Unexpectedly real

Algebra Level 3

Let a,b,ca,b,c be real numbers so that a+b+c=1a+b+c=1. Find the maximum value of a1+a2+b1+b2+c1+c2\frac { a }{ 1+a^{ 2 } } +\frac { b }{ 1+{ b }^{ 2 } } +\frac { c }{ 1+{ c }^{ 2 } } .

If the answer can be written as xy\frac{x}{y}, where x,yx,y are positive integers so that gcd(x,y)=1\gcd(x,y)=1, find xy\left| x-y \right| .

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