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Let a1,...,a2048a_1,...,a_{2048}a1,...,a2048 be non-negative real numbers so that ∑i=12048ai=1\sum _{ i=1 }^{ 2048 }{ { a }_{ i } }=1 ∑i=12048ai=1. Find the maximum value of ∑i=12048aiai2+1\sum _{ i=1 }^{ 2048 }{ \frac { { a }_{ i } }{ { a }_{ i }^{ 2 }+1 } } ∑i=12048ai2+1ai.
If the answer can be written as xy\frac{x}{y}yx, where x,yx,yx,y are positive integers so that (x,y)=1(x,y)=1(x,y)=1, find ∣x−y∣\left| x-y \right| ∣x−y∣.
Bonus: Generalize!
Inspiration.
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