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Which is the biggest and which the smallest of $2000^{2002}, 2001^{2001} \text{and } 2002^{2000}$? ### Infinite Continued Fractions

In this article we are going to look at infinite continued fractions - continued fractions that do not terminate. ### Complex Sine

Solve the equation sin z = 2 for complex z. You only need the formula you are given for sin z in terms of the exponential function, and to solve a quadratic equation and use the logarithmic function.

# Gosh Cosh

##### Age 16 to 18Challenge Level

You do not need to have met hyperbolic trig functions before in order to do this question as the definitions are given and you are led through the question step by step. You could look up the results and the proofs of the hyperbolic trig identities but you will gain much more from thinking through these proofs for yourself.

Possible extension:
Prove, by induction or otherwise, that for $x> 0$ and positive integral values of $n> 1$ $$\sinh nx > n\sinh x.$$