Draw a square. A second square of the same size slides around the
first always maintaining contact and keeping the same orientation.
How far does the dot travel?
Points A, B and C are the centres of three circles, each one of
which touches the other two. Prove that the perimeter of the
triangle ABC is equal to the diameter of the largest circle.
An AP rectangle is one whose area is numerically equal to its perimeter. If you are given the length of a side can you always find an AP rectangle with one side the given length?
Tom and Jerry must each cut their piece of paper in half. Suppose the sides of the original piece of paper have length $2x$ and $2y$, with $x\geq y$.
Then $2x+4y=40$ and $4x+2y=50$ which implies that $6x+6y=90$, therefore the perimeter of the original piece, $4x+4y=60$.
This problem is taken from the UKMT Mathematical Challenges.