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Let \(S_{k}\) be the area bounded by the curve \(y=x^2 (1-x)^k \) and the lines \(x=0\), \(y=0\) and \(x=1\). If ...

\[ \large{{\underbrace{6666666\ldots68}_{10^5 \text{ digits } }}^2} \]

Find the sum of the digits of the number above (when it is expanded).

Welcome all to the first ever Brilliant Mechanics Contest. Like the Brilliant Integration Contest, the aim of the Mechanics Contest is to improve skills and techniques often used in Olympiad ...

\[\large \dfrac{3 \tan^4 \left( \dfrac\pi7 \right) + 1}{35+ \tan^4\left( \dfrac\pi7 \right)} = \dfrac{A}{B} \tan^{C} \left( \dfrac\pi7 \right) \]

Given that the equation ...

\[ \large \dfrac{3 - \tan^2\left( \dfrac\pi7 \right)}{1 - \tan^2\left( \dfrac\pi7 \right)} \div \cos\left( \dfrac\pi7 \right) = \, ? \]

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