Machine generated contents note: 1.Top-Tier Math Circles -- 2.Why, What, and for Whom? -- 3.Notation and Technicalities -- 4.The Art of Being a Mathematician and Problem Solving -- 5.Acknowledgments -- 1.Experimenting and Conjecturing -- 2.A Triangle Workout -- 3.Walking Along an Optimal Path -- 4.W...

Contents note continued: 4.Hints and Solutions to Selected Problems -- 1.Puzzling It Out -- 2.A Polynomial Prelude -- 3.Action Groups -- 4.General Groups -- 5.Some More Examples of Groups -- 6.Permutation (or Symmetric) Groups -- 7.The 15-Puzzle Puzzled Out -- 8.Hints and Solutions to Selected Probl...

Contents note continued: 10.Which are the "Closest" Lines -- 11.Hints and Solutions to Selected Problems -- 1.The Language of Inequalities -- 2.Arithmetic Mean - Geometric Mean Inequality -- 3.Power Mean Inequality -- 4.The Land of the Convex -- 5.Applications of Convexity to Inequalities -- 6.Geome...

Contents note continued: 5.Hints and Solutions to Selected Problems -- 1.What Comes from Within -- 2.The Culture of Circles -- 3.Eastern European vs. USA Math Circles -- 4.History and Power -- 5.Does the U.S. Need Top-Tier Math Circles?.