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The document contains mathematical problems related to discrete mathematics and combinatorics. Let me transcribe the key parts of the problems and outline their contents:

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**1.** Prove that if \( F \subset \{0, 1\}^n \) is a non-empty set, then there are at least \( |F| \cdot \log_2 \frac{2n}{|F|} \) edges connecting the points of \( F \), where an edge connects two points \( u, v \in \{0, 1\}^n \) if and only if they differ in exactly one coordinate.

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**2.** For a function \( f : \{0,1\}^n \to \mathbb{R} \), define its expected value (arithmetic mean) as:
\[
Ef = \frac{1}{2^n} \sum_{a \in \{0,1\}^n} f(a).
\]
A function \( f \) is called monotone if its value does not decrease when any of the coordinates of \( a \) are set to 1 from 0.

Prove that for two monotone functions \( f, g : \{0,1\}^n \to \mathbb{R} \), the inequality \( Ef \geq Eg \) holds if and only if \( f(a) \geq g(a) \) for all \( a \in \{0,1\}^n \).

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**3.** Given vertices \( x_1 < x_2 < \dots < x_k \) of a cube \( \{0,1\}^n \), prove that for any chain of non-intersecting paths connecting these vertices, if \( x < y < z \) are three vertices of one of these chains, then there is another element \( w \) on one of the chains that lies between \( x \) and \( z \), such that \( x < w < z \).

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**4.** Prove that the number of monotone Boolean functions from \( \{0,1\}^n \) does not exceed \( 3^{(n^2)/4} \).

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**5.** In any graph where the degree of every vertex is no less than 3, prove that it contains a simple cycle of length divisible by 2.

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**6.** In any tournament (complete oriented graph), corresponding to a round-robin volleyball tournament, show that there is a vertex from which every other vertex can be reached via a directed path in the graph with no more than two edges.

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**7.** In any directed graph, prove that one can choose several pairs of non-adjacent vertices such that for every remaining pair of adjacent vertices, one can reach the other in no more than two steps.

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**8.** In any graph without isolated vertices, prove that there is a pair of vertices such that the subgraph induced by this pair and the other vertices contains no cycles.

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Let me know if you'd like to dive into the details of any specific question or need further clarification! Here are some related questions:

1. How do you calculate the number of edges between vertices in a cube?
2. What properties define a monotone function in the Boolean cube?
3. Can you explain the pigeonhole principle and how it's used in combinatorics?
4. How does one approach counting monotone Boolean functions?
5. What are the key steps to prove the existence of cycles in high-degree graphs?

**Tip:** When dealing with combinatorial problems, try breaking down the structure into smaller, manageable parts to spot patterns and relationships more easily.

The image contains several discrete mathematics problems related to hypercubes, monotone Boolean functions, graph theory, and combinatorics.

Explore a set of advanced discrete mathematics problems covering hypercubes, monotone Boolean functions, graph cycles, and tournament graphs. These problems are suitable for undergraduate students studying combinatorics and graph theory.

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