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Problem: Given is the goup \( G=(a, b) \) and its subgroup \( H \). 1. Find the order of elements \( a, b, a b \). 2. Determine the order \( H \) and make a Cayley table for it, 3. Check that \( b H=H b \), and deduce from this that the subgroup \( H \) is normal in \( G \). 4. Describe the cosets \( G \) by \( H \). 5. Check that \( G / H \) is cyclic, and find its order. 6. Determine the order of the group \( G \). 7. Determine whether a cyclic subgroup with generator \( b \) is normal in \( G \), 8. Find all subgroups \( Z(b) \). \[ a=\left(\begin{array}{lll} 1 & 2 & 3 \end{array}\right), \quad b=\left(\begin{array}{ll} 12 \end{array}\right)(45), \quad H=(a) \text {. } \]

1.3.7 Permutation group

12.85 $

Problem: Do permutations of order 11 generate a group \( S_{11} \) ?

1.3.8 Permutation group

2.57 $

Problem: Prove that the group \( A_{5} \) is simple, that is it has no proper normal subgroups.

1.3.9 Permutation group

5.14 $

Problem: Let \( G- \) a group, \( |G|=6 \). Prove that \( G \) is commutative, or \( G \cong S_{3} \) (is isomorphic).

1.3.10 Permutation group

3.85 $

Problem: Labour productivity tends to increase, the growth rate is \( f(t)=\frac{2 t}{t^{2}+1} \). Find the law of change of labour productivity \( F(t) \).

18.9 Mathematical methods and models in economics

1.28 $

Problem: 11 workers produce products in a workshop. The productivity of the \( i-t h \) worker is equal to \( 1+0,1(i-1) \mathrm{kg} \) of production per hour, \( i=1, \ldots, 11 \). Each worker has produced \( 7 \mathrm{~kg} \) of products. Determine the total working time of all workers of the workshop and the labour productivity of the whole workshop.

18.11 Mathematical methods and models in economics

1.28 $

Problem: The rate of change in labour productivity is directly proportional to the value \( t^{2}-t \), the coefficient of proportionality is \( k, k<0 \). Let's find the law of change in labour productivity, if in case of \( t=0 \) the productivity was 1 c.u.

18.12 Mathematical methods and models in economics

1.28 $

Problem: On sides \( A B, B C, C D, D E, E F, F A \) of the regular hexagon \( A B C D E F \) with the area \( S \) points \( A_{1}, B_{1}, C_{1}, D_{1}, E_{1}, F_{1} \) are marked so, that \[ \frac{A A_{1}}{A_{1} B}=\frac{B B_{1}}{B_{1} C}=\frac{C C_{1}}{C_{1} D}=\frac{D D_{1}}{D_{1} E}=\frac{E E_{1}}{E_{1} F}=\frac{F F_{1}}{F_{1} A}=\frac{1}{4} . \]

12.1.7 Olympic geometry

5.14 $

Problem: Annual income \( f(t) \) is a time function, \( i \) is a specific rate of share: Find the discounted volume of the income for \( T \) years (the interest is calculated continuously).

18.10 Mathematical methods and models in economics

1.28 $

Problem: The children ate sandwiches and sweets at the New Year's table, and each of them ate something, and someone might have eaten both. It is known that the boys who ate sandwiches were no more than \( 5 / 16 \) of the total number of children who ate sandwiches, and the boys who ate sweets were no more than \( 2 / 5 \) of the total number of children who ate sweets. a) Could there be 13 boys at the table if it is additionally known that there were 25 children in total at the table? b) What is the largest number of boys that could be at the table, if it is additionally known that there were 25 children at the table? c) What was the smallest share of girls in the total number of children without taking into account the additional conditions of points \( a \) and \( b \) ?

17.24 USE problems

5.14 $

Problem: Is \( \rho(x, y)=\max _{1 \leq k \leq n} k\left(x_{k}-y_{k}\right) \) a metric on \( \mathbb{R}^{n} \), where \( x, y \in \mathbb{R}^{n}, x=\left(x_{1}, \ldots, x_{n}\right), y=\left(y_{1}, \ldots, y_{n}\right) \) ?

19.1.1.1 Properties of metric spaces

2.57 $

Problem: Let the metric on \( \mathbb{R} \) be \( \rho(x, y)=\mid \tan ^{-1} x- \) \( -\tan ^{-1} y \mid \). Prove that the obtained space is not complete. Indication: consider the sequence \( x_{n}=n \).

19.1.1.4 Properties of metric spaces

2.57 $

Problem: The metric \( \rho(x, y)=\tan ^{-1}|x-y| \) is given on \( \mathbb{R} \). Find out if this space is complete.

19.1.1.5 Properties of metric spaces

2.57 $

Problem: The metric \( \rho(x, y)=\left|x^{3}-y^{3}\right| \) is given on \( \mathbb{R} \). Find out if this space is complete.

19.1.1.6 Properties of metric spaces

2.57 $

Problem: Prove that the set of rational numbers is not complete on space \( \mathbb{R} \).

19.1.1.7 Properties of metric spaces

5.14 $

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