We solve problems

Problem #PRU-110157

Problems Geometry Plane geometry Plane transformations Similarity Homothetic transformations, rotations of homothetic transformations Circles Connection between the arc length, the angle, and the chord Calculus Derivative Derivative and multiple roots

14–16 5.0

The circles $σ_{1}$ and $σ_{2}$ intersect at points $A$ and $B$ . At the point $A$ to $σ_{1}$ and $σ_{2}$ , respectively, the tangents $l_{1}$ and $l_{2}$ are drawn. The points $T_{1}$ and $T_{2}$ are chosen respectively on the circles $σ_{1}$ and $σ_{2}$ so that the angular measures of the arcs $T_{1} A$ and $A T_{2}$ are equal (the arc value of the circle is considered in the clockwise direction). The tangent $t_{1}$ at the point $T_{1}$ to the circle $σ_{1}$ intersects $l_{2}$ at the point $M_{1}$ . Similarly, the tangent $t_{2}$ at the point $T_{2}$ to the circle $σ_{2}$ intersects $l_{1}$ at the point $M_{2}$ . Prove that the midpoints of the segments $M_{1} M_{2}$ are on the same line, independent of the positions of the points $T_{1}, T_{2}$ .

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-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-61021

Problems Calculus Derivative Derivative and multiple roots

15–16 4.0

Prove that the following polynomial does not have any identical roots: $P (x) = 1 + x + x^{2} / 2! + \dots + x^{n} / n!$

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-61023

Problems Calculus Derivative Derivative and multiple roots

15–16 4.0

Prove that the polynomial $x^{2 n} - n x^{n + 1} + n x^{n - 1} - 1$ for $n > 1$ has a triple root of $x = 1$ .

Problem #PRU-61213

Problems Calculus Derivative Increasing and decreasing. Researching functions Algebra and arithmetic Trigonometry Trigonometry (other)

14–15 3.0

Find the largest and smallest values of the functions

a) $f_{1} (x) = a \cos x + b \sin x$ ; b) $f_{2} (x) = a \cos^{2} x + b \cos x \sin x + c \sin^{2} x$ .

Problem #PRU-61332

Problems Calculus Derivative Derivative and tangent line Methods Examples and counterexamples. Constructive proofs

15–16 3.0

The Newton method (see Problem 61328) does not always allow us to approach the root of the equation $f (x) = 0$ . Find the initial condition $x_{0}$ for the polynomial $f (x) = x (x - 1) (x + 1)$ such that $f (x_{0}) \neq x_{0}$ and $x_{2} = x_{0}$ .

Problem #PRU-61403

Problems Calculus Derivative Increasing and decreasing. Researching functions Algebra and arithmetic Surds. Rational powers. Irrational equations.

15–16 3.0

Prove the inequality: $\frac{(b_{1} + \dots b_{n})^{b_{1} + \dots b_{n}}}{(a_{1} + \dots a_{n})^{b_{1} + \dots + b_{n}}} \leq {(\frac{b_{1}}{a_{1}})}^{b_{1}} \dots {(\frac{b_{n}}{a_{n}})}^{b_{n}}$ where all variables are considered positive.

Problem #PRU-61407

Problems Calculus Derivative Convexity and concavity

15–16 3.0

Inequality of Jensen. Prove that if the function $f (x)$ is convex upward on $[a, b]$ , then for any distinct points $x_{1}, x_{2}, \dots, x_{n}$ ( $n \geq 2$ ) from $[a; b]$ and any positive $α_{1}, α_{2}, \dots, α_{n}$ such that $α_{1} + α_{2} + \dots + α_{n} = 1$ , the following inequality holds: $f (α_{1} x_{1} + \dots + α_{n} x_{n}) > α_{1} f (x_{1}) + \dots + α_{n} f (x_{n})$ .