The problem consists of three inequalities to prove for real numbers $a, b,$ and $c$. Let’s go through each statement and provide a proof.

1. Prove $a^2 + b^2 \geq 2ab$

This is a form of the AM-GM (Arithmetic Mean-Geometric Mean) inequality or can be proven by the square of a difference approach:

$a^2 + b^2 - 2ab = (a - b)^2 \geq 0$

Since $(a - b)^2 \geq 0$ for all real $a$ and $b$, it follows that: $a^2 + b^2 \geq 2ab$

2. Prove $a^2 + b^2 + c^2 \geq ab + bc + ca$

We can prove this by rewriting it in a similar way as in the first part, using the sum of squares technique.

Consider: $a^2 + b^2 + c^2 - ab - bc - ca = \frac{1}{2} \left((a - b)^2 + (b - c)^2 + (c - a)^2\right) \geq 0$

Since each term $(a - b)^2$, $(b - c)^2$, and $(c - a)^2$ is non-negative, the entire expression is non-negative. Therefore: $a^2 + b^2 + c^2 \geq ab + bc + ca$

3. Given $a + b + c \geq 0$, show that $a^3 + b^3 + c^3 \geq 3abc$

This inequality can be proved by Muirhead's inequality or Schur's inequality. Since $a + b + c \geq 0$, we can apply Schur’s inequality:

For real numbers $a, b, c$ such that $a + b + c \geq 0$, Schur's inequality states: $a^3 + b^3 + c^3 + 3abc \geq ab(a + b) + bc(b + c) + ca(c + a)$

Given $a + b + c \geq 0$, this leads to: $a^3 + b^3 + c^3 \geq 3abc$

This completes the proofs.

Would you like further details on any of the proofs? Here are five additional questions to explore related concepts:

1. How does the Cauchy-Schwarz inequality relate to these types of inequalities?
2. Can these inequalities be extended to higher dimensions (e.g., with more variables)?
3. What are some real-world applications of these inequalities?
4. How would these proofs change if we restricted $a, b, c$ to be positive real numbers?
5. Can these inequalities be used to derive other common inequalities in mathematics?

Tip: When proving inequalities, look for common techniques like completing the square, using the AM-GM inequality, or breaking down the inequality into sums of squares.