### 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?

### 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.

### Escriptions

For any right-angled triangle find the radii of the three escribed circles touching the sides of the triangle externally.

# Overturning Fracsum

##### Age 16 to 18Challenge Level

Solve the following system of equations to find the values of $x$, $y$ and $z$. $${\frac {xy}{x+y}}=\frac 1 2$$ $${\frac {yz} {y+z}} =\frac 1 3$$ $${\frac{xz}{x+z}} = \frac 1 7$$

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