Over and Over Again

1. Transformations and their iteration; 2. Arithmetic and geometric means; 3. Isoperimetric inequality for triangles; 4. Isoperimetric quotient; 5. Colored marbles; 6. Candy for school children; 7. Sugar rather than candy; 8. Checkers on a circle; 9. Decreasing sets of positive integers; 10. Matrix manipulations; 11. Nested traingles; 12. Morley's theorem and Napoleon's theorem; 13. Complex numbers in geometry; 14. Birth of an IMO problem; 15. Barycentric coordinates; 16. Douglas-Neumann theorem.