The circles \(\sigma_1\) and \(\sigma_2\) intersect at points \(A\) and \(B\). At the point \(A\) to \(\sigma_1\) and \(\sigma_2\), respectively, the tangents \(l_1\) and \(l_2\) are drawn. The points \(T_1\) and \(T_2\) are chosen respectively on the circles \(\sigma_1\) and \(\sigma_2\) so that the angular measures of the arcs \(T_1A\) and \(AT_2\) 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 \(\sigma_1\) intersects \(l_2\) at the point \(M_1\). Similarly, the tangent \(t_2\) at the point \(T_2\) to the circle \(\sigma_2\) intersects \(l_1\) at the point \(M_2\). Prove that the midpoints of the segments \(M_1M_2\) are on the same line, independent of the positions of the points \(T_1, T_2\).
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.
Prove that the root a of the polynomial \(P (x)\) has multiplicity greater than 1 if and only if \(P (a) = 0\) and \(P '(a) = 0\).
Prove that the following polynomial does not have any identical roots: \(P(x) = 1 + x + x^2/2! + \dots + x^n/n!\)
For which \(A\) and \(B\) does the polynomial \(Ax^{n + 1} + Bx^n + 1\) have the number \(x = 1\) at least two times as its root?
Prove that the polynomial \(x^{2n} - nx^{n + 1} + nx^{n - 1} - 1\) for \(n > 1\) has a triple root of \(x = 1\).
Find the largest and smallest values of the functions
a) \(f_1 (x) = a \cos x + b \sin x\); b) \(f_2 (x) = a \cos^2x + b \cos x \sin x + c \sin^2x\).
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 \(x_2 = x_0\).
Prove the inequality: \[\frac{(b_1 + \dots b_n)^{b_1 + \dots b_n}}{(a_1 + \dots a_n)^{b_1 + \dots + b_n}}\leq \left(\frac{b_1}{a_1}\right)^{b_1}\dots \left( \frac{b_n}{a_n}\right)^{b_n}\] where all variables are considered positive.
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 \(\alpha_{1}, \alpha_{2}, \dots , \alpha_{n}\) such that \(\alpha_ {1} + \alpha_{2} + \dots + \alpha_{n} = 1\), the following inequality holds: \(f (\alpha_{1} x_1 + \dots + \alpha_{n} x_n) > \alpha_{1} f (x_1) + \dots + \alpha_{n} f (x_n)\).