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The exercise asks to use contrapositive reasoning ("raisonnement par la contraposée") to prove several mathematical statements. Here's a breakdown of each problem:

1. **Statement 1:**
   \[
   \forall (x,y) \in \mathbb{R}^2 : x \neq y \Rightarrow (x+1)(y-1) \neq (x-1)(y+1)
   \]
   You need to show that if \( x \neq y \), then the expressions \( (x+1)(y-1) \) and \( (x-1)(y+1) \) are not equal. Using contrapositive reasoning, you would instead prove that if \( (x+1)(y-1) = (x-1)(y+1) \), then \( x = y \).

2. **Statement 2:**
   \[
   b \neq \frac{1}{8}a \Rightarrow \frac{a+2b}{2a-b} \neq \frac{2}{3}, \quad \text{avec} \, a,b \in \mathbb{R}, \text{tel que} \, b \neq 2a.
   \]
   Here, using contrapositive reasoning, you would show that if \( \frac{a+2b}{2a-b} = \frac{2}{3} \), then \( b = \frac{1}{8}a \), given that \( b \neq 2a \).

3. **Statement 3:**
   \[
   \forall (x,y,z) \in \mathbb{R}^3 : (xy \neq 1 \, \text{et} \, x \neq y) \Rightarrow \frac{x}{x^2 + x + 1} \neq \frac{y}{y^2 + y + 1}
   \]
   Prove that if \( xy \neq 1 \) and \( x \neq y \), then the fractions \( \frac{x}{x^2 + x + 1} \) and \( \frac{y}{y^2 + y + 1} \) are unequal. Using contrapositive reasoning, you would prove that if \( \frac{x}{x^2 + x + 1} = \frac{y}{y^2 + y + 1} \), then either \( xy = 1 \) or \( x = y \).

4. **Statement 4:**
   \[
   \forall x \in \left( \frac{2}{3}, +\infty \right], \forall y \in \left( \frac{2}{3}, +\infty \right], x \neq y \Rightarrow 3x^2 - 4x \neq 3y^2 - 4y.
   \]
   Here, the goal is to prove that if \( x \neq y \), then the quadratic expressions \( 3x^2 - 4x \) and \( 3y^2 - 4y \) are not equal. Using the contrapositive, you need to prove that if \( 3x^2 - 4x = 3y^2 - 4y \), then \( x = y \).

5. **Statement 5:**
   \[
   \forall a \in \mathbb{R}^+ : a^4 + 8a^3 + 18a^2 + 8a - 2 > 0 \Rightarrow \sqrt{3} + \sqrt{3} - 2 < a.
   \]
   Finally, you need to show that the given polynomial expression in \( a \) is positive only if \( a \) is greater than \( \sqrt{3} + \sqrt{3} - 2 \). The contrapositive approach would involve proving that if \( a \leq \sqrt{3} + \sqrt{3} - 2 \), then the polynomial is non-positive.

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Would you like to dive into one of these proofs in detail? 

Here are 5 related questions to deepen your understanding:
1. How does the contrapositive approach simplify proving a statement?
2. Can you provide a step-by-step example of how to apply contrapositive reasoning to a mathematical equation?
3. What are some common scenarios where contrapositive reasoning is more efficient than direct proof?
4. How can inequalities involving polynomials be handled using contrapositive reasoning?
5. How would you approach cases where both contrapositive and direct proof seem equally complex?

**Tip:** When working with contrapositive reasoning, remember that proving the negation of the conclusion often simplifies algebraic manipulations.

En utilisant, le raisonnement par la contraposée, démontrer que: various mathematical proofs are provided in the exercise list.

Explore detailed solutions and discussions on using contrapositive reasoning to solve advanced mathematical problems, including algebraic expressions, inequalities, and polynomial relations. These exercises are designed for university-level students.

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