We solve problems

Problems Algebra and arithmetic Algebraic equations and systems of equations Diophantine equations

Determine all the integer solutions for the equation $21 x + 48 y = 6$ .

Problem #PRU-32088

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

It is known that a certain polynomial at rational points takes rational values. Prove that all its coefficients are rational.

Problem #PRU-32132

Problems Methods Extremal principle Extremal principle (other) Set theory and logic Order relations Algebraic methods Proof by exhaustion

13–15 3.0

The order of books on a shelf is called wrong if no three adjacent books are arranged in order of height (either increasing or decreasing). How many wrong orders is it possible to construct from $n$ books of different heights, if: a) $n = 4$ ; b) $n = 5$ ?

Problem #PRU-34872

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations.

14–15 3.0

On a function $f (x)$ , defined on the entire real line, it is known that for any $a > 1$ the function $f (x) + f (a x)$ is continuous on the whole line. Prove that $f (x)$ is also continuous on the whole line.

Problem #PRU-35079

Problems Methods Proof by contradiction Calculus Real numbers Rational and irrational numbers

14–15 3.0

Prove that the infinite decimal $0.1234567891011121314 \dots$ (after the decimal point, all of the natural numbers are written out in order) is an irrational number.

Problem #PRU-35154

Problems Methods Examples and counterexamples. Constructive proofs Pigeonhole principle Pigeonhole principle (other) Set theory and logic Theory of algotithms Theory of algorithms (other)

13–16 4.0

Prove that multiplying the polynomial $(x + 1)^{n - 1}$ by any polynomial different from zero, we obtain a polynomial having at least $n$ nonzero coefficients.

Problem #PRU-35298

Problems Algebra and arithmetic Algebraic equations and systems of equations Higher order equations. Palindromic polynomial equations Equations of higher order (other)

13–15 3.0

Determine all solutions of the equation $(n + 2)! - (n + 1)! - n! = n^{2} + n^{4}$ in natural numbers.

Problem #PRU-35438

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

2.0

In a group of friends, each two people have exactly five common acquaintances. Prove that the number of pairs of friends is divisible by 3.

Problem #PRU-35457

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Prove that the equation $a_{1} \sin x + b_{1} \cos x + a_{2} \sin 2 x + b_{2} \cos 2 x + \dots + a_{n} \sin n x + b_{n} \cos n x = 0$ has at least one root for any values of $a_{1}, b_{1}, a_{2}, b_{2}, \dots, a_{n}, b_{n}$ .

Problem #PRU-35541

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Methods Pigeonhole principle Pigeonhole principle (angles and lengths)

13–15 3.0

Solve the equation $2 x^{x} = \sqrt{2}$ for positive numbers.