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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-65318

Problems Algebra and arithmetic 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-111345

Problems Calculus Integral Applications of integration Area by integration Algebra and arithmetic 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.