We solve problems

Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Classical inequalities

We are given rational positive numbers $p, q$ where $1 / p + 1 / q = 1$ . Prove that for positive $a$ and $b$ , the following inequality holds: $a b \leq \frac{a^{p}}{p} + \frac{b^{q}}{q}$ .

Problem #PRU-61403

Problems Calculus Derivative Increasing and decreasing. Researching functions Algebra and arithmetic Surds. Rational powers. Irrational equations.

15–16 3.0

Prove the inequality: $\frac{(b_{1} + \dots b_{n})^{b_{1} + \dots b_{n}}}{(a_{1} + \dots a_{n})^{b_{1} + \dots + b_{n}}} \leq {(\frac{b_{1}}{a_{1}})}^{b_{1}} \dots {(\frac{b_{n}}{a_{n}})}^{b_{n}}$ where all variables are considered positive.

Problem #PRU-61407

Problems Calculus Derivative Convexity and concavity

15–16 3.0

Inequality of Jensen. Prove that if the function $f (x)$ is convex upward on $[a, b]$ , then for any distinct points $x_{1}, x_{2}, \dots, x_{n}$ ( $n \geq 2$ ) from $[a; b]$ and any positive $α_{1}, α_{2}, \dots, α_{n}$ such that $α_{1} + α_{2} + \dots + α_{n} = 1$ , the following inequality holds: $f (α_{1} x_{1} + \dots + α_{n} x_{n}) > α_{1} f (x_{1}) + \dots + α_{n} f (x_{n})$ .

Problem #PRU-61411

Problems Algebra and arithmetic Algebraic inequalities and systems of inequalities Algebraic inequalities (other)

13–15 3.0

Let $p$ and $q$ be positive numbers where $1 / p + 1 / q = 1$ . Prove that $a_{1} b_{1} + a_{2} b_{2} + \dots + a_{n} b_{n} \leq (a_{1}^{p} + \dots a_{n}^{p})^{1 / p} (b_{1}^{q} + \dots + b_{n}^{q})^{1 / q}$ The values of the variables are considered positive.

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-61457

Problems Calculus Functions of several variables

13–16 4.0

Liouville’s discrete theorem. Let $f (x, y)$ be a bounded harmonic function (see the definition in problem number 11.28), that is, there exists a positive constant $M$ such that $\forall (x, y) \in Z^{2}$ $| f (x, y) | \leq M$ . Prove that the function $f (x, y)$ is equal to a constant.

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