### 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?

### 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?

### Pair Squares

The sum of any two of the numbers 2, 34 and 47 is a perfect square. Choose three square numbers and find sets of three integers with this property. Generalise to four integers.

# Pick's Quadratics

##### Stage: 5 Challenge Level:

Count the $k$-points for $k = 1, 2, 3, 4,$ and $5$ both inside the rectangle and on the perimeter. Use your results to find $A$, $B$ and $C$ and to verify that the two quadratic formulae are satisfied for these values of $k$.