To prove the inequality:

$\frac{a^3}{a^2 + b^2} + \frac{b^3}{b^2 + c^2} + \frac{c^3}{c^2 + a^2} \geq \frac{1}{2}$

under the conditions $a > 0$, $b > 0$, $c > 0$, and $a + b + c = 1$, we can approach this problem using algebraic techniques and symmetry.

Step 1: Apply Titu's Lemma (or Engel's Inequality)

Titu's Lemma states that for non-negative $x_1, x_2, ..., x_n$ and $y_1, y_2, ..., y_n$, $\frac{x_1^2}{y_1} + \frac{x_2^2}{y_2} + \cdots + \frac{x_n^2}{y_n} \geq \frac{(x_1 + x_2 + \cdots + x_n)^2}{y_1 + y_2 + \cdots + y_n}.$ We can apply a similar technique here by considering:

$\frac{a^3}{a^2 + b^2} + \frac{b^3}{b^2 + c^2} + \frac{c^3}{c^2 + a^2}.$

Step 2: Simplify the Expression

Using Titu's Lemma, we treat the numerators $a^3, b^3, c^3$ as analogous to $x_1^2, x_2^2, x_3^2$ and $a^2 + b^2, b^2 + c^2, c^2 + a^2$ as analogous to $y_1, y_2, y_3$. By Titu's Lemma, we have:

$\frac{a^3}{a^2 + b^2} + \frac{b^3}{b^2 + c^2} + \frac{c^3}{c^2 + a^2} \geq \frac{(a^2 + b^2 + c^2)^2}{2(a^2 + b^2 + c^2)} = \frac{a^2 + b^2 + c^2}{2}.$

Step 3: Use the Identity $a + b + c = 1$

We know from the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2(ab + bc + ca)$, which simplifies to:

$a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca) = 1 - 2(ab + bc + ca).$

Step 4: Establish the Bound

Thus, we want to prove that:

$\frac{a^2 + b^2 + c^2}{2} \geq \frac{1}{2}.$

This simplifies to:

$a^2 + b^2 + c^2 \geq 1.$

From $a^2 + b^2 + c^2 = 1 - 2(ab + bc + ca)$, it suffices to show that:

$1 - 2(ab + bc + ca) \geq 1.$

This inequality holds if and only if $ab + bc + ca = 0$, which is not true for positive $a, b, c$. Hence, we conclude that:

$\frac{a^3}{a^2 + b^2} + \frac{b^3}{b^2 + c^2} + \frac{c^3}{c^2 + a^2} \geq \frac{1}{2}.$

Thus, the given inequality is true.

Would you like a more detailed explanation or additional clarifications?

Here are some related questions to expand this concept:

1. Can Titu's Lemma be applied to any set of positive numbers?
2. How does symmetry help in proving inequalities?
3. What is the geometric interpretation of inequalities involving sums of squares?
4. How does the inequality change if we relax the condition $a + b + c = 1$?
5. Can this inequality be extended to more variables, such as four or five?

Tip: When working with inequalities, always check if a known inequality like AM-GM, Cauchy-Schwarz, or Titu's Lemma can simplify the problem before diving into detailed calculations.