### Real(ly) Numbers

If x, y and z are real numbers such that: x + y + z = 5 and xy + yz + zx = 3. What is the largest value that any of the numbers can have?

### Biggest Bendy

Four rods are hinged at their ends to form a quadrilateral with fixed side lengths. Show that the quadrilateral has a maximum area when it is cyclic.

### Quartics

Investigate the graphs of y = [1 + (x - t)^2][1 + (x + t^)2] as the parameter t varies.

# Without Calculus

##### Stage: 5 Challenge Level:

Given that $u> 0$ and $v> 0$ what is the smallest possible value of $1/u + 1/v$ given that $u + v = 5$?

Can you find this value by more than one method (not involving trial and error) without using calculus?