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Problem: A real number \( a \neq-1 \) is given. Let's define the sequence \( x_{n} \) as follows: \( x_{1}=a, x_{n+1}=x_{n}^{2}+x_{n}, n \geq 1 \). Let's also define the sequence \( y_{n} \), as well as the sum and product of its first \( n \) elements: \[ y_{n}=\frac{1}{1+x_{n}}, \quad S_{n}=\sum_{k=1}^{n} y_{k}, \quad p_{n}=\prod_{k=1}^{n} y_{k} . \]

12.6.5 Higher mathematics

6.47 $

Problem: Find the probability that the divisor of \( 10^{99} \) is a multiple of \( 10^{88} \).

12.6.6 Higher mathematics

2.59 $

Problem: The integral operator \( (A \varphi)(x)=\int_{0}^{1} \varphi(y) d y \) is considered as acting from \( L_{2}[0 ; 1] \) in \( L_{2}[0 ; 1] \). Is the operator \( A \) completely continuous? Is the operator \( A \) self-adjoint? Find all characteristic values of the operator \( A \) and the corresponding eigenfunctions. For what functions \( f \in L_{2}[0 ; 1] \) is the equation \( (A \varphi)(x)=f(x), 0 \leq x \leq 1 \) solvable, with respect to \( \varphi \in L_{2}[0 ; 1] \) ?

19.8.1 Integral equations

6.47 $

Problem: Is the equation \( \int_{0}^{2 \pi} \cos (x+y) \cdot \varphi(y) d y=\pi \cos x \), \( 0 \leq x \leq 2 \pi \) solvable? If yes, is there only one solution?

19.8.2 Integral equations

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Problem: In the three-dimensional space \( \mathbb{R}^{3} \) with standard metric, let's consider the set \( A=\mathbb{Q} \times \mathcal{T} \times \mathbb{R} \) and the point \( x_{0}=(1 ; \sqrt{2} ; 0) \). Which of the following statements are correct: 1) \( x_{0} \) is an interior point of the set \( A \), 2) \( x_{0} \) is the adherent point of the set \( A \), 3) \( x_{0} \) is a limit point of the set \( A \), 4) \( x_{0} \) is an isolated point of the set \( A \), 5) \( x_{0} \) is a boundary point of the set \( A \).

19.7.1 Properties of sets

3.11 $

Problem: In the metric space \( (\mathbb{R} ; \rho) \) with a natural metric \( \rho(x, y)=|x-y| \) the following set is given: \( A=(-\infty ; 0) \cup[1 ; 2] \cup\{3\} \). For the set \( A \) find: 1) the interior, 2) the closure, 3) the set of limit points, 4) the set of isolated points, 5) the set of boundary points.

19.7.2 Properties of sets

5.18 $

Problem: Let the set \( A \subset \mathbb{R} \) is open, and \( B \subset \mathbb{R} \) is closed. Prove that the set \( A \backslash B \) is open, and \( B \backslash A \) is closed.

19.7.3 Properties of sets

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Problem: Any system of non-intersecting intervals on the line is at most countable.

19.7.4 Properties of sets

2.07 $

Problem: Every open subset of a line that does not coincide with \( \mathbb{R} \) is a union of at most a countable set of nonoverlapping intervals and (possibly) one or two open rays.

19.7.5 Properties of sets

3.88 $

Problem: Prove that no interval can be represented as a union of two non-intersecting non-empty open subsets of \( \mathbb{R} \).

19.7.6 Properties of sets

2.59 $

Problem: Let \( A \subset \mathbb{R} \) be an arbitrary set, let \( \operatorname{Int} A \) is the set of its interior points. Prove the equivalence of the following three statements: 1) Int \( \bar{A}=\emptyset ; 2 \) ) the set \( A \) is nowhere dense; 3 ) the set \( \bar{A} \) is nowhere dense.

19.7.7 Properties of sets

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Problem: Let such a set of segments \( \left[a_{j}, b_{j}\right] \subset \mathbb{R} \) be given, that every two of them have at least one common point. Prove that there is a point, belonging to each of these segments.

19.7.8 Properties of sets

2.59 $

Problem: Prove that there exists \( \sqrt[3]{7} \in \mathbb{R} \), that is to say a real number, which is equal to 7 when cubed.

19.7.9 Properties of sets

3.88 $

Problem: Prove that the set of rational numbers \( \mathbb{Q} \) is countable.

19.7.10 Properties of sets

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Problem: Using the theorem of nested intervals, prove that the segment is uncountable.

19.7.11 Properties of sets

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