### 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?

### Medallions

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?

# Square Mean

##### Stage: 4 Challenge Level:

Well done Freddie from Packwood Haugh School and Danny from Milliken High School, Canada.

Is the mean of the squares of two numbers greater than, or less than, the square of their means? Let the two numbers be $p$ and $q$.

$$\text{Square of mean } = {( p + q )^2\over 4} = {{p^2 + 2pq + q^2}\over 4}$$

$$\text{Mean of squares } = {{p^2 + q^2}\over 2}$$.

$${{p^2 + q^2}\over 2} - {{p^2 + 2pq + q^2}\over 4} = {{p^2 -2pq + q^2}\over 4}= {(p - q)^2\over 4}\geq 0.$$.

Note that this difference, ${(p - q)^2\over 4}$, is zero if $p=q$ and positive for all other choices of $p$ and $q$. So if the numbers are equal then the mean of the squares is equal to the square of the mean. Otherwise the mean of the squares is greater than the square of the mean.