AIME Prep for High-School Mathletes
1-on-1 Socratic method coaching with former IMO medalist tutors

Polynomial division, the factor theorem, the remainder theorem. Rational root theorem and its applications. The conjugate root theorem for both complex and irrational conjugates. Multiplicity of roots. Polynomial interpolation: given n + 1 points, there is a unique polynomial of degree n passing through them. Lagrange interpolation as the explicit construction.

For a polynomial a_nxⁿ + a_n−1xⁿ⁻¹ + … + a₀ with roots r₁, r₂, …, r_n, the elementary symmetric polynomials of the roots equal (−1)^k a_n−k/a_n. Using Vieta’s to compute symmetric functions of roots without solving the polynomial. Newton’s identities relating power sums to elementary symmetric polynomials: p_k = e₁p_k−1 − e₂p_k−2 + … ± k · e_k.

Solving Cauchy functional equations: f(x + y) = f(x) + f(y), f(xy) = f(x)f(y), f(x + y) = f(x)f(y), and similar. Substitution techniques: setting x = y, x = 0, x = 1, x = −y to extract structural information. Recognizing when continuity or monotonicity constraints force linear or exponential solutions.

The complex plane, modulus, argument, conjugate. Polar form z = re^iθ and Euler’s formula. De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). The nth roots of unity ω_k = e^2πik/n as vertices of a regular n-gon on the unit circle. Geometric properties of polynomials with complex roots.

The roots-of-unity filter for extracting coefficients from generating functions: if f(x) = Σ a_nxⁿ, then Σ_{n ≡ r (mod k)} a_n = (1/k) Σ_j ω^−jr f(ω^j) where ω = e^2πi/k. Trigonometric sums via roots of unity. The classic result that Σ_j cos(2πj/n) = 0 for n ≥ 2. Sum of fifth roots, eighth roots, twelfth roots — appearing on AIME regularly.

Encoding a sequence a_n as a formal power series A(x) = Σ a_nxⁿ. Generating function manipulations: multiplication corresponds to convolution, differentiation corresponds to shifting and scaling. Solving counting problems by setting up and manipulating the generating function. Closed forms for common generating functions: 1/(1 − x), 1/(1 − x)², (1 + x)ⁿ. Recognizing when a problem calls for generating functions.

Linear recurrences and the characteristic equation. For a_n = c₁a_n−1 + c₂a_n−2, find the roots r₁, r₂ of x² − c₁x − c₂ = 0; the general solution is a_n = A · r₁ⁿ + B · r₂ⁿ. Repeated roots, complex roots, and their interpretations. Non-homogeneous recurrences and particular solutions.

The AM-GM, GM-HM, and QM-AM inequalities and the full QM-AM-GM-HM chain. The Cauchy-Schwarz inequality: (Σ a_ib_i)² ≤ (Σ a_i²)(Σ b_i²). The rearrangement inequality. Jensen’s inequality for convex and concave functions. The substitution and normalization techniques for setting up inequality problems. When to use weighted AM-GM versus standard AM-GM.

Symmetric polynomial techniques. Schur-like inequalities at the AIME level. Substitution patterns: a = x + y, b = y + z, c = z + x for triangle inequalities. SOS (sum of squares) decompositions for proving non-negative expressions are non-negative.

Fermat’s Little Theorem: a^p−1 ≡ 1 (mod p) for prime p with gcd(a, p) = 1. Euler’s theorem: a^φ(n) ≡ 1 (mod n) where φ is Euler’s totient. Wilson’s theorem: (p − 1)! ≡ −1 (mod p). Computing inverses modulo n via the Euclidean algorithm.

The Chinese Remainder Theorem: solving systems of congruences with coprime moduli. The order of an element ord_n(a) and its relationship to φ(n) and the discrete logarithm. Primitive roots and their structure. Using order to solve sophisticated AIME modular problems.

The Lifting the Exponent Lemma (LTE) for prime p: under conditions, v_p(aⁿ − bⁿ) = v_p(a − b) + v_p(n) where v_p denotes p-adic valuation. Conditions: p odd, p | (a − b), p ∤ a, p ∤ b. Modified version for p = 2 and for aⁿ + bⁿ. Recognizing LTE on AIME problems involving divisibility of large numbers.

Linear Diophantine equations ax + by = c and their general solutions. Pell’s equation x² − Dy² = 1 and its solutions via continued fractions. Pythagorean triples and their full parameterization. Equations of the form xy = ax + by + c and similar AIME staples.

