Let \(m\) and \(n\) be integers. Prove that \(mn(m + n)\) is an even number.
Write the following rational numbers in the form of decimal fractions: a) \(\frac {1}{7}\); b) \(\frac {2}{7}\); c) \(\frac{1}{14}\); d) \(\frac {1}{17}\).
There are 4 weights and scales. How many loads that are different by weight can be accurately weighed using these weights, if
a) weights can be placed only on one side of the scales;
b) weights can be placed on both sides of the scales?
Will thought of a number: 1, 2 or 3. You can ask him only one question, to which he can answer “yes”, “no” or “I do not know”. Can you guess the number by asking just one question?
Prove the following formulae are true: \[\begin{aligned} a^{n + 1} - b^{n + 1} &= (a - b) (a^n + a^{n-1}b + \dots + b^n);\\ a^{2n + 1} + b^{2n + 1} &= (a + b) (a^{2n} - a^{2n-1}b + a^{2n-2}b^2 - \dots + b^{2n}). \end{aligned}\]
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) \(\overline {z} z = |z|^2\).
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) \(\operatorname{arg} \frac{z_1 - z}{z - z_2} = 0\).
Prove the formulae: \(\arcsin (- x) = - \arcsin x\), \(\arccos (- x) = \pi - \arccos x\).