Building Tetrahedra

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

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

Three circular medallions fit in a rectangular box. Can you find 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.