Bài 12

To solve the problems presented in the text, we can break them down into two parts.

### Part 1:
You need to prove the inequality:

\[
\frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \geq \frac{3}{2}
\]

This is a well-known inequality that can be approached using techniques such as the Cauchy-Schwarz inequality or Nesbitt's inequality.

### Part 2:
Given the conditions:

\[
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 3
\]

we are required to find the value of \( P \) defined as:

\[
P = \frac{1}{\sqrt{a^2 + ab + b}} + \frac{1}{\sqrt{b^2 + bc + c}} + \frac{1}{\sqrt{c^2 + ca + a}}
\]

To evaluate \( P \), we substitute \( a, b, c \) based on the condition provided and apply the appropriate inequalities or algebraic manipulations.

### Steps to Solve:
1. **Prove Part 1**:
Use techniques like the Cauchy-Schwarz inequality to demonstrate:
\[
\left( \frac{a}{b+c} + \frac{b}{c+a} + \frac{c}{a+b} \right) \left( (b+c) + (c+a) + (a+b) \right) \geq (a+b+c)^2
\]

Manipulate this to show that \( \frac{3}{2} \) holds.

2. **Evaluate Part 2**:
From the condition \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 3 \), we can express \( a, b, c \) in terms of a common variable, say \( k \) so that they sum to a specific value or equalize them.

Use symmetry and known inequalities (like Cauchy-Schwarz) to simplify \( P \).

### Conclusion:
After these steps, you would arrive at the required proofs and evaluations. If you'd like to see any particular derivation or deeper exploration into one of the problems, just let me know!