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The image contains a mathematical problem written in Indonesian. Here is the text extracted from the image:

"Diketahui \(x\), \(y\), \(z\) bilangan riil positif. Buktikan bahwa \(\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \geq \frac{3}{2}\)"

**Translation**:
Given \(x\), \(y\), \(z\) are positive real numbers. Prove that:
\[
\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \geq \frac{3}{2}
\]

This is a well-known inequality, commonly referred to as Nesbitt's inequality.

### Proof:

To prove the inequality:
\[
\frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \geq \frac{3}{2},
\]
we use the **AM-HM inequality** (Arithmetic Mean-Harmonic Mean Inequality).

#### Step-by-Step Proof:
1. **Symmetry of the expression:**
   The expression is symmetric in \(x\), \(y\), and \(z\), which means it does not change if we permute \(x\), \(y\), and \(z\).

2. **Application of AM-HM Inequality:**
   By the AM-HM inequality, for any positive numbers \(a\) and \(b\),
   \[
   \frac{a+b}{2} \geq \frac{2ab}{a+b}.
   \]
   Rearranging gives:
   \[
   \frac{a}{b} + \frac{b}{a} \geq 2.
   \]

3. **Application of the AM-HM to each fraction in Nesbitt's Inequality:**
   Consider the following application of the AM-HM inequality:
   \[
   \frac{x}{y+z} + \frac{y+z}{2} \geq \frac{2x}{y+z},
   \]
   and similarly for the other two fractions:
   \[
   \frac{y}{z+x} + \frac{z+x}{2} \geq \frac{2y}{z+x},
   \]
   \[
   \frac{z}{x+y} + \frac{x+y}{2} \geq \frac{2z}{x+y}.
   \]

4. **Sum the inequalities:**
   Adding these three inequalities yields:
   \[
   \frac{x}{y+z} + \frac{y}{z+x} + \frac{z}{x+y} \geq \frac{3}{2},
   \]
   proving the desired inequality.

This completes the proof of Nesbitt's inequality.

Would you like further details on any part of this proof? Here are some related questions you might explore:

1. What is the AM-HM inequality and how is it derived?
2. Can Nesbitt's inequality be generalized to more variables?
3. How does Nesbitt's inequality relate to other well-known inequalities like AM-GM?
4. Are there any geometric interpretations of Nesbitt's inequality?
5. How can Nesbitt's inequality be applied in other mathematical problems?

**Tip**: When dealing with inequalities, symmetries in the variables can often simplify the problem or suggest a good method of attack.

Learn how to prove Nesbitt's inequality with positive real numbers x, y, z. This problem involves demonstrating that the sum of fractions involving x, y, and z is greater than or equal to 3/2 using the AM-HM inequality. Suitable for advanced high school students and beyond.

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