# Unexpectedly real

Algebra Level 3

Let $a,b,c$ be real numbers so that $a+b+c=1$. Find the maximum value of $\frac { a }{ 1+a^{ 2 } } +\frac { b }{ 1+{ b }^{ 2 } } +\frac { c }{ 1+{ c }^{ 2 } }$.

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

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