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As \(n\) approaches infinity, \(\dbinom{2n}{n}\) is asymptotic to \(k\dfrac{A^n}{n^B}\) for some positive constant \(k.\) Find the value of \(\dfrac{A}{B}.\)

\[x^{1729}\equiv 1\pmod{2017}\]

How many positive integer solutions \(x<2017\) does the congruency above have?

\[\large \int_{-1/\sqrt{3}}^{1/\sqrt{3}} \dfrac{x^2(1-x^2)}{(e^x+1)(1+x^2)^4} \, dx \]

If the above integral is equal to ...

Let the function \(\Theta (n)\) denote the sum of all natural numbers less than or equal to \(n\). However, this function has one trick - if the number to be added ...

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