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?

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.

Can you show how the first equation can be arranged to get $\dfrac 1 x + \dfrac 1 y =2$?

Can you rearrange the other two equations in a similar way?

ItÂ might be helpful to have a look at the problem Symmetricality.