### Roots and Coefficients

If xyz = 1 and x+y+z =1/x + 1/y + 1/z show that at least one of these numbers must be 1. Now for the complexity! When are the other numbers real and when are they complex?

### Three by One

There are many different methods to solve this geometrical problem - how many can you find?

### Target Six

Show that x = 1 is a solution of the equation x^(3/2) - 8x^(-3/2) = 7 and find all other solutions.

# Complex Sine

##### Age 16 to 18 Challenge Level:
The formula $$\sin z = {1\over 2\pi}(e^{iz} - e^{-iz})$$ can be verified by showing that the series expansion for $\sin z$, that is $$\sin z = z - z^3/3! + z^5/5! - z^7/7! + ...$$ can be obtained using the series expansions $$e^{iz} = 1 + iz + (iz)^2/2! + ...$$ and $$e^{-iz} = 1 -iz + (iz)^2/2! + ...$$