We solve problems

Problem #PRU-61350

Problems Algebra Algebraic equations and systems of equations Discrete Mathematics Set theory and logic Mathematical logic Weightings

15–17 5.0

There are 13 weights. It is known that any 12 of them could be placed in 2 scale cups with 6 weights in each cup in such a way that balance will be held.

Prove the mass of all the weights is the same, if it is known that:

a) the mass of each weight in grams is an integer;

b) the mass of each weight in grams is a rational number;

c) the mass of each weight could be any real (not negative) number.

Problem #PRU-61353

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities

11–14 2.0

Prove that $\sqrt{\frac{a^{2} + b^{2}}{2}} \geq \frac{a + b}{2}$ .

Problem #PRU-61392

Problems Algebra Algebraic inequalities and systems of inequalities Classical inequalities

13–15 3.0

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-61399

Problems Number Theory Divisibility Equations in integer numbers Methods Extremal principle Sorting in increasing (decreasing) order

13–15 2.0

Prove that the equation $\frac{x}{y} + \frac{y}{z} + \frac{z}{x} = 1$ is unsolvable using positive integers.

Problem #PRU-61403

Problems Calculus Derivative Increasing and decreasing. Researching functions Algebra 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-61406

Problems Calculus Derivative Convexity and concavity

15–16 3.0

Prove that if the function $f (x)$ is convex upwards on the line $[a, b]$ , then for any distinct points $x_{1}, x_{2}$ in $[a; b]$ and for any positive $α_{1}, α_{2}$ such that $α_{1} + α_{2} = 1$ the following inequality holds: $f (α_{1} x_{1} + α_{2} x_{2}) > α_{1} f (x_{1}) + α_{2} f (x_{2})$ .

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 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 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-61429

Problems Algebra Sequences Sums of number sequences and difference series

13–16 2.0

Let the sequences of numbers ${a_{n}}$ and ${b_{n}}$ , that are associated with the relation $Δ b_{n} = a_{n}$ ( $n = 1, 2, \dots$ ), be given. How are the partial sums $S_{n}$ of the sequence ${a_{n}}$ $S_{n} = a_{1} + a_{2} + \dots + a_{n}$ linked to the sequence ${b_{n}}$ ?