Problems

Let \(z = x + iy\), \(w = u + iv\). Find a) \(z + w\); b) \(zw\); c) \(z/w\).

Prove the equalities:

a) \(\overline{z+w} = \overline{z} + \overline{w}\); b) \(\overline{zw} = \overline{z} \overline{w}\); c) \(\overline{\frac{z}{w}} = \frac{\overline{z}}{\overline{w}}\); d) \(|\overline{z}| = |z|\); d) \(\overline{\overline{z}} = z\).

Prove the equalities:

a) \(z + \overline {z} = 2 \operatorname{Re} z\);

b) \(z - \overline {z} = 2i \operatorname{Im} z\);

c) \(\overline {z} z = |z|^2\).

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: \[0 < \frac{x-y}{1+xy} < \frac{1}{\sqrt3}.\]

An iterative polyline serves as a geometric interpretation of the iteration process. To construct it, on the \(Oxy\) plane, the graph of the function \(f (x)\) is drawn and the bisector of the coordinate angle is drawn, as is the straight line \(y = x\). Then on the graph of the function the points \[A_0 (x_0, f (x_0)), A_1 (x_1, f (x_1)), \dots, A_n (x_n, f (x_n)), \dots\] are noted and on the bisector of the coordinate angle – the points \[B_0 (x_0, x_0), B_1 (x_1, x_1), \dots , B_n (x_n, x_n), \dots.\] The polygonal line \(B_0A_0B_1A_1 \dots B_nA_n \dots\) is called iterative.

Construct an iterative polyline from the following information:

a) \(f (x) = 1 + x/2\), \(x_0 = 0\), \(x_0 = 8\);

b) \(f (x) = 1/x\), \(x_0 = 2\);

c) \(f (x) = 2x - 1\), \(x_0 = 0\), \(x_0 = 1{,}125\);

d) \(f (x) = - 3x/2 + 6\), \(x_0 = 5/2\);

e) \(f (x) = x^2 + 3x - 3\), \(x_0 = 1\), \(x_0 = 0{,}99\), \(x_0 = 1{,}01\);

f) \(f (x) = \sqrt{1 + x}\), \(x_0 = 0\), \(x_0 = 8\);

g) \(f (x) = x^3/3 - 5x^2/x + 25x/6 + 3\), \(x_0 = 3\).

The sequence of numbers \(a_n\) is given by the conditions \(a_1 = 1\), \(a_{n + 1} = a_n + 1/a^2_n\) (\(n \geq 1\)).

Is it true that this sequence is limited?

Prove that for a monotonically increasing function \(f (x)\) the equations \(x = f (f (x))\) and \(x = f (x)\) are equivalent.

Prove that \(\sqrt{\frac{a^2 + b^2}{2}} \geq \frac{a+b}{2}\).

Prove that the equation \(\frac {x}{y} + \frac {y}{z} + \frac {z}{x} = 1\) is unsolvable using positive integers.