The Answer Is Not 0.225

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

\[ \frac{ a}{ \sqrt{1-c} } + \frac{b} { \sqrt{1-a} } + \frac{ c} { \sqrt{1-b} } \leq 1 + k ? \]

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