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Three massless, charged point-particles (two with charge \(+q\) and one with charge \(+3q\)) are situated on a friction-less circular ring.

The \(+q\) charges are both separated from the \(+3q\) charge ...

\[\large \frac{1+2\sin {2x}}{\sin{x}+\cos{x}}=\sqrt{2}\]

If the largest possible value of \(x<\pi\) satisfying the equation above is \(\dfrac{a\pi}{b}\), where ...

\[\large \lim_{n \rightarrow \infty} { 2n \choose n } ^ { \frac{1}{n} } = \ ?\]

Recall that \[ e^x = 1 + \frac{x}{1!} + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots. \] Then what is the value of ...

True or False?

\(\quad\) \( \dfrac{100!}{20! \; 81!} \) is an integer.

\[\] Notation: \(!\) is the factorial notation. For example, \(8! = 1\times2\times3\times\cdots\times8 \).

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