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Problem: A weighted graph is defined by a matrix of arc lengths. Plot the graph. Find: a) the spanning tree of minimum weight; b) the shortest distance from vertex \( v_{4} \) to the remaining vertices of the graph, using Dijkstra's algorithm.

6.4.29 Graph theory

6.24 $

Problem: Prove the equalities. Illustrate using Euler-Venn diagrams. a) \( (A \cap B) \backslash(A \cap C)=(A \cap B) \backslash C \), b) \( (A \cup B) \times C=(A \times C) \cup(B \times C) \).

6.8.13 Set theory

1.25 $

Problem: Solve the Cauchy problem using Duhamel's formula: \[ y^{\prime \prime}+4 y=2 \tan x, y(0)=0, y^{\prime}(0)=0 \text {. } \]

8.2.1.10 Differential equations

3.75 $

Problem: Solve the equation \[ \int_{0}^{t} \sin (t-\tau) x(\tau) d \tau=x^{\prime \prime}+x+2 \cos t-4 \] under the given initial conditions \( x(0)=0, x^{\prime}(0)=0 \) (with respect to the function \( x(t) \) ).

8.2.1.11 Differential equations

2.5 $

Problem: Solve the differential equation, using the operational method. \[ y^{\prime \prime}+4 y=\cos t, \quad y(0)=1, \quad y^{\prime}(0)=0 . \]

8.2.1.12 Differential equations

2 $

Problem: Calculate and determine the absolute and relative errors of the result. \[ \begin{array}{l} \frac{\sqrt{a} \cdot b}{c}, \text { if } a=228.60 \pm 0.06, \\ b=86.40 \pm 0.02, \quad c=68.70 \pm 0.05 . \end{array} \]

14.1.3 Approximate numbers

2 $

Problem: Implement 3 steps of the golden-section search method in order to find the minimum value of the function \( y=x^{2}+2 x \) on the interval \( [-2 ; 0] \). Estimate the error of the obtained approximation.

14.2.2 Golden section search method

2.5 $

Problem: For the function \( f:[a, b] \rightarrow R \) a) find out if it's bounded; b) find the measure of the set of discontinuity points; c) find out if there is a proper or improper Riemann integral for it; d) find out if it's measurable \( f \); e) find the Lebesgue integral \( \int_{[a, b]} f(t) d t \), if it exists. \( a=-1, \quad b=1, \quad K \) is the Cantor set, \( f(t)=\left\{\begin{array}{cc}n, \quad t \in\left(\frac{1}{3^{n+1}}, \frac{1}{3^{n}}\right) \backslash K, n \in \mathbb{N}, \\ & {\left[e^{t^{2}}\right], \quad t \in K,} \\ \frac{1}{\sqrt{1+t}}, \quad & t \in\left([-1,0) \cup\left(\frac{1}{3}, 1\right)\right) \backslash K .\end{array}\right. \)

19.6.1.6 Lebesgue measure and integration

4.99 $

Problem: Let the functions \( f, g: \mathbb{R} \rightarrow \mathbb{R} \), be continuous. Prove that the set \( \{x \in \mathbb{R}: f(x)

19.7.13 Properties of sets

2.5 $

Problem: Establish a one-to-one correspondence between \( [-7 ; 4] \) and \( (-\infty ; 5] \).

19.7.14 Properties of sets

3.25 $

Problem: Prove the statement using mathematical induction: \( \left(n^{3}+11 n\right) \vdots 6 \), for all natural numbers \( n \geq 1 \).

6.6.33 Combinatorics

0 $

Problem: Find a sequence \( \left\{a_{n}\right\} \), satisfying the recurrence relation \( a_{n+2}-3 a_{n+1}+2 a_{n}=0 \) and the initial conditions \( a_{1}=3, a_{2}=7 \).

6.6.34 Combinatorics

1 $

Problem: How many positive three-digit numbers are there that are divisible by exactly one of these three numbers?

6.6.35 Combinatorics

2.5 $

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

6.6.36 Combinatorics

2 $

30 Problem: Calculate the indefinite integral: \[ \int\left(\cos ^{2} x \sin x+x \sqrt{1+x^{2}}\right) d x \]

9.3.1.8 Indefinite integrals

0.75 $

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