If x, y, z are distinct positive real numbers then find the range of {9(x²+y²)÷4z²}+{9(y²+z²)÷4x²}+{9(x²+z²)÷4y²}

We are tasked with finding the range of the following expression involving three distinct positive real numbers \( x \), \( y \), and \( z \):

\[
S = \frac{9(x^2 + y^2)}{4z^2} + \frac{9(y^2 + z^2)}{4x^2} + \frac{9(x^2 + z^2)}{4y^2}.
\]

### Step 1: Symmetry in the expression
The expression \( S \) is symmetric in terms of \( x \), \( y \), and \( z \), so we expect the minimum value to occur when \( x = y = z \), and then we can explore the general behavior.

### Step 2: Substituting \( x = y = z \)
Let \( x = y = z = t \). Substituting this into the expression for \( S \):

\[
S = \frac{9(t^2 + t^2)}{4t^2} + \frac{9(t^2 + t^2)}{4t^2} + \frac{9(t^2 + t^2)}{4t^2}
= 3 \times \frac{9(2t^2)}{4t^2} = 3 \times \frac{18}{4} = 3 \times 4.5 = 13.5.
\]

So, when \( x = y = z \), \( S = 13.5 \).

### Step 3: Investigating the range of \( S \)
Now, let's explore the behavior when \( x \), \( y \), and \( z \) are distinct positive real numbers.

Since \( x^2 + y^2 \geq 2xy \) by the arithmetic mean-geometric mean inequality (AM-GM inequality), each term in the expression \( S \) is positive. Additionally, since the denominator involves the squares of \( x \), \( y \), and \( z \), making one of the variables very small compared to the others will cause one of the terms to grow without bound.

Thus, as \( x \), \( y \), or \( z \) approach zero or infinity, the expression \( S \) increases indefinitely.

### Step 4: Conclusion
- The minimum value of \( S \) occurs when \( x = y = z \), and it is \( 13.5 \).
- As \( x \), \( y \), or \( z \) become large or small, \( S \) can become arbitrarily large.

Therefore, the range of \( S \) is:

\[
[13.5, \infty).
\]

Would you like further clarifications, or have additional questions? Here are 5 related questions you might find interesting:

1. What is the AM-GM inequality, and how is it applied in optimization problems?
2. Can this technique be generalized to higher-order symmetric expressions?
3. What is the behavior of the expression if \( x \), \( y \), and \( z \) are not distinct?
4. How would the range change if \( x \), \( y \), and \( z \) are constrained within a specific interval?
5. Can similar methods be used to explore other types of inequalities involving symmetric functions?

**Tip:** Symmetry in mathematical expressions often provides a way to simplify and minimize or maximize functions, making it easier to solve optimization problems.

If x, y, z are distinct positive real numbers, then find the range of {9(x²+y²)÷4z²}+{9(y²+z²)÷4x²}+{9(x²+z²)÷4y²}.

Explore the range of the expression {9(x²+y²)/4z²} + {9(y²+z²)/4x²} + {9(x²+z²)/4y²} for distinct positive real numbers x, y, and z using algebraic symmetry and the AM-GM inequality. Learn how to solve this problem with a detailed step-by-step solution.

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