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The document you're working with appears to be a math problem that involves integer parts and fractional parts of real numbers, particularly focusing on a function \( E(x) = \left\lfloor x \right\rfloor \), which gives the greatest integer less than or equal to \( x \). Let's analyze the content and steps based on the image:

### Exercise 13 Overview:
- You are dealing with the integer part \( E(x) \) and the fractional part \( x - E(x) \) of a real number \( x \).
- There are inequalities and functional relations described using \( d(x) \), which seems to involve periodic properties.
  
### Key Mathematical Notations:
1. \( E(x) = \left\lfloor x \right\rfloor \), the greatest integer less than or equal to \( x \).
2. \( x = p + d(x) \), where \( p = E(x) \) is the integer part and \( d(x) \) is a periodic fractional part that satisfies \( 0 \leq d(x) < 1 \).
3. The inequalities \( x - 1 < E(x) \leq x \) and \( 0 \leq d(x) < 1 \) are fundamental.

### Tasks (translated from the problem):
1. **Task 1**: Prove that for any real number \( x \), the integer part \( p = E(x) \) satisfies these inequalities.
   
2. **Task 2**: Prove that for any real number \( x \), the periodic part \( d(x) \) satisfies \( 0 \leq d(x) < 1 \).

3. **Task 3**: Determine the period of the function \( d(x) \), given that \( d(12x) = d(13x) \). This involves finding periodicity properties of the fractional part function.

4. **Task 4**: Solve the system of equations for the two functions \( E(2x - 1) = \left\lfloor 2x - 1 \right\rfloor \) and \( x \leq \left\lfloor x \right\rfloor + \left\lceil x \right\rceil \), distinguishing the cases where \( x = 4 \).

5. **Task 5**: Show that for any \( x \in \mathbb{R} \), the fractional part function \( d(x) \) satisfies certain conditions over a period of time (possibly related to the functions defined earlier).

Would you like to go into more detail for any specific task, or would you prefer I walk you through solving one of them step by step?

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### Related Questions:
1. What is the significance of the fractional part \( d(x) \) in the periodicity context?
2. How can we prove that \( E(x) \leq x \)?
3. What mathematical methods are used to solve periodic functions like \( d(x) \)?
4. How does the system of equations in Task 4 relate to the properties of the floor and ceiling functions?
5. What conditions must be satisfied for \( d(x) \) to have a specific period?

**Tip**: When working with floor functions \( \left\lfloor x \right\rfloor \), remember that \( x \) is always between \( \left\lfloor x \right\rfloor \) and \( \left\lfloor x \right\rfloor + 1 \). This can help in solving inequalities.

Given a real number x, the problem explores the relationship between its integer part and fractional part, especially focusing on the function E(x) = floor(x) and related periodic functions.

This problem involves analyzing real numbers by separating their integer and fractional parts using the floor function E(x). The task includes proving inequalities, finding periodic functions, and solving equations involving periodicity. Ideal for advanced high school or undergraduate students.

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