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?
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 right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.
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 one of these numbers can have?