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?
Investigate how you can work out what day of the week your birthday will be on next year, and the year after...
Let the radius of the circle be r. This implies that the radius
of the semicircle is 2r. The area of the semi circle is $1/2 \times
\pi \times(2r)^2$, which is twice the area of the small circle.
This problem is taken from the UKMT Mathematical Challenges.