AMC 12 Prep for Grades 11–12
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Expansion, factoring, and the standard algebraic identities. Difference of squares: a² − b² = (a − b)(a + b). Sum and difference of cubes: a³ ± b³ = (a ± b)(a² ∓ ab + b²). The binomial theorem (a + b)ⁿ = Σ C(n, k) a^n−k b^k. Symmetric polynomial expressions and Newton’s identities at the AMC 12 level: relating power sums p_k = a^k + b^k + … to elementary symmetric polynomials.

The discriminant D = b² − 4ac as the central object of quadratic analysis. Three cases: D > 0 gives two distinct real roots, D = 0 gives one repeated root, D < 0 gives two complex conjugate roots. Vieta’s formulas for polynomials of arbitrary degree: for a_nxⁿ + … + a₀ = 0, the elementary symmetric functions of the roots equal (−1)^k a_n−k/a_n. Using Vieta’s to compute symmetric expressions of roots (sums of squares, sums of cubes, sums of reciprocals) without finding the roots themselves.

Polynomial division and the factor theorem. The rational root theorem. Conjugate root theorem: if a polynomial has real coefficients, complex roots come in conjugate pairs; if rational coefficients, irrational roots involving √n come in conjugate pairs. Multiplicity of roots and its relationship to derivatives.

The complex plane and basic operations. Modulus, argument, and conjugate. Polar form: z = r(cos θ + i sin θ) = re^iθ. De Moivre’s theorem: (cos θ + i sin θ)ⁿ = cos(nθ) + i sin(nθ). Computing powers and roots of complex numbers via polar form.

The nth roots of unity as solutions to zⁿ = 1, geometrically vertices of a regular n-gon on the unit circle. The primitive nth roots and the cyclotomic polynomials. Using roots of unity to evaluate trigonometric sums and prove combinatorial identities. The classic identity Σ cos(2πk/n) = 0 derived from roots of unity.

Arithmetic and geometric sequences. Telescoping techniques. Solving linear recurrences via the characteristic equation: for a recurrence like a_n = c₁a_n−1 + c₂a_n−2, find the roots of x² − c₁x − c₂ = 0 and write a_n as a linear combination of those roots to the nth power. Fibonacci-style problems and Binet’s formula.

Function notation, domain, range, and composition. Inverse functions and the condition for invertibility. Even and odd functions. Periodic functions. Basic functional equations at the AMC 12 level: solving f(x + y) = f(x) + f(y), f(xy) = f(x) + f(y), and similar Cauchy-style equations.

Properties of logarithms: log(ab) = log(a) + log(b), log(aⁿ) = n · log(a), change of base log_b(x) = log(x)/log(b). Solving exponential and logarithmic equations and inequalities. Logarithmic identities used in competition problems.

The AM-GM inequality: for non-negative reals, (a₁ + a₂ + … + a_n)/n ≥ ⁿ√(a₁a₂…a_n), with equality when all terms are equal. The Cauchy-Schwarz inequality: (Σ a_ib_i)² ≤ (Σ a_i²)(Σ b_i²). The QM-AM-GM-HM chain. Recognizing when an AMC 12 problem requires an inequality technique versus a direct computation.

Divisibility rules. Prime factorization. Number of factors: if n = p₁^a₁ · p₂^a₂ · … · p_k^a_k, then d(n) = (a₁ + 1)(a₂ + 1)…(a_k + 1). Sum of factors σ(n) = Π (p_i^a_i+1 − 1)/(p_i − 1). Multiplicative arithmetic functions.

Solving congruences. 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) for gcd(a, n) = 1, where φ is Euler’s totient function. Wilson’s theorem: (p − 1)! ≡ −1 (mod p) for prime p.

Solving systems of congruences via the Chinese Remainder Theorem. The order of an element modulo n. Primitive roots and their existence. Using order to solve advanced modular problems.

Linear Diophantine equations ax + by = c and the conditions for solutions (gcd(a, b) divides c). Solving via the Euclidean algorithm. Introduction to Pell’s equation x² − Dy² = 1 at the AMC 12 level. Pythagorean triples and their parameterization.

Place value problems. Number base conversions. Digit-sum problems and divisibility tests derived from them. Patterns in digit manipulation at competition level.

