All AMC topics

Equations involving |x|; solved by case analysis on sign, squaring carefully, or interpreting geometrically as distance on the number line.

The inequality between the arithmetic and geometric means of non-negative reals: (a1+…+an)/n ≥ (a1·…·an)^(1/n), with equality iff all ai are equal; a core tool for bounding sums and products.

Sequences with constant common difference d: a_n = a_1 + (n-1)d; includes sum S_n = n(a_1+a_n)/2 and problems involving term counting and inserted means.

States that (Σ a_i b_i)^2 ≤ (Σ a_i^2)(Σ b_i^2); used to bound dot products, sums of fractions (via Engel form / Titu's lemma), and many competition inequalities.

Strategies for comparing a^b vs c^d and simplifying expressions with common bases or exponents by rewriting, taking logarithms, or normalizing exponents.

For z = a+bi, the conjugate z̄ = a−bi; properties include z·z̄ = |z|^2 and conjugate-closure of polynomials with real coefficients (roots come in conjugate pairs).

The magnitude |a+bi| = √(a^2+b^2), representing distance from the origin in the complex plane; used in geometry problems, polar form, and multiplicative properties.

The operation (f ∘ g)(x) = f(g(x)); includes domain restrictions, decomposition of complex expressions, and behavior of iterated compositions.

Problems introducing a novel binary operation or function rule (e.g., a ★ b = a^2 − b) and requiring evaluation, property testing, or solving equations under the definition.

(cos θ + i sin θ)^n = cos(nθ) + i sin(nθ); used to compute powers and roots of complex numbers and to derive multiple-angle trigonometric identities.

Identities a^2−b^2 = (a−b)(a+b), a^n−b^n = (a−b)(a^(n−1)+…+b^(n−1)), and a^n+b^n for odd n; used in algebra, factoring, and number theory.

Problems involving the sum, product, reversal, or positional rearrangement of the base-10 digits of a number; often tied to divisibility rules and modular arithmetic.

For ax^2+bx+c, Δ = b^2−4ac determines the nature of the roots: two real distinct (Δ>0), one real double (Δ=0), or two complex conjugate (Δ<0); also tests for perfect squares.

e^(iθ) = cos θ + i sin θ, unifying exponential and trigonometric functions; the special case e^(iπ)+1 = 0 is the cornerstone of complex analysis and many identity proofs.

Functions of the form f(x) = a·b^x with b>0; includes growth/decay behavior, domain and range, graphing, and algebraic manipulation of exponential expressions.

Methods for decomposing polynomials into products of lower-degree factors, including common-factor extraction, grouping, trinomial splitting, substitution, and identity-based factoring — a foundational tool for solving equations and simplifying expressions.

Triangular, square, pentagonal, and other polygonal numbers, plus pattern problems where a geometric/visual configuration grows; requires finding a closed-form or recurrence.

Using ⌊x⌋ and ⌈x⌉ in equations and identities; includes step-function behavior, Hermite's identity, and bounding techniques with fractional parts {x}.

Equations where the unknown is a function, e.g., f(x+y) = f(x)+f(y); solved by substitution of specific values, exploiting symmetry, and identifying function class.

Sequences with constant common ratio r: a_n = a_1·r^(n-1); includes finite sum formula, infinite geometric series when |r|<1, and geometric-mean insertion.

Functions f^(-1) satisfying f(f^(-1)(x)) = x; covers existence (one-to-one requirement), algebraic derivation, and the reflection symmetry about y = x.

Core log identities: log(ab) = log a + log b, log(a^n) = n log a, and log_b a = log_c a / log_c b; used to simplify and solve logarithmic equations.

Inverses of exponential functions, f(x) = log_b(x); covers definition, domain, graphs, and use in solving equations where the unknown is in an exponent.

Word problems combining solutions, alloys, or quantities with differing concentrations; solved by tracking conserved amounts (mass of solute) through weighted averages.

Recognizing that an iterated function or mod-sequence eventually cycles, determining cycle length, and using it to compute far-out terms or remainders.

Dividing polynomials to obtain quotient and remainder; synthetic division is a compact scheme for division by x − a, useful for factoring and evaluating polynomials.

Analysis of y = ax^2+bx+c: vertex at x = -b/(2a), axis of symmetry, direction of opening, maximum/minimum value, and intercepts — central to optimization problems.

