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Big, Bigger, Biggest

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

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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.

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Sierpinski Triangle

What is the total area of the triangles remaining in the nth stage of constructing a Sierpinski Triangle? Work out the dimension of this fractal.

Equation Attack

Age 16 to 18 Challenge Level:
Let $f(x) = a^x + b^x - 1$, find $f(0)$ and $f(1)$ and consider the graph of $f(x)$.