Advanced angle chasing: angles from parallel lines, inscribed angles in a circle, tangent-chord angles. Triangle congruence and similarity at the AMC 12 level. The triangle centers (centroid, circumcenter, incenter, orthocenter) and the relationships between them. The Euler line: centroid, circumcenter, and orthocenter are collinear.

Multiple formulas for triangle area: (1/2)bh, (1/2)ab · sin(C), abc/(4R) where R is the circumradius, and rs where r is the inradius and s is the semi-perimeter. Heron’s formula: area = √(s(s − a)(s − b)(s − c)). Choosing the right formula for the given information.

Properties of chords, tangents, and secants. The inscribed angle theorem and its consequences. Power of a Point: for a point P and circle, PA · PB is constant for all chords through P; equals PT² for a tangent from P. The radical axis of two circles.

Properties of cyclic quadrilaterals: opposite angles sum to 180°. Ptolemy’s theorem: for a cyclic quadrilateral ABCD, AC · BD = AB · CD + BC · AD. Brahmagupta’s formula for the area of a cyclic quadrilateral.

Right triangle trigonometry and the unit circle. The Law of Sines: a/sin(A) = b/sin(B) = c/sin(C) = 2R. The Law of Cosines: c² = a² + b² − 2ab · cos(C), derived from the dot product rather than memorized. Trigonometric identities: angle addition formulas sin(α ± β) = sin α cos β ± cos α sin β and cos(α ± β) = cos α cos β ∓ sin α sin β. Double-angle, half-angle, and product-to-sum identities.

Solving trigonometric equations. Sum-to-product and product-to-sum identities. The relationship between trigonometry and complex numbers via Euler’s formula e^iθ = cos θ + i sin θ, and the resulting identities sin θ = (e^iθ − e^−iθ)/(2i) and cos θ = (e^iθ + e^−iθ)/2. Using complex exponentials to derive trigonometric identities cleanly.

Distance and midpoint formulas. Equations of lines: slope-intercept y = mx + b, point-slope y − y₁ = m(x − x₁), and the two-point slope formula m = (y₂ − y₁)/(x₂ − x₁). Equation of a circle: (x − h)² + (y − k)² = r². Parallel and perpendicular line relationships. The Russian Math approach of using vector parameterization for harder coordinate problems.

The parabola, ellipse, and hyperbola: their definitions via focus-directrix, foci, and asymptotes. Standard equations and their geometric meaning. The discriminant of a general conic Ax² + Bxy + Cy² + Dx + Ey + F = 0: B² − 4AC determines the type (negative → ellipse, zero → parabola, positive → hyperbola). Eccentricity and its geometric significance.

Volumes and surface areas of prisms, pyramids, cylinders, cones, and spheres. Volume of a sphere: (4/3)πr³. Volume of a cone: (1/3)πr²h. Dot product and cross product of vectors and their geometric meaning. Distances between points, lines, and planes in 3D.

The fundamental counting principle. Permutations: P(n, k) = n!/(n − k)!. Combinations: C(n, k) = n!/(k!(n − k)!). Factorials, binomial coefficients, and Pascal’s triangle. Multinomial coefficients for partitioning into more than two groups.

Complementary counting. Casework and systematic enumeration. Counting with restrictions. The Principle of Inclusion-Exclusion: |A ∪ B ∪ C| = |A| + |B| + |C| − |A ∩ B| − |A ∩ C| − |B ∩ C| + |A ∩ B ∩ C|, extending to arbitrary unions.

Stars and bars: the number of ways to put n identical objects into k distinct bins is C(n + k − 1, k − 1). Counting lattice paths and using the reflection principle. The bijection principle: showing two sets have the same size by constructing a bijection between them.

Setting up recursive counting arguments. Solving linear recurrences via the characteristic equation. Introduction to generating functions: encoding a sequence a_n as the coefficients of a formal power series Σ a_nxⁿ and using algebraic manipulations of the series to extract combinatorial information.

Basic probability. Probability with combinations. Conditional probability: P(A | B) = P(A ∩ B)/P(B). Bayes’ theorem at the AMC 12 level. Independence and its consequences.

Linearity of expectation: E[X + Y] = E[X] + E[Y], even when X and Y are dependent. The power of linearity for counting expected occurrences of events. Calculating expected values via indicator random variables.