Solving ax^2+bx+c = 0 using factoring, completing the square, the quadratic formula, or Vieta's; includes analyzing the nature of roots via the discriminant.

Word problems linking rate × time = quantity, including combined-work problems, motion with constant/variable speed, and upstream/downstream setups.

Sequences defined by recurrence relations such as F_n = F_(n-1)+F_(n-2); includes characteristic equation methods, closed forms (Binet), and pattern recognition mod m.

States that the remainder of dividing a polynomial p(x) by (x - a) equals p(a); paired with the Factor Theorem to test roots and factor polynomials quickly.

Finding and characterizing roots of polynomials of degree three or more using Rational Root Theorem, synthetic division, symmetry, substitutions, and Vieta's formulas; often requires factoring into lower-degree pieces.

Rewriting xy + ax + by + ab as (x+b)(y+a) by adding/subtracting a constant; turns bilinear Diophantine or algebraic equations into factorable form.

Standard expansion and factoring identities such as (a+b)^2 = a^2+2ab+b^2, (a-b)^2, (a+b)(a-b) = a^2-b^2, (a+b)^3, a^3±b^3, and (a+b+c)^2; used to simplify expressions, reveal hidden structure, and shortcut algebraic manipulation.

Closed forms for Σ k, Σ k^2, Σ k^3, Σ 1/(k(k+1)), and similar standard sums; used for evaluating finite sums that appear in counting, algebra, and number theory.

Systems where swapping variables leaves the equations unchanged; solved by introducing elementary symmetric polynomials (s = x+y, p = xy) to reduce variables.

Techniques for solving multiple simultaneous linear equations in several unknowns via substitution, elimination, matrix methods, or Cramer's rule; includes detecting inconsistent and dependent systems.

Rewriting a_k as b_k − b_(k+1) (or b_(k+1)/b_k for products) so that intermediate terms cancel, leaving a short closed form for the sum or product.

Relations expressing the elementary symmetric functions of a polynomial's roots in terms of its coefficients (e.g., sum and product of roots for quadratics and cubics); widely used to extract information about roots without solving the equation.

Performing and converting between the three representations, including addition, subtraction, multiplication, division, and percent-change computations.

Applying ⌈x⌉ or ⌊x⌋ when quantities must be whole (e.g., buses, boxes, people) even though arithmetic yields a non-integer result.

Computing positions and angles of clock hands and related rotational problems; hour hand moves 0.5°/min, minute hand 6°/min, with periodic coincidence and alignment questions.

Problems requiring several linked operations or intermediate quantities; emphasizes careful translation of language into algebra and checking of units/reasonableness.

PEMDAS/BODMAS convention governing the order to evaluate parentheses, exponents, multiplication/division, and addition/subtraction in an expression.

Expressing relative quantities a:b and solving equations of the form a/b = c/d; used in scaling, mixture, and similar-figure problems.

Converting between measurement units (length, area, volume, mass, time, currency) using conversion factors and dimensional analysis.

If two choices are mutually exclusive with m and n options respectively, the total is m+n; used to combine disjoint cases.

Counting technique that establishes a one-to-one correspondence between two sets to conclude they have equal size; a powerful proof technique in combinatorics.

(x+y)^n = Σ C(n,k) x^(n−k) y^k; yields coefficients for polynomial expansions and is the basis for many combinatorial identities.

The number of orbits under a group action equals the average number of fixed points: (1/|G|) Σ |Fix(g)|; used to count configurations up to symmetry (rotations, reflections).

Probability defined as (favorable outcomes)/(total outcomes) when all outcomes are equally likely; the starting point of elementary probability.

Number of unordered k-subsets of an n-set: C(n,k) = n!/(k!(n−k)!); satisfies Pascal's identity and symmetry C(n,k) = C(n,n−k).

Counting the complement of the desired set and subtracting from the total, useful when direct counting is more complex than the complement.

P(A|B) = P(A∩B)/P(B); probability of A given that B has occurred, foundational for Bayes' theorem and event dependence.

E[X] = Σ x·P(X=x); linearity states E[X+Y] = E[X]+E[Y] regardless of independence — a powerful tool for computing expected values by decomposition.

Probability computed as a ratio of lengths, areas, or volumes rather than counts; used when outcomes form a continuous region.

Events A and B are independent iff P(A∩B) = P(A)P(B); independence simplifies joint probability computations.

