### Building Tetrahedra

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

A 1 metre cube has one face on the ground and one face against a wall. A 4 metre ladder leans against the wall and just touches the cube. How high is the top of the ladder above the ground?

### Areas and Ratios

What is the area of the quadrilateral APOQ? Working on the building blocks will give you some insights that may help you to work it out.

# HCF Expression

##### Stage: 4 Short Challenge Level:

For all positive integer values of $p$ and $q$, $2p^2 q$ and $3pq^2$ have a common factor of $pq$.
They will also have an additional common factor of $2$ if $q =2$ and an additional common factor of $3$ if $p=3$.
As the values of $p$ and $q$ are to be chosen from $2, 3$ and $5$, the largest possible value of the highest common factor will occur when $p=3$ and $q=5$.
For these values of $p$ and $q$, $2p^2 q$ and $3pq^2$ have values $90$ and $225$ respectively, giving a highest common factor of $45$.

This problem is taken from the UKMT Mathematical Challenges.