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\[\begin{cases} a=\sqrt{4-\sqrt{5-a}} \\ b=\sqrt{4+\sqrt{5-b}} \\ c=\sqrt{4-\sqrt{5+c}} \\ d=\sqrt{4+\sqrt{5+d}} \end{cases} \]

Find the product ...

Given three positive integers \(a, b,\) and \(c,\)

Estimate the time it takes for a pencil of length \(l=\si{10\ \centi\meter}\) to fall if the pencil starts from rest at the angle ...

\[\frac{1}{2!} + \frac{2}{3!} + \frac{3}{4!} + \frac{4}{5!} + \cdots = \ ?\]

Let \(a, b, c,\) and \(d\) be real numbers such that \(a^2+b^2=1,\) \(c^2+d^2=1,\) and \(ac+bd=0 \). Determine \( ab+cd \).

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