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Let \(a >1\) be a real number which is not an integer, and let \(k\) be the smallest positive integer such that ...

\[\large \sum_{k=1}^\infty \dfrac{\sin k}k = \dfrac{\pi^a - b } c \]

If the equation above holds true for positive integers \(a,b\) and \(c\), find ...

\[ \large \int_{0}^{\frac{\pi}{2}} (\ln\sin{x})^2 dx = \frac{\pi^A}{B}+\pi\ln^2(\sqrt{C}^{\sqrt{C}}) \] The equation above is satisfied for positive integers ...

For an integer \(p\), define \[f_p(\alpha) = e^{i\alpha /p^2} . e^{2i\alpha /p^2} . e^{3i\alpha /p^2} ........ e^{pi\alpha /p^2}\]

Evaluate...

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