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\[\large \lim _{ n\rightarrow \infty }{ { 2 }^{ n }\underbrace { \sqrt { 2-\sqrt { 2+\sqrt { 2+\dots +\sqrt { 2 } } } } }_{ n-1 ~~ \text{roots}} } = \ ?\]

If \(f\) is a real valued function such that \(f(x+f(x))=4f(x)\) and \(f(1)=4\), then find \(f(21)\).

An ant finds itself trapped in the \(xy\)-plane, and its initial position is \((1,0).\)

Let \(S_k\) denote the circle with radius \(k\) centered around the origin. Starting from ...

Let \(x, y\) and \(z\) be complex numbers such that

\[x + y + z = 2\]

\[x^2 + y^2 + z^2 = 3\]

\[xyz = 4\].

Evaluate

...

\[\large \int_\frac \pi 4^ {\frac {65\pi}4} \frac{dx}{(1+ 2^{ \sin x} )(1+ 2^{\cos x})} = k \pi \]

Find \(k\)?

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