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Problem #PRU-107987

Problems Calculus Functions of several variables Algebra and arithmetic Polynomials Algebraic identities for polynomials Identical transformations

13–15 4.0

To each pair of numbers $x$ and $y$ some number $x * y$ is placed in correspondence. Find $1993 * 1935$ if it is known that for any three numbers $x, y, z$ , the following identities hold: $x * x = 0$ and $x * (y * z) = (x * y) + z$ .

Problem #PRU-109787

Problems Algebra and arithmetic Polynomials Algebraic identities for polynomials Factoring polynomials Calculus Real numbers Rational and irrational numbers

14–16 4.0

A numeric set $M$ containing 2003 distinct numbers is such that for every two distinct elements $a, b$ in $M$ , the number $a^{2} + b \sqrt{2}$ is rational. Prove that for any $a$ in $M$ the number $q \sqrt{2}$ is rational.

Problem #PRU-32884

Problems Methods Extremal principle Extremal principle (other) Mathematical induction Mathematical induction (other) Algebra and arithmetic Polynomials Polynomial remainder theorem (little Bezout's theorem). Factorisation. Polynomials with integer coefficients and integer values Proof by contradiction Calculus Real numbers Rational and irrational numbers

12–12 3.0

Prove that if the irreducible rational fraction $p / q$ is a root of the polynomial $P (x)$ with integer coefficients, then $P (x) = (q x - p) Q (x)$ , where the polynomial $Q (x)$ also has integer coefficients.

Problem #PRU-60855

Problems Algebra and arithmetic Polynomials Quadratic polynomials Quadratic equations. Formula for the roots. Quadratic equations. Vieta's rule. Calculus Real numbers Rational and irrational numbers

13–15 3.0

One of the roots of the equation $x^{2} + a x + b = 0$ is $1 + \sqrt{3}$ . Find $a$ and $b$ if you know that they are rational.

Problem #PRU-61013

Problems Algebra and arithmetic Polynomials Polynomials with integer coefficients and integer values Calculus Real numbers Rational and irrational numbers Number theory. Divisibility The greatest common divisor (GCD) and the least common multiplier (LCM). Mutually prime numbers

13–15 3.0

Prove that if $(p, q) = 1$ and $p / q$ is a rational root of the polynomial $P (x) = a_{n} x^{n} + \dots + a_{1} x + a_{0}$ with integer coefficients, then

a) $a_{0}$ is divisible by $p$ ;

b) $a_{n}$ is divisible by $q$ .

Problem #PRU-61018

Problems Calculus Derivative Derivative and multiple roots Algebra and arithmetic Polynomials Polynomial remainder theorem (little Bezout's theorem). Factorisation.

15–16 3.0

Prove that the root a of the polynomial $P (x)$ has multiplicity greater than 1 if and only if $P (a) = 0$ and $P^{'} (a) = 0$ .

Problem #PRU-61022

Problems Calculus Derivative Derivative and multiple roots Algebra and arithmetic Polynomials Division of polynomials with remainders. The greatest common divisor (GCD) and the least common multiple (LCM) for polynomilas.

14–16 3.0

For which $A$ and $B$ does the polynomial $A x^{n + 1} + B x^{n} + 1$ have the number $x = 1$ at least two times as its root?

Problem #PRU-65726

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Algebra and arithmetic Polynomials Quadratic polynomials Investigating the quadratic polynomial

13–15 3.0

Author: A. Khrabrov

Do there exist integers $a$ and $b$ such that

a) the equation $x^{2} + a x + b = 0$ does not have roots, and the equation $⌊ x^{2} ⌋ + a x + b = 0$ does have roots?

b) the equation $x^{2} + 2 a x + b = 0$ does not have roots, and the equation $⌊ x^{2} ⌋ + 2 a x + b = 0$ does have roots?

Note that here, square brackets represent integers and curly brackets represent non-integer values or 0.

Problem #PRU-77989

Problems Methods Examples and counterexamples. Constructive proofs Algebra and arithmetic Polynomials Quadratic polynomials Quadratic equations and systems of equations

13–16 4.0

The equations $\begin{matrix} (1) & a x^{2} + b x + c = 0 \end{matrix}$ and $\begin{matrix} (2) & - a x^{2} + b x + c \end{matrix}$ are given. Prove that if $x_{1}$ and $x_{2}$ are, respectively, any roots of the equations (1) and (2), then there is a root $x_{3}$ of the equation $\frac{1}{2} a x^{2} + b x + c$ such that either $x_{1} \leq x_{3} \leq x_{2}$ or $x_{1} \geq x_{3} \geq x_{2}$ .

Problem #PRU-78003

Problems Algebra and arithmetic Number theory. Divisibility Arithmetic functions Polynomials Polynomials(other)

15–16 3.0

Prove that if $x_{0}^{4} + a_{1} x_{0}^{3} + a_{2} x_{0}^{2} + a_{3} x_{0} + a_{4}$ and $4 x_{0}^{3} + 3 a_{1} x_{0}^{2} + 2 a_{2} x_{0} + a_{3} = 0$ then $x^{4} + a_{1} x^{3} + a_{2} x^{2} + a_{3} x + a_{4}$ is divisible by $(x - x_{0})^{2}$ .