An investigation involving adding and subtracting sets of consecutive numbers. Lots to find out, lots to explore.
A circle is inscribed in an equilateral triangle. Smaller circles
touch it and the sides of the triangle, the process continuing
indefinitely. What is the sum of the areas of all the circles?
Rectangle PQRS has X and Y on the edges. Triangles PQY, YRX and XSP
have equal areas. Prove X and Y divide the sides of PQRS in the
Inspector Remorse estimates that he can solve the average murder in $x$ hours, a bank robbery in half that time, and a car theft in one third of the time he takes to solve a bank robbery.
This problem is taken from the UKMT Mathematical Challenges.