### 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?

### Target Six

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

### 8 Methods for Three by One

This problem in geometry has been solved in no less than EIGHT ways by a pair of students. How would you solve it? How many of their solutions can you follow? How are they the same or different? Which do you like best?

# Thousand Words

##### Stage: 5 Challenge Level:

Draw a simple diagram which makes it clear that the following inequality holds for any complex numbers $z$ and $w$ $$|z-w|\geq |z| -|w|$$

Now draw another simple diagram which makes it clear that the following inequality holds for any real numbers $\alpha> \beta$

$$|e^{i\alpha}-e^{i\beta}|\leq \alpha - \beta$$