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Problem: Calculate the integral of the function of the complex variable \( f(z)=u(x, y)+i v(x, y) \) along the given arc \( \widetilde{A B} \) : \[ \begin{array}{l} f(z)=(2 x y-x-y+1)+i(3 x y+2 x+2 y+1), \\ \widetilde{A B}: y=x^{2}-2 x-3, A(-4 ; 5), B(0 ;-3) . \end{array} \]

10.1.4 Integral of a complex variable

2.57 $

Problem: Calculate the integral of the function of the complex variable \( f(z)=u(x, y)+i v(x, y) \) along the given arc \( \widetilde{A B} \) : \[ \begin{array}{l} f(z)=(2 x y-x-y+1)+i(3 x y+2 x+2 y+1), \\ \widetilde{A B}: y=x^{2}-2 x-3, A(-1 ; 0), B(0 ;-3) . \end{array} \]

10.1.5 Integral of a complex variable

2.57 $

Problem: Calculate the integral of the analytic function: \[ \int_{4-3 i}^{1+4 i}\left(3 z^{2}-2 z+1\right) d z . \]

10.1.6 Integral of a complex variable

1.28 $

Problem: Calculate the integral of the analytic function: \[ \int_{2+2 i}^{-5+2 i}\left(3 z^{2}-4 z-1\right) d z \text {. } \]

10.1.7 Integral of a complex variable

1.28 $

Problem: Apply Cauchy's integral formula to calculate a closed loop integral. \[ \oint_{|z+1|=2} \frac{z-2}{(z-3)(z+1-i)} d z . \]

10.1.8 Integral of a complex variable

1.54 $

Problem: Apply Cauchy's integral formula to calculate a closed loop integral. \[ \oint_{|z+2|=\frac{3}{2}} \frac{z-1}{(z+4)(z+2-i)} d z . \]

10.1.9 Integral of a complex variable

1.54 $

Problem: A regular 85-gon is inscribed in a circle, at the vertices of which different natural numbers are written. A pair of non-neighboring vertices of a polygon \( A \) and \( B \) is called interesting if at least on one of the two arcs of \( A B \) at all vertices of the arc there are numbers greater than the numbers, written at vertices \( A \) and \( B \). What is the least number of interesting pairs of vertices that this polygon can have?

12.4.8 Various Olympiad problems

5.14 $

Problem: Construct a section through the diagonal of the face of a cube, which is equal to the face of this cube.

12.1.6 Olympic geometry

4.37 $

Problem: The platoon has 3 sergeants and 30 soldiers. How many ways are there to assign one sergeant and three soldiers to patrol?

6.6.7 Combinatorics

0.77 $

Problem: How many positive numbers between 20 to 1000 are divisible by exactly one of the numbers 7,11 or 13 ?

6.6.8 Combinatorics

1.54 $

Problem: Car license plates in this region consist of 3 digits (the total number of digits is 10) and three letters of the alphabet \( X=\{A, B, C, D, E, H, K, M, O, P, T, X, Y\} \). How many cars can be numbered with different numbers?

6.6.9 Combinatorics

1.28 $

Problem: There are \( n \) lines drawn on the plane, among which there is not a single pair of parallel lines and not a single triple of lines intersecting at one point. How many parts is the plane divided into by these lines?

6.6.10 Combinatorics

3.08 $

Problem: How many integers are there in the range 0 to 100000 , that contain exactly one digit 7 and one digit 3 ?

6.6.12 Combinatorics

3.08 $

Problem: Find the coefficients of \( a=x^{3} y^{2} z^{2}, b=x^{2} y^{2} z^{2} \), \( c=x^{4} z^{4} \) in the expansion \( \left(2 x+3 y+5 z^{2}\right)^{6} \).

6.6.13 Combinatorics

1.54 $

Problem: Prove that the binomial coefficient \( C(n, k) \) increases in \( n \) for fixed \( k \).

6.6.11 Combinatorics

1.28 $

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