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. How can you maximise its area?

Quartics

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

Little and Large

Age 16 to 18 Challenge Level:

The point $X$ moves around inside a rectangle of dimension $p$ units by $q$ units. The distances of $X$ from the vertices of the rectangle are $a$, $b$, $c$ and $d$ units. What are the least and the greatest values of

$a^2 + b^2 + c^2 + d^2$

and where is the point $X$ when these values occur?