We solve problems

Problem #PRU-104104

Problems Calculus Functions of one variable. Continuity Certain properties of a function and recurrence relations. Algebra Polynomials Quadratic polynomials

12–14 2.0

Find all functions $f (x)$ such that $f (2 x + 1) = 4 x^{2} + 14 x + 7$ .

Problem #PRU-109946

Problems Algebra Algebraic equations and systems of equations Algebraic equations and systems of equations (other) Calculus Real numbers Integer and fractional parts. Archimedean property Polynomials Quadratic polynomials Investigating the quadratic polynomial

14–15 3.0

Solve the equation $(x + 1)^{3} = x^{3}$ .

Problem #PRU-111345

Problems Calculus Integral Applications of integration Area by integration Algebra Polynomials Quadratic polynomials Quadratic equations. Vieta's rule.

16–16 3.0

The numbers $p$ and $q$ are such that the parabolas $y = - 2 x^{2}$ and $y = x^{2} + p x + q$ intersect at two points, bounding a certain figure.

Find the equation of the vertical line dividing the area of this figure in half.

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-116445

Problems Methods Methods of calculus Continuity considerations Algebra Polynomials Quadratic polynomials Investigating the quadratic polynomial

13–15 2.0

Is it true that, if $b > a + c > 0$ , then the quadratic equation $a x^{2} + b x + c = 0$ has two roots?

Problem #PRU-60855

Problems Algebra 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-65318

Problems Algebra Polynomials Quadratic polynomials Investigating the quadratic polynomial Mean values Algebraic inequalities and systems of inequalities Quadratic inequalites (several variables) Probability and statistics Statistics

14–16 3.0

In a numerical set of $n$ numbers, one of the numbers is 0 and another is 1.

a) What is the smallest possible variance of such a set of numbers?

b) What should be the set of numbers for this?

Problem #PRU-65726

Problems Calculus Real numbers Integer and fractional parts. Archimedean property Algebra 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 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-78043

Problems Methods Methods of calculus Monotonicity and boundedness Pigeonhole principle Pigeonhole principle (other) Algebra Polynomials Quadratic polynomials

13–15 4.0

The expression $a x^{2} + b x + c$ is an exact fourth power for all integers $x$ . Prove that $a = b = 0$ .