### 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.

'For any planar polygon with vertices at lattice points the quadratic formula $i(k)=Ak^2 - Bk +C$ gives the number of $k$-points inside the polygon and the quadratic formula $g(k)= Ak^2 + Bk +C$ gives the number of $k$-points in the closed polygon (including the boundary and the interior points), where $A$ is the area of the polygon.'