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Let \(a, b\) be positive integers and \((a, b) = 1\). Prove that the quantity cannot be a real number except in the following cases \((a, b) = (1, 1)\), \((1,3)\), \((3,1)\).

Let \(f (x)\) be a polynomial of degree \(n\) with roots \(\alpha_1, \dots , \alpha_n\). We define the polygon \(M\) as the convex hull of the points \(\alpha_1, \dots , \alpha_n\) on the complex plane. Prove that the roots of the derivative of this polynomial lie inside the polygon \(M\).

a) Using geometric considerations, prove that the base and the side of an isosceles triangle with an angle of \(36^{\circ}\) at the vertex are incommensurable.

b) Invent a geometric proof of the irrationality of \(\sqrt{2}\).

Let \(z_1\) and \(z_2\) be fixed points of a complex plane. Give a geometric description of the sets of all points \(z\) that satisfy the conditions:

a) \(\operatorname{arg} \frac{z - z_1}{z - z_2} = 0\);

b) \(\operatorname{arg} \frac{z_1 - z}{z - z_2} = 0\).

Find the largest and smallest values of the functions

a) \(f_1 (x) = a \cos x + b \sin x\); b) \(f_2 (x) = a \cos^2x + b \cos x \sin x + c \sin^2x\).

Prove the formulae: \(\arcsin (- x) = - \arcsin x\), \(\arccos (- x) = \pi - \arccos x\).

Prove that amongst any 7 different numbers it is always possible to choose two of them, \(x\) and \(y\), so that the following inequality was true: \[0 < \frac{x-y}{1+xy} < \frac{1}{\sqrt3}.\]