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Suppose [tex]X[/tex] is a set of 16 distinct positive integers, [tex]X=\left\{{x_{1}, \cdots , x_{16}}\right\}[/tex].

Then, for every [tex]X[/tex], there exists some integer [tex]k\in\left\{{1, \cdots , 8}\right\}[/tex] and disjoint subsets [tex]A,B\subset X[/tex]

[tex]A=\left\{a_{1},\cdots\ ,a_{k}\right\}[/tex] and [tex]B=\left\{b_{1},\cdots\ ,b_{k}\right\}[/tex]

such that [tex]\left|\alpha - \beta\right|<.00025[/tex],

where [tex]\alpha= \frac{1}{a_{1}}+\cdots+\frac{1}{a_{k}}[/tex] and [tex]\beta= \frac{1}{b_{1}}+\cdots+\frac{1}{b_{k}}[/tex].

I know that .00025 is pretty close to 2^-12.

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# Summing sets of inverses of integers

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