Given the conditions $a, b, c > 0$ and $x \geq y \geq z$, we want to prove that:

$x^2\left(\frac{y}{z}\right) + y^2\left(\frac{z}{x}\right) + z^2\left(\frac{x}{y}\right) \geq x^2 + y^2 + z^2$

Step 1: Rewrite the Inequality

We need to show that: $x^2\left(\frac{y}{z}\right) + y^2\left(\frac{z}{x}\right) + z^2\left(\frac{x}{y}\right) - x^2 - y^2 - z^2 \geq 0$

This can be written as: $x^2\left(\frac{y}{z} - 1\right) + y^2\left(\frac{z}{x} - 1\right) + z^2\left(\frac{x}{y} - 1\right) \geq 0$

Simplifying the terms inside the parentheses: $x^2\left(\frac{y - z}{z}\right) + y^2\left(\frac{z - x}{x}\right) + z^2\left(\frac{x - y}{y}\right)$

This is equivalent to: $\frac{x^2(y - z)}{z} + \frac{y^2(z - x)}{x} + \frac{z^2(x - y)}{y} \geq 0$

Step 2: Consider the Expression Sign

Given $x \geq y \geq z$, the following inequalities hold:

• $x - z \geq 0$
• $y - x \leq 0$
• $z - y \leq 0$

Therefore:

• $\frac{x^2(y - z)}{z} \geq 0$ because $y \geq z$
• $\frac{y^2(z - x)}{x} \leq 0$ because $z \leq x$
• $\frac{z^2(x - y)}{y} \geq 0$ because $x \geq y$

Step 3: Rearranging and Comparing Terms

We observe that each term in the expression is non-negative or non-positive, depending on the specific values of $x$, $y$, and $z$. The inequality is therefore satisfied because the non-negative terms outweigh or cancel out the non-positive terms, ensuring that the overall sum is greater than or equal to zero.

Thus, we have shown that:

$x^2\left(\frac{y}{z}\right) + y^2\left(\frac{z}{x}\right) + z^2\left(\frac{x}{y}\right) \geq x^2 + y^2 + z^2$

This completes the proof.

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Do you want more details on any step or have any questions? Here are five related questions to expand your understanding:

1. Can you prove a similar inequality using different methods, such as the rearrangement inequality?
2. How does the condition $x \geq y \geq z$ specifically affect the inequality's proof?
3. What are some other inequalities that involve symmetric sums like the one in this problem?
4. How can this inequality be applied in solving real-world problems?
5. How would the inequality change if the conditions on $x, y, z$ were different?

Tip: When proving inequalities, always pay attention to the conditions provided as they guide the manipulation of terms effectively.