Multiplicative arithmetic functions: τ(n), σ(n), φ(n), μ(n). The Möbius function and Möbius inversion. Computing sums of arithmetic functions over divisors. Recognizing problems that decompose multiplicatively.

Number base conversions at the AIME level. Digit-sum problems and divisibility tests derived from them. Casting out nines and its generalizations. Patterns in digit manipulation over large numbers.

The centroid, circumcenter, incenter, orthocenter, and their properties. The Euler line: centroid, circumcenter, and orthocenter are collinear, with the centroid dividing the segment from orthocenter to circumcenter in ratio 2:1. The nine-point circle and its properties. Triangle centers in coordinates.

Power of a Point: PA · PB = PC · PD for chords through P, equals PT² for tangent T. The radical axis of two circles as the locus of points with equal power. The radical center of three circles. Using power of a point to prove concurrence and collinearity.

Cyclic quadrilateral properties: opposite angles sum to 180°, exterior angle equals opposite interior angle. Ptolemy’s theorem: AC · BD = AB · CD + BC · AD for cyclic ABCD. Brahmagupta’s formula for area. Generalized Ptolemy inequality for non-cyclic quadrilaterals.

The Law of Sines a/sin A = 2R and Law of Cosines c² = a² + b² − 2ab cos C, derived from vectors and the dot product. Stewart’s theorem: for a cevian of length d in triangle with sides a, b, c dividing side a into segments m and n, b²m + c²n − a · d² = a · m · n. Mass point geometry as an alternative.

Distance formulas, slope formulas (point-slope, two-point: m = (y₂ − y₁)/(x₂ − x₁), slope-intercept). Equations of lines and circles. Vector parameterization of lines and planes. Dot product geometry: v · w = |v||w|cos θ. Cross product in 3D and its geometric meaning.

Parabola, ellipse, hyperbola defined via focus-directrix and via foci. Eccentricity. The discriminant B² − 4AC of a general conic Ax² + Bxy + Cy² + … = 0 determining its type. Reflection properties of conics. Parametric equations for conics.

Rotations, reflections, translations, dilations, and spiral similarities. Composition of transformations. Inversion in a circle: properties and applications. Using transformations to reduce hard geometry problems to simpler configurations.

Configurations involving multiple cyclic quadrilaterals. The Apollonius circle. Isogonal conjugates and isogonal lines. Harmonic conjugates and the cross ratio at an introductory level. Recognizing recurring configurations in AIME and olympiad problems.

Volumes and surface areas in 3D. Cross-sections of polyhedra and curved surfaces. Distances between skew lines via vectors. Inscribed and circumscribed spheres in tetrahedra. The Cayley-Menger determinant for tetrahedral volumes.

Permutations, combinations, multinomial coefficients. Stars and bars. Lattice path counting. Counting subsets, sequences, and arrangements with constraints. Recognizing the right counting framework for a given problem.

The general Principle of Inclusion-Exclusion: |A₁ ∪ A₂ ∪ … ∪ A_n| = Σ|A_i| − Σ|A_i ∩ A_j| + … + (−1)ⁿ⁺¹|A₁ ∩ … ∩ A_n|. Derangements (permutations with no fixed points): D_n = n! Σ (−1)^k/k!. Surjective function counts via inclusion-exclusion.

The bijection principle: showing two sets have the same size by constructing an explicit bijection. Catalan numbers: C_n = (1/(n+1)) · C(2n, n), counting balanced parenthesizations, Dyck paths, binary trees, triangulations. Recognizing Catalan structures in seemingly unrelated problems.

Setting up generating functions for combinatorial problems. The product principle for generating functions: choosing from independent sets. Common generating function techniques: partitions, compositions, restricted sums. Extracting coefficients via algebraic manipulation.

Setting up recurrences for counting problems. The characteristic equation method for linear recurrences. Recognizing when problems decompose recursively. Connecting recurrences to generating functions.

The pigeonhole principle and its generalizations. The extremal principle: in many problems, considering the extremal element (largest, smallest, leftmost) forces a structural property. Both are foundational to olympiad-style reasoning and appear regularly on AIME.

Probability with combinations. Conditional probability and Bayes’ theorem. Independence. Expected value and linearity of expectation. Computing expected values via indicator variables — the most powerful technique on AIME probability problems.

Introduction to the probabilistic method: proving existence by showing positive probability. Markov’s and Chebyshev’s inequalities at an introductory level. Random walks and their expected behavior. Indicator-variable techniques for harder problems.