### 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?

### Overturning Fracsum

Can you solve the system of equations to find the values of x, y and z?

### Building Tetrahedra

Can you make a tetrahedron whose faces all have the same perimeter?

# Symmetricality

##### Age 14 to 18Challenge Level

Here is a set of five equations:

$$b+c+d+e=4\\ a+c+d+e=5\\ a+b+d+e=1\\ a+b+c+e=2\\ a+b+c+d=0$$

What do you notice when you add the five equations?

Can you now find the values of $a, b, c, d$ and $e$?

Here is a different set of equations:

$$xy = 1\\ yz = 4\\ zx = 9$$

What do you notice when you multiply the three equations given above?

Can you now find the values of $x, y$ and $z$?
Is there more than one possible set of values?

Here is a third set of equations:

$$ab = 1\\ bc = 2\\ cd = 3\\ de = 4\\ ea = 6$$

Can you find all the sets of values ${a, b, c, d, e}$ that satisfy these equations?

Extension

You may like to have a go at Overturning Fracsum.

Can you create your own set of symmetrical equations?