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-102995

Problems Algebra Functional equations

14–16 4.0

Can there exist two functions $f$ and $g$ that take only integer values such that for any integer $x$ the following relations hold:

a) $f (f (x)) = x$ , $g (g (x)) = x$ , $f (g (x)) > x$ , $g (f (x)) > x$ ?

b) $f (f (x)) < x$ , $g (g (x)) < x$ , $f (g (x)) > x$ , $g (f (x)) > x$ ?

Problem #PRU-107992

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Algebra Sequences Algebraic methods Symmetry and involutions

14–16 5.0

For each pair of real numbers $a$ and $b$ , consider the sequence of numbers $p_{n} = ⌊ 2 {a n + b} ⌋$ . Any $k$ consecutive terms of this sequence will be called a word. Is it true that any ordered set of zeros and ones of length $k$ is a word of the sequence given by some $a$ and $b$ for $k = 4$ ; when $k = 5$ ?

Note: $⌊ c ⌋$ is the integer part, ${c}$ is the fractional part of the number $c$ .

Problem #PRU-109599

Problems Calculus Number sequences Boundedness, monotonicity Algebra Polynomials Algebraic identities for polynomials Identical transformations Sequences

14–16 5.0

Prove that for any natural number $a_{1} > 1$ there exists an increasing sequence of natural numbers $a_{1}, a_{2}, a_{3}, \dots$ , for which $a_{1}^{2} + a_{2}^{2} + \dots + a_{k}^{2}$ is divisible by $a_{1} + a_{2} + \dots + a_{k}$ for all $k \geq 1$ .

Problem #PRU-111763

Problems Calculus Derivative Calculating a derivative Algebra Absolute value Inequalities containing absolute values Polynomials Quadratic polynomials

15–16 4.0

The quadratic trinomials $f (x)$ and $g (x)$ are such that $f^{'} (x) g^{'} (x) \geq | f (x) | + | g (x) |$ for all real $x$ . Prove that the product $f (x) g (x)$ is equal to the square of some trinomial.

Problem #PRU-35621

Problems Calculus Integral Calculating integrals Algebra Trigonometry Identical transformations (trigonometry) Methods Algebraic methods Symmetry and involutions

16–16 4.0

Calculate $\int_{0}^{π / 2} (\sin^{2} (\sin x) + \cos^{2} (\cos x)) d x$ .

Problem #PRU-60389

Problems Algebra Algebraic equations and systems of equations Binomial theorem

14–16 3.0

How many rational terms are contained in the expansion of

a) $(\sqrt{2} + \sqrt[4]{3})^{100}$ ;

b) $(\sqrt{2} + \sqrt[3]{3})^{300}$ ?

Problem #PRU-79400

Problems Algebra Algebra Functional equations Calculus Functions of one variable. Continuity Periodicity and aperiodicity

16–16 3.0

We consider a function $y = f (x)$ defined on the whole set of real numbers and satisfying $f (x + k) \times (1 - f (x)) = 1 + f (x)$ for some number $k \neq 0$ . Prove that $f (x)$ is a periodic function.

Problem #PRU-79595

Problems Algebra Algebra Functional equations

14–16 4.0

The function $f (x)$ for each real value of $x \in (- \infty, + \infty)$ satisfies the equality $f (x) + (x + 1 / 2) \times f (1 - x) = 1$ .

a) Find $f (0)$ and $f (1)$ . b) Find all such functions $f (x)$ .

Problem #PRU-97817

Problems Calculus Number sequences Boundedness, monotonicity Algebra Sequences

15–18 4.0

$a_{1}, a_{2}, a_{3}, \dots$ is an increasing sequence of natural numbers. It is known that $a_{a_{k}} = 3 k$ for any $k$ . Find a) $a_{100}$ ; b) $a_{2022}$ .