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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.

When water is drained from a pool, the water level \(h\) in it varies depending on the time \(t\) according to the function \(h (t) = at^2 + bt + c\), and at the time \(t_0\) of when the draining is ending, the equalities \(h (t_0) = h' (t_0) = 0\) are satisfied. For how many hours does the pool drain completely, if in the first hour the water level in it is reduced by half?

For a given polynomial \(P (x)\) we describe a method that allows us to construct a polynomial \(R (x)\) that has the same roots as \(P (x)\), but all multiplicities of 1. Set \(Q (x) = (P(x), P'(x))\) and \(R (x) = P (x) Q^{-1} (x)\). Prove that

a) all the roots of the polynomial \(P (x)\) are the roots of \(R (x)\);

b) the polynomial \(R (x)\) has no multiple roots.

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

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

Prove that for \(n> 0\) the polynomial \(nx^{n + 1} - (n + 1) x^n + 1\) is divisible by \((x - 1)^2\).