The Answer Is Not 0.225

Algebra Level 5

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

a1c+b1a+c1b1+k? \frac{ a}{ \sqrt{1-c} } + \frac{b} { \sqrt{1-a} } + \frac{ c} { \sqrt{1-b} } \leq 1 + k ?

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