### 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.

# Blood Buffers

### Why do this problem?

This problem provides a meaningful logarithmic challenge leading up to the derivation of the experimentally observable titration curve. It will introduce along the way many fascinating ideas from chemistry and human biology whilst keeping the focus entirely mathematical.

### Possible approach

This question is best studied individually. There is a lot of information to process and make sense of.

### Key questions

At each stage, how does the science have a clear, precise mathematical meaning?

### Possible extension

Students who make it to the end of this question are doing very well. They might like to pursue the ideas raised in undergraduate science books.

### Possible support

You might try and get started using the pH formula with the problem Temperature pH