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?

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.