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There are 15 (not necessarily distinct) integers chosen uniformly at random from the range from 0 to 999, inclusive. Albert then computes the sum of their units digits, while Bob ...

Let \(S\) be the set of pairs of real numbers \((x,y)\) such that for at least one pair of positive integers \(m,n\),

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\[ \large \prod _{ n=1 }^{ \infty }{ \left( 1+\frac { 1 }{ \left\lfloor \sqrt [ 100 ]{ n } \right\rfloor } \right) }^{(-1)^{n-1}} =\frac { \pi }{ a } \]

Find the value of \(a\) that satisfies ...

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