The inequality to be proven is:

$(x + y + z) \cdot 2 \left( \frac{1}{2} + \frac{1}{x^2 + y^2} + \frac{1}{y^2 + z^2} + \frac{1}{z^2 + x^2} \right) + \frac{36}{x + y + z} \geq \frac{63}{4}$

Given:

• $x, y, z$ are non-negative real numbers.
• No two of $x, y, z$ are equal to zero.

Approach:

1. Simplify the Expression: First, you can consider simplifying the expression by factoring out common terms and finding symmetry within the variables.

2. Use Symmetry and Inequalities: Since the expression involves sums of squares and reciprocal terms, you could apply classical inequalities like the AM-GM inequality, Cauchy-Schwarz inequality, or Titu's lemma to get bounds on the terms.

3. Estimate the Bound: You want to show that the left-hand side is always greater than or equal to $\frac{63}{4}$. Focus on the individual terms that might lead to a contradiction if the inequality does not hold, and try using substitutions for easier terms such as letting $x = y = z$ as a special case to test.

Suggestions for Next Steps:

• Do you want a detailed step-by-step solution using one of the classical inequalities?
• Would you like assistance with a specific part of the inequality manipulation?

Related Questions:

1. How can the AM-GM inequality help simplify expressions with sums of squares?
2. What are the applications of the Cauchy-Schwarz inequality in inequalities involving multiple terms?
3. Can we test this inequality by setting specific values for $x, y, z$ and see if it holds?