Well, I like 2048!

Algebra Level 4

Let a1,...,a2048a_1,...,a_{2048} be non-negative real numbers so that i=12048ai=1\sum _{ i=1 }^{ 2048 }{ { a }_{ i } }=1 . Find the maximum value of i=12048aiai2+1\sum _{ i=1 }^{ 2048 }{ \frac { { a }_{ i } }{ { a }_{ i }^{ 2 }+1 } } .

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

Bonus: Generalize!

Inspiration.

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