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The image contains the following mathematical problem involving three real numbers \( a \), \( b \), and \( c \), which satisfy:

\[
a^2 + b^2 + c^2 = 2
\]

There are two questions:

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 real \( a \), \( b \), and \( c \).**

Let's break down these problems.

### 1. Proving the inequality when \( a \leq 0 \):

Given the condition \( a^2 + b^2 + c^2 = 2 \), the first task is to show that:

\[
a + b + c \leq abc + 2
\]

under the additional assumption that \( a \leq 0 \). This will likely involve manipulating both sides of the inequality using the given condition \( a^2 + b^2 + c^2 = 2 \) and exploiting properties of real numbers when \( a \leq 0 \).

### 2. Proving the general inequality:

The second part is a more general claim that for any real numbers \( a \), \( b \), and \( c \) satisfying \( a^2 + b^2 + c^2 = 2 \), the inequality:

\[
a + b + c \leq abc + 2
\]

always holds. This will likely involve a more general approach, potentially using techniques from inequalities, such as symmetry or optimization over real numbers.

Would you like me to walk through the detailed solution for either or both of these problems?

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Here are 5 related questions to deepen your understanding:

1. How would the solution change if \( a^2 + b^2 + c^2 = 3 \)?
2. Can this type of inequality be extended to four variables?
3. What other inequalities can be derived from the condition \( a^2 + b^2 + c^2 = 2 \)?
4. Can the inequality \( a + b + c \leq abc + 2 \) be proven using geometric methods?
5. What role does symmetry play in solving inequalities like these?

**Tip**: When dealing with inequalities, always check if specific cases (e.g., \( a = 0 \), \( a = b = c \)) can simplify the problem or provide insight into the general behavior.

Let a, b, and c be real numbers such that: a^2 + b^2 + c^2 = 2. 1. Prove that if a ≤ 0 then a + b + c ≤ abc + 2. 2. Show that a + b + c ≤ abc + 2 for all a, b, and c in ℝ.

Explore how to prove inequalities involving real numbers a, b, and c with the condition a^2 + b^2 + c^2 = 2. This problem addresses two cases: when a is non-positive and the general case. Learn the step-by-step approach to handling such inequalities.

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