Proving x y z Using Inequalities and Substitutions

In this article, we will explore a detailed step-by-step process to prove that x y z given the equation:

(frac{x^2}{y^2} cdot frac{y^2}{z^2} cdot frac{z^2}{x^2} frac{x}{y} cdot frac{y}{z} cdot frac{z}{x})

Introduction

We will introduce new variables and manipulate the equation using inequalities and substitutions to prove that x y z. This approach showcases a methodical and rigorous mathematical proof technique.

Introducing New Variables

Let's define three new variables:

a frac{x}{y} b frac{y}{z} c frac{z}{x}

These substitutions will help simplify our original equation and make the subsequent steps more manageable.

Substituting and Simplifying

Given the substitutions, we can rewrite our original equation as:

(a^2 cdot b^2 cdot c^2 a cdot b cdot c)

This can be further simplified to:

(a^2 cdot b^2 cdot c^2 - a cdot b cdot c 0)

Completing the Square

To further simplify the equation, we can complete the square:

((a - frac{1}{2})^2 (b - frac{1}{2})^2 (c - frac{1}{2})^2 - frac{3}{4} 0)

This simplifies to:

((a - frac{1}{2})^2 (b - frac{1}{2})^2 (c - frac{1}{2})^2 frac{3}{4})

Analyzing the Equation

The expression on the left is a sum of squares, which is always non-negative. For the equality to hold, each term must be equal to zero:

(a - frac{1}{2} 0)

(b - frac{1}{2} 0)

(c - frac{1}{2} 0)

Therefore, we have:

(a frac{1}{2}, b frac{1}{2}, c frac{1}{2})

Returning to Original Variables

Reverting back to the original variables, we get:

(frac{x}{y} frac{1}{2}, frac{y}{z} frac{1}{2}, frac{z}{x} frac{1}{2})

From (frac{x}{y} frac{1}{2}), we find:

(x frac{1}{2} y)

From (frac{y}{z} frac{1}{2}), we find:

(y frac{1}{2} z)

From (frac{z}{x} frac{1}{2}), we find:

(z frac{1}{2} x)

Finding the Relationship

Substituting (y frac{1}{2} z) into (x frac{1}{2} y), we get:

(x frac{1}{2} left( frac{1}{2} z right) frac{1}{4} z)

Substituting (z frac{1}{2} x) into (x frac{1}{4} x), we get:

(x frac{1}{4} left( frac{1}{2} x right) frac{1}{8} x)

This implies that (x 0) or (x y z).

Conclusion

Therefore, the only solution satisfying the given equation is:

(x y z)

We have thus shown that all three variables must be equal.