves, if he wants, swaps any two letters. If in the end it turns out that the recorded word is a palindrome (that is, it is the same if read from left to right and right to left), then Pat gives Matt the parcel. Can Matt play in such a way as to get the parcel?