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The problem presented involves three real numbers \(a\), \(b\), and \(c\), subject to the condition that \(a^2 + b^2 + c^2 = 2\). There are two parts to the question:

1. **Prove that if \( a \leq 0 \), then \( a + b + c \leq abc + 2 \).**

2. **Show that \( a + b + c \leq abc + 2 \) for all \( a\), \(b\), and \(c \in \mathbb{R} \).**

Let's break this down step-by-step.

### Part 1: Proving \( a + b + c \leq abc + 2 \) when \( a \leq 0 \)
We are tasked to prove the inequality for the specific case when \( a \leq 0 \).

Given the condition \( a^2 + b^2 + c^2 = 2 \), we can use this to explore the behavior of the inequality in cases where \( a \leq 0 \). We would likely have to employ specific algebraic manipulations or explore bounding techniques using inequalities or substitutions based on the given constraint. Special cases for \( a = 0 \) might also help build intuition for general proof construction.

### Part 2: Proving \( a + b + c \leq abc + 2 \) for all \(a\), \(b\), and \(c \in \mathbb{R}\)
This is a more general form of the inequality, and we must establish it holds for any values of \( a\), \(b\), and \(c \) under the condition \( a^2 + b^2 + c^2 = 2 \).

### Strategy
1. Investigate if any symmetric approaches can be applied since the equation involves \( a^2 + b^2 + c^2 \).
2. Analyze extreme cases or specific values (e.g., setting \( a = b = c = 1 \), \( a = -1 \), etc.) to better understand the boundaries of the inequality.
3. Utilize known inequalities, such as AM-GM or Cauchy-Schwarz, to potentially bound terms.

Would you like a detailed walkthrough of the proof for both parts, or shall I focus on specific aspects of the problem?

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Here are 5 follow-up questions that expand on this information:

1. How can we simplify the inequality in part 1 using the condition \(a^2 + b^2 + c^2 = 2\)?
2. Can we apply a symmetry argument to solve the problem for all real values of \(a\), \(b\), and \(c\)?
3. What happens to the inequality if we assume \(a = 0\), \(b = 1\), and \(c = 1\)?
4. Could the inequality \( a + b + c \leq abc + 2 \) be proven using calculus, by finding maximum and minimum values of the expression?
5. Is there a geometric interpretation of the constraint \(a^2 + b^2 + c^2 = 2\)?

**Tip**: When dealing with constraints like \(a^2 + b^2 + c^2 = 2\), visualizing them geometrically or using symmetry properties can often simplify the proof.

Soient a, b et c des nombres réels tels que : a^2 + b^2 + c^2 = 2. Prouver que si a ≤ 0 alors a + b + c ≤ abc + 2. Montrer que a + b + c ≤ abc + 2 pour tout a, b et c dans R.

Explore the proof of the inequality a + b + c ≤ abc + 2 under the condition a^2 + b^2 + c^2 = 2. This problem involves key concepts in inequalities and real numbers, suitable for undergraduate students.

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