Inequality Masterpiece

Algebra Level 4

\[\large \frac{ab+1}{(a+b)^2} + \frac{bc+1}{(b+c)^2} + \frac{ca+1}{(c+a)^2}\]

Positive real numbers \(a\), \(b\) and \(c\) are such that \(a^2 + b^2 + c^2 + (a+b+c)^2 \le 4\). Find the minimum value of the expression above.


Question from USAMO 2011 (slightly modified).

Bonus: Prove it and post your solution.
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