# Well, I like 2048!

Algebra Level 4

Let $a_1,...,a_{2048}$ be non-negative real numbers so that $\sum _{ i=1 }^{ 2048 }{ { a }_{ i } }=1$. Find the maximum value of $\sum _{ i=1 }^{ 2048 }{ \frac { { a }_{ i } }{ { a }_{ i }^{ 2 }+1 } }$.

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

Bonus: Generalize!

Inspiration.

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