Counting integer-coordinate points inside geometric regions (circles, polygons); tools include Pick's theorem, floor-sum identities, and direct enumeration.

Generalization of binomial coefficients: n!/(n1!·n2!·…·nk!) counts arrangements of n items into groups of sizes n1,…,nk or expansions of (x1+…+xk)^n.

If one choice has m options and an independent follow-up choice has n options, the combined sequence has m·n options; the foundation of counting.

Number of ordered arrangements of k items from n: P(n,k) = n!/(n−k)!; used when order matters.

|A1∪…∪An| = Σ|Ai| − Σ|Ai∩Aj| + … ± |A1∩…∩An|; corrects overcounting when unioning overlapping sets.

Defining a counting function by a recurrence (e.g., a_n = a_(n-1)+a_(n-2)) and solving or evaluating it; common for tilings, paths, and structured arrangements.

Counts solutions to x1+…+xk = n in non-negative integers as C(n+k−1,k−1); models distributing identical objects into distinct bins.

Model a stochastic process as states with transition probabilities, then use a recurrence or matrix power to compute absorption/hitting probabilities or long-run behavior.

In a right triangle, the altitude to the hypotenuse creates two similar sub-triangles, yielding geometric-mean relations h^2 = pq, a^2 = pc, b^2 = qc.

The bisector of an angle of a triangle divides the opposite side in the ratio of the adjacent sides: BD/DC = AB/AC; used for length ratios and mass-point setups.

Computing areas of complex shapes by splitting into triangles/rectangles or rearranging pieces into a simpler figure of equal area.

Standard area formulas for triangles, quadrilaterals, and regular polygons, plus Heron's formula √(s(s-a)(s-b)(s-c)) for a triangle given three sides.

Area = (1/2)·perimeter·apothem, or (1/4)n·s^2·cot(π/n); includes derivations by splitting into n isoceles triangles.

Intersection of a triangle's three medians, dividing each in ratio 2:1; also the geometric center (average of vertex coordinates) and the balance point.

Intersection of the perpendicular bisectors of the three sides; equidistant from all three vertices and the center of the circumscribed circle.

Formulas C = 2πr, arc length = rθ (θ in radians), and sector area = (1/2)r^2θ; also degree-based variants for AMC-style problems.

Quadrilaterals inscribed in a circle; opposite angles sum to 180°, Ptolemy's theorem applies, and angle-chasing shortcuts abound.

Distance between (x1,y1) and (x2,y2) equals √((x2−x1)^2 + (y2−y1)^2); a direct corollary of the Pythagorean theorem.

Formula d = |Ax0+By0+C|/√(A^2+B^2) giving perpendicular distance from (x0,y0) to line Ax+By+C = 0; essential in coordinate geometry.

(x−h)^2 + (y−k)^2 = r^2 for a circle centered at (h,k) with radius r; general form x^2+y^2+Dx+Ey+F = 0 reduces by completing the square.

Standard form (x−h)^2/a^2 − (y−k)^2/b^2 = 1; characterized by difference-of-distances to foci, with asymptotes y−k = ±(b/a)(x−h).

Standard form (x−h)^2/a^2 + (y−k)^2/b^2 = 1; foci, eccentricity, and the sum-of-distances property characterize the curve.

Core side-ratio facts for equilateral (1:1:1 with height (√3/2)s) and 30-60-90 triangles (1 : √3 : 2); fast shortcuts for geometry and trigonometry problems.

Intersection of the three angle bisectors; equidistant from all three sides and the center of the inscribed circle (incircle).

An inscribed angle equals half the central angle subtending the same arc; consequences include Thales' theorem and relations for cyclic quadrilaterals.

Each interior angle of a regular n-gon equals (n−2)·180°/n; sum of interior angles equals (n−2)·180°; exterior angles always sum to 360°.

c^2 = a^2 + b^2 − 2ab·cos C; generalizes the Pythagorean theorem and solves SAS and SSS triangle configurations.

In any triangle, a/sin A = b/sin B = c/sin C = 2R; used to relate sides and opposite angles and to find the circumradius.

Identifying axes of reflection for 2D shapes and planes of reflection for 3D solids; used in counting, geometry, and invariant-preserving transformations.

Assigning masses to vertices so that balanced cevians reveal segment ratios; a lightweight alternative to coordinates for problems involving Ceva/Menelaus configurations.

Center of the nine-point circle, which passes through the three midpoints of sides, three feet of altitudes, and three midpoints of segments from each vertex to the orthocenter.

Intersection of a triangle's three altitudes; has rich relationships with the circumcenter and centroid via the Euler line.

Quadrilaterals with both pairs of opposite sides parallel; opposite sides and angles are equal, diagonals bisect each other, and area = base × height.

Pick's theorem A = I + B/2 − 1 for lattice polygons; Shoelace formula gives signed area of a polygon from its vertex coordinates — two tools for coordinate-based area.

For two chords through a point P inside a circle meeting it at A,B and C,D: PA·PB = PC·PD; the signed value is the power of the point.

For a point outside a circle, PT^2 = PA·PB where PT is the tangent length and PAB is a secant; connects tangent-length to secant products.

For a right triangle, a^2+b^2 = c^2 where c is the hypotenuse; extended to distance formulas, Pythagorean triples, and used ubiquitously in geometry.

Applying isometries — reflections over lines, rotations about points, translations — to coordinates; used in symmetry arguments and transformational geometry.

m = (y2−y1)/(x2−x1); parallel lines have equal slope, perpendicular lines have slopes multiplying to −1 (when defined).

Interpreting 3D figures and their plane cross-sections; includes nets, unfoldings, and dihedral-angle reasoning — key for solid geometry problems.

Identities like sin A + sin B = 2 sin((A+B)/2)cos((A−B)/2) and sin A·sin B = (1/2)[cos(A−B)−cos(A+B)]; used to rewrite sums/products for simplification and integration.

Cone: V = (1/3)πr^2h, lateral area = πrℓ (ℓ = slant height). Frustum: V = (h/3)(R^2+Rr+r^2)π; lateral area = π(R+r)ℓ.

V = πr^2h; lateral surface area = 2πrh; total surface area = 2πrh + 2πr^2 for a closed right circular cylinder.

For a prism, V = base_area × height and lateral surface area = perimeter × height; total surface area adds two congruent base areas.

V = (1/3)·base_area·height; lateral surface area sums the triangular faces, using slant height for right regular pyramids.

Sphere of radius r: V = (4/3)πr^3 and surface area = 4πr^2; often paired with inscribed/circumscribed solid problems.

A tangent to a circle is perpendicular to the radius at the point of tangency; external tangents from a point have equal length; tangent-chord angle equals inscribed angle.

Quadrilaterals with at least one pair of parallel sides; area = (1/2)(b1+b2)h, and the midsegment equals the average of the parallel sides.

Interior angles of a triangle sum to 180°; systematic labeling and propagation of angles through a figure to deduce unknown angles.

Identical triangles (same shape and size); criteria SSS, SAS, ASA, AAS, HL; used to transfer side and angle equalities across figures.

For any triangle, the sum of any two side lengths exceeds the third; used to bound side lengths and determine feasibility of triangle configurations.

Two triangles with equal corresponding angles and proportional sides; criteria AA, SAS, SSS similarity; widely used to derive length ratios and prove geometric facts.

Core identities including Pythagorean (sin^2+cos^2=1), angle sum/difference, double/half-angle, and co-function relations; foundation for trig manipulation.

Defining a function or sequence in terms of its earlier values with a base case; used to solve counting problems and structure induction proofs.

Splitting a problem into mutually exclusive, exhaustive cases and solving each separately; important to avoid double-counting and to ensure completeness.

Assigning colors (often two-coloring or parity) to a board or set and using invariant color counts to prove impossibility or structural constraints.

Analysis of combinatorial games with perfect information and no chance; covers winning/losing positions, Sprague-Grundy values, and the Nim XOR rule.

Quantities preserved under the allowed moves of a process or puzzle; identifying an invariant is a standard way to prove impossibility or bound outcomes.

Puzzles where a set of rules determines a unique ordering or assignment; solved by systematic elimination, contradiction, and case analysis.

Arithmetic average (sum divided by count); the balance point of a data set, sensitive to outliers.

The middle value of an ordered data set (or the average of the two middle values when the count is even); robust to outliers.

The most frequently occurring value(s) in a data set; applicable to categorical and numerical data.

If n+1 objects go into n boxes, at least one box contains at least two; generalizations include ⌈n/k⌉ for n objects into k boxes.

Difference between the maximum and minimum values of a data set; a simple measure of spread.