Question 2
Let $a$, $b$, $c$ be positive real numbers
such that $a^2+b^2+c^2+(a+b+c)^2 \le 4$. Prove that
\[\frac{ab+1}{(a+b)^2} + \frac{bc+1}{(b+c)^2} + \frac{ca+1}{(c+a)^2} \ge 3.\]
Proposed by Titu Andreescu (credit in the supplied notes).
such that $a^2+b^2+c^2+(a+b+c)^2 \le 4$. Prove that
\[\frac{ab+1}{(a+b)^2} + \frac{bc+1}{(b+c)^2} + \frac{ca+1}{(c+a)^2} \ge 3.\]
Proposed by Titu Andreescu (credit in the supplied notes).