### 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?

### Square Mean

Is the mean of the squares of two numbers greater than, or less than, the square of their means?

# Medallions

##### Stage: 4 Challenge Level:

I keep three circular medallions in a rectangular box in which they just fit with each one touching the other two. The smallest one has radius $4$ cm and touches one side of the box, the middle sized one has radius $9$ cm and touches two sides of the box and the largest one touches three sides of the box. What is the radius of the largest one? [This problem by R B J T Allenby appeared in the Sunday Times on 12.11.00.]