The answer is not 0

Algebra Level 5

Let a,b,ca,b,c be non-negative reals. Find the maximum value of kk such that (a+b+c)(1a+1b+1c)+k(ab+bc+ca)a2+b2+c29+k.(a+b+c)\left(\frac { 1 }{ a } +\frac { 1 }{ b } +\frac { 1 }{ c } \right)+\frac { k(ab+bc+ca) }{ { a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } } \ge 9+k. Write your answer to 3 decimal places.

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