The Answer Is Not 0.5

Algebra Level 5

What is the smallest real number $k$ (to 3 decimal places), such that for all ordered triples of non-negative reals $(a,b,c)$ which satisfy $a + b + c = 1$ , we have

$\frac{ a^2 + bc} { a+1} + \frac{ b^2+ca} { b+1} + \frac{ c^2+ab} { c+1} \leq k?$

If you want a similar inequality, you can try The Answer Is Not 0.225.