Prove that the triangle \(ABC\) is regular if and only if, by turning it by \(60^{\circ}\) (either clockwise or anticlockwise) with respect to point A, its vertex B moves to \(C\).
Prove that the midpoints of the sides of a regular polygon form a regular polygon.
Prove that if you rotate through an angle of \(\alpha\) with the center at the origin, the point with the coordinates \((x, y)\), it goes to the point \((x \cos \alpha - y \sin \alpha, x \sin \alpha + y \cos \alpha)\).
a) In Wonderland, there are three cities \(A\), \(B\) and \(C\). 6 roads lead from city \(A\) to city \(B\), and 4 roads lead from city \(B\) to city \(C\). How many ways can you travel from \(A\) to \(C\)?
b) In Wonderland, another city \(D\) was built as well as several new roads – two from \(A\) to \(D\) and two from \(D\) to \(C\). In how many ways can you now get from city \(A\) to city \(C\)?
A car registration number consists of three letters of the Russian alphabet (that is, 30 letters are used) and three digits: first we have a letter, then three digits followed by two more letters. How many different car registration numbers are there?
How many ways can you choose four people for four different positions, if there are nine candidates for these positions?
How many rational terms are contained in the expansion of
a) \((\sqrt 2 + \sqrt[4]{3})^{100}\);
b) \((\sqrt 2 + \sqrt[3]{3})^{300}\)?
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?
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) \(\operatorname{arg} \frac{z_1 - z}{z - z_2} = 0\).