### Big, Bigger, Biggest

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

### 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.

### 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.

# Log Attack

##### Stage: 5 Challenge Level:
Solve the equation $a^x + b^x = 1$ where $0< a, b < 1$ and $a + b < 1$, in the special cases:
(i) $a = b\quad$ (ii) $a = {1\over 2}, \ b={1\over 4}\quad$
You can find exact solutions to the equation $a^x + b^x = 1$ in special cases like (i) and (ii).
More generally you will need to use a numerical method for finding approximate solutions. See Equation Attack.