Topics: algebra, number theory, logic, geometry, trigonometry, counting, probability, miscellaneous topics.

Style: The class is very interactive, students may ask questions and the instructors will discuss them in class. Though a number of new concepts will be introduced, the class relies a large number of pre-requisite notions. Some students may need to purchase one or more curriculum classes in addition to this class. For the 2012-2013 season we will combine AMC 10/12 preparation with AIME preparation. Subsequently, the AIME class that will be offered around January 2013 will incorporate USA(J)MO topics. This class prepares ahead of the next exam!

Pre-requisite concepts for geometry: Lines, planes, points, parallel and perpendicular lines, angles, circle definition and elements, triangles, quadrilaterals, medians, midlines, perpendicular bisector, angle bisector, altitude, area of parallelogram, triangle, circle, cube, box, volume of box, cube, similarity, special triangles, Pythagorean theorem. We will review some these concepts in class.

Concepts we teach for geometry: Concurrence of important lines in triangles (with proofs), properties of tangent lines and circles, power of a point with respect to a circle, mass point geometry, vectors, cyclic quadrilaterals, Stewart's theorem, van Aubel's theorem, Ptolemy's theorem, plane and 3D lattices, using mathematical induction in geometry, a variety of problem solving strategies.

Pre-requisite concepts for arithmetic: Factor, multiple, writing integers in expanded form, place value, equivalent fractions, operations with fractions, decimals, percentages, factor trees, integer exponents, least common multiple, greatest common divisor/factor, averages, fundamental theorem of arithmetic, integer division algorithm, Euclid's algorithm, testing for primality, remainder (modular) arithmetic, divisibility rules, cryptarithms, digit patterns, bases of numeration, direct and inverse proportionality.

Concepts we teach for arithmetic: divisibility, sum of digits, rational and irrational numbers, arithmetic functions, congruences, Fermat's little theorem, Diophantine equations, repunits, strategies based on inequalities.

Pre-requisite concepts for counting: Definition of discrete and geometric probability, sum and product rules, notions of set theory, inclusion-exclusion principle, linear and circular permutations, Pigeonhole principle, counting factors, triangular numbers, Fibonacci numbers, arithmetic and geometric sequences and series, binomial theorem.

Concepts we teach for counting: number of divisors of an integer, sum of divisors of an integer, combinations, properties of combinations, combinatorial sums, parametrization of problems with continuous sample space, derangements, generating functions, other miscellaneous counting methods.

Pre-requisite concepts for algebra: linear and quadratic equations, Viete's relations, polynomials, fundamental theorem of algebra, factor theorem, exponential and logarithmic functions, functions, properties of functions, functional equations, algebraic identities, factoring polynomials, absolute value equations and inequalities, rational expressions, irrational expressions.

Concepts we teach for algebra: higher degree polynomial equations, rational, real, complex roots, advanced factoring techniques, factoring of multivariable polynomials, advanced changes of variable, logarithmic inequalities, special cases of exponential and logarithmic equations, complex numbers, polar coordinates, complex number representations, cyclotomic equations, telescoping sums, exponential Diophantine equations, functional equations, inequalities.

Pre-requisite concepts for trigonometry: trigonometry in the unit circle, trigonometric functions, properties and graphs of functions, inverse trigonometric functions, trigonometric identities, trigonometric equations.

Concepts we teach for trigonometry: advanced identities, telescoping advanced trigonometric sums, trigonometric equations and inequalities, triangle trigonometry.

Other miscellaneous topics will be taught: combinatorial geometry, analytic geometry, graph theory, a variety of connections with physics - especially Fermat's and Huygens' principles, geodesic paths, trilinear coordinates, and more as time permits.

Homework there is no homework for this class.

Materials are selected from math competitions world wide, as well as constructed by the instructors. We selectively use a very large base of problems and quizzes that do not repeat sooner than 2 years. Students who re-take this class will encounter different practice problems each time around. We constantly add new problems and monitor trends in the formats of the major math contests.