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In this exercise, we are given that \( a \), \( b \), and \( c \) are positive real numbers. The goal is to prove the inequality:

\[
\frac{1}{a^2 + b^2 + c^2} \leq \frac{1}{2ab}
\]

### Approach to the solution:

We can break this proof into steps, likely using standard inequality techniques such as:

1. **AM-GM inequality (Arithmetic Mean - Geometric Mean)**: 
   - This inequality states that for any non-negative real numbers \(x\) and \(y\), the arithmetic mean is always greater than or equal to the geometric mean:
     \[
     \frac{x + y}{2} \geq \sqrt{xy}
     \]
     Applied to the terms involving \(a\), \(b\), and \(c\).

2. **Other useful inequalities**:
   - There might be a way to use symmetry or algebraic manipulation of the terms to simplify the problem.

Would you like a detailed step-by-step solution for this inequality proof?

Here are some related questions to expand on this:
1. What is the AM-GM inequality, and how can it be applied here?
2. Can we use other inequalities such as the Cauchy-Schwarz inequality in this context?
3. What assumptions about \(a\), \(b\), and \(c\) might simplify the problem?
4. Can this inequality hold in the case when \(a = b = c\)?
5. Are there any other inequalities that can be useful in this type of problem?

**Tip**: Inequalities often rely on algebraic simplifications or transformations, so always check for symmetry or standard forms like sums of squares.

Démontre que \( \frac{1}{a^2 + b^2 + c^2} \leq \frac{1}{2ab} \), where \(a\), \(b\), and \(c\) are positive real numbers.

This problem focuses on proving the inequality \( \frac{1}{a^2 + b^2 + c^2} \leq \frac{1}{2ab} \) using algebraic techniques and the Arithmetic Mean - Geometric Mean Inequality. Suitable for students in Grades 10-12.

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