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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?


Age 14 to 18
Challenge Level

Here is a set of five equations:


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?


You may like to have a go at Overturning Fracsum.

Can you create your own set of symmetrical equations?