### Bang's Theorem

If all the faces of a tetrahedron have the same perimeter then show that they are all congruent.

### Rudolff's Problem

A group of 20 people pay a total of £20 to see an exhibition. The admission price is £3 for men, £2 for women and 50p for children. How many men, women and children are there in the group?

### Polycircles

Show that for any triangle it is always possible to construct 3 touching circles with centres at the vertices. Is it possible to construct touching circles centred at the vertices of any polygon?

# Overturning Fracsum

##### Stage: 4 Challenge Level:

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