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.

Collatz-ish

Stage: 4 Short Challenge Level:

The terms of the sequence are

$$6, 3, 14, 7, 34, 17, 84, 42, 21, 104, 52, 26, 13, 64, 32, 16, 8, 4, 2, 1, 4, 2, 1, \dots$$

As can be seen, there will now be no other terms in the sequence other than $4$, $2$, and $1$. It can also be seen that the only values of $n$ for which the $n$th term equals $n$ are $13$ and $16$.

This problem is taken from the UKMT Mathematical Challenges.