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The given image shows a mathematical expression with a proof task, written in French. The task says:

**1) Montrer que: ∀ (x, y) ∈ ℝ² :**

\[
\frac{\sqrt{x^2 + y^2}}{2} \geq \sqrt{\frac{xy}{2}} \geq \frac{2xy}{x + y}
\]

Translation:  
**1) Show that: For all (x, y) ∈ ℝ²:**

\[
\frac{\sqrt{x^2 + y^2}}{2} \geq \sqrt{\frac{xy}{2}} \geq \frac{2xy}{x + y}
\]

This problem involves proving the inequality that applies to real numbers \( x \) and \( y \). The inequality compares the geometric mean, arithmetic mean, and a certain combination of the variables. 

### Approach:
1. **First Inequality**: 
   Prove that:
   \[
   \frac{\sqrt{x^2 + y^2}}{2} \geq \sqrt{\frac{xy}{2}}
   \]
   This might involve the **AM-GM inequality** (Arithmetic Mean - Geometric Mean inequality), which typically states that for any non-negative real numbers \( a \) and \( b \):
   \[
   \frac{a + b}{2} \geq \sqrt{ab}
   \]
   In this case, we need to adjust for the square root and fractions.

2. **Second Inequality**:
   Prove that:
   \[
   \sqrt{\frac{xy}{2}} \geq \frac{2xy}{x + y}
   \]
   This step will likely involve algebraic manipulation, possibly including factoring, and working with proportions to show the relationship.

Would you like me to go step by step in solving this inequality proof for you?

Also, here are 5 related questions for exploration:
1. What is the AM-GM inequality, and how is it applied to inequalities like this?
2. What are other common inequalities involving arithmetic and geometric means?
3. How do you handle square roots when proving inequalities?
4. What techniques can be used to simplify algebraic expressions involving square roots and fractions?
5. What are the conditions required for an inequality to hold true for all real numbers?

**Tip**: When dealing with inequalities involving square roots, try squaring both sides to simplify and then manipulate algebraically.

Montrer que ∀(x, y) ∈ ℝ²: (√(x² + y²) / 2) ≥ √(xy / 2) ≥ (2xy / (x + y))

Explore the step-by-step solution to proving the inequality ∀(x, y) ∈ ℝ²: (√(x² + y²) / 2) ≥ √(xy / 2) ≥ (2xy / (x + y)). This problem involves applying the AM-GM inequality, working with square roots and algebraic manipulation, suitable for advanced algebra or calculus students.

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