The points P, Q, R and S are the midpoints of the edges of a convex
quadrilateral. What do you notice about the quadrilateral PQRS as
the convex quadrilateral changes?
Prove Pythagoras' Theorem using enlargements and scale factors.
Triangle ABC is equilateral. D, the midpoint of BC, is the centre
of the semi-circle whose radius is R which touches AB and AC, as
well as a smaller circle with radius r which also touches AB and
AC. What is the value of r/R?
The time taken for six gardeners to dig six flower beds is the
same as the time it takes one gardener to dig one bed of the same
size, and hence the answer is $9$ hours.
This problem is taken from the UKMT Mathematical Challenges.