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Problem: How many positive three-digit numbers are there: a) not divisible by any of the numbers \( 5,15,25 \) ? b) divisible by exactly one of these three numbers?

6.6.22 Combinatorics

3.06 $

Problem: Find the coefficients when \[ a=x^{2} \cdot y^{4} \cdot z^{6}, b=x^{2} \cdot y^{2} \cdot z^{2}, c=x^{2} \cdot y^{8} \] in the expansion \( \left(4 \cdot x+5 \cdot y^{2}+z^{3}\right)^{6} \).

6.6.23 Combinatorics

2.04 $

Problem: Find a sequence \( \left\{a_{n}\right\} \), satisfying the recurrence relation \( 2 a_{n+2}-10 a_{n+1}+8 a_{n}=0 \) and the initial conditions \( a_{1}=0, a_{2}=-12 \).

6.6.24 Combinatorics

1.02 $

Problem: Prove the statement using mathematical induction: \[ 1^{2}+2^{2}+3^{2}+\cdots+n^{2}=n \cdot(n+1)(2 n+1) / 6 \text {. } \]

6.6.25 Combinatorics

1.02 $

Problem: Seven employees of the company are sent to study a foreign language, and they need to be distributed to study English, German and French (each learns only one language). a) How many different ways of such distribution are there? b) In how many ways can they be arranged to study in two completely identical library rooms (at least one in a room)?

6.6.26 Combinatorics

3.83 $

Problem: How many positive three-digit numbers are there: a) divisible by the numbers 5,18 or 21 ? b) divisible by exactly one of these three numbers?

6.6.27 Combinatorics

3.06 $

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

6.6.28 Combinatorics

2.04 $

Problem: Find a sequence \( \left\{a_{n}\right\} \), satisfying the recurrence relation \( 2 a_{n+2}-10 a_{n+1}+12 a_{n}=0 \) and the initial conditions \( a_{1}=3, a_{2}=27 \).

6.6.29 Combinatorics

1.02 $

Problem: Nine employees of the company are sent to study a foreign language. They need to be distributed to study English, Spanish, German and French (each learns only one language). a) How many different ways of such a distribution exist? b) In how many ways can they be arranged to study in three completely identical rooms of the library (at least one in a room)?

6.6.30 Combinatorics

3.06 $

Problem: How many positive three-digit numbers are there: a) not divisible by any of the numbers \( 9,14,21 \) ? b) divisible by exactly one of these three numbers?

6.6.31 Combinatorics

3.06 $

Problem: Construct a linear homogeneous difference equation (of the minimum possible order) with constant real coefficients, the partial solutions of which are the functions \[ (-3)^{s}, 5^{s}, s \cdot 5^{s}, 12^{s} \cos \frac{11 \pi(s+1)}{6} \text {. } \]

6.6.32 Combinatorics

2.04 $

(3) Problem: Calculate the integral using Stokes formula. \[ \int_{\Gamma} x y z d x+y^{2} z d y+z x^{2} d z \] where \( \Gamma \) is the curve: \( \left\{\begin{array}{l}x^{2}+z^{2}=a^{2}, a>0 \\ y^{2}+z^{2}=a^{2}, x \geq 0\end{array}\right. \) is positively oriented on the outer side of the first cylinder.

9.8.3 Surface integrals

3.06 $

3) Problem: Calculate the surface integral of the \( 1^{\text {st }} \) kind: \[ \iint_{S} \sqrt{x^{2}+y^{2}} d S \] where \( S \) is the part of the conical surface \( z=\sqrt{x^{2}+y^{2}} \), distinguished by the condition \( z \leq 1 \).

9.8.4 Surface integrals

2.55 $

(3) Problem: Calculate the surface integral of the 2nd kind: \[ \iint_{S}\left(2 x^{2}+y^{2}+z^{2}\right) d y d z \] where \( S \) is the outer side of the lateral surface of the cone \( \sqrt{y^{2}+z^{2}} \leq x \leq H \).

9.8.5 Surface integrals

2.04 $

(a) Problem: Calculate the surface integral of the 2nd kind: \[ \iint_{S}(5 x+y) d y d z+z d x d y \] where \( S \) is the inside of the ellipsoid \( x^{2} / 4+ \) \( +y^{2} / 9+z^{2}=1 \).

9.8.6 Surface integrals

3.32 $

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