AIME Syllabus
The AIME is a 15-question exam where every answer is an integer from 0 to 999. The syllabus covers four main areas of pre-calculus mathematics, with a focus on clever problem-solving rather than advanced topics.
Algebra
- Polynomials Vieta's formulas, roots of unity, and relationships between roots and coefficients.
- Sequences and Series Arithmetic and geometric progressions, telescoping sums, and recurrence relations.
- Inequalities AM-GM, Cauchy-Schwarz, and other classical inequalities.
- Functional Equations Determining function properties or forms by solving equations involving the function itself.
- Logarithms and Exponents Advanced manipulation of logarithmic and exponential expressions and equations.
Geometry
- Euclidean Geometry Advanced circle properties (power of a point), cyclic quadrilaterals, and complex triangle theorems.
- Trigonometry Identities, laws of sines and cosines, often applied to solve geometric problems.
- Coordinate Geometry Using coordinates and algebraic methods to solve problems about shapes and positions.
- 3D Geometry Properties of polyhedra, spheres, and spatial reasoning.
Number Theory
- Divisibility and Primes Properties of prime numbers, prime factorization, and number of divisors.
- Modular Arithmetic Congruences, Fermat's Little Theorem, Euler's Totient Theorem, and Chinese Remainder Theorem.
- Diophantine Equations Finding integer solutions to polynomial equations.
- Number Bases Problems involving representation of numbers in different bases.
Combinatorics
- Counting Principles Advanced permutations, combinations, and the Principle of Inclusion-Exclusion.
- Probability Calculating probabilities in complex scenarios, often involving geometric or counting techniques.
- Pigeonhole Principle Applying the principle to prove existence or establish bounds.
- Recurrence Relations Setting up and solving recurrence relations to model counting problems.