The inequality problem in the image reads:

Given $a, b, c > 0$ such that:

$(2a + b + c)(a + 2b + c)(a + b + 2c) = 64$

Prove that:

$\frac{\sqrt{2a^4b^3}}{\sqrt{a^5 - a + b^2 + 1}} + \frac{\sqrt{2b^4c^3}}{\sqrt{b^5 - b + c^2 + 1}} + \frac{\sqrt{2c^4a^3}}{\sqrt{c^5 - c + a^2 + 1}} \leq 3$

Outline for the proof:

1. Symmetry in the problem:

   • The symmetry among $a$, $b$, and $c$ suggests that applying an inequality like AM-GM (Arithmetic Mean - Geometric Mean inequality) may simplify the proof.

2. AM-GM Inequality:

   • Consider applying the AM-GM inequality to each term in the sum. Since all terms are positive and symmetric, AM-GM inequality might help to bound each expression.

3. Bound the individual terms:

   • Each individual fraction can be simplified to check if there are any straightforward bounds, likely related to properties of symmetric polynomials or inequalities involving products of $a, b, c$.

4. Handling the condition:

   • The given condition $(2a + b + c)(a + 2b + c)(a + b + 2c) = 64$ could simplify certain terms or help derive necessary bounds on the individual variables.

5. Final comparison:

   • Summing the bounds for each term and checking if they collectively are less than or equal to 3 will lead to the final result.

Would you like me to work out the steps for solving this inequality using these ideas, or do you have a specific part you'd like to focus on?

Related Questions:

1. How can we derive bounds on expressions with multiple variables like $a$, $b$, and $c$?
2. What are some typical strategies for solving symmetrical inequalities?
3. Can the condition $(2a + b + c)(a + 2b + c)(a + b + 2c) = 64$ be generalized for other values?
4. How does AM-GM inequality help in solving inequalities involving sums of radicals?
5. What are the challenges of proving inequalities with three variables?

Tip:

When solving inequalities, always look for symmetrical patterns between variables. Symmetry can simplify your problem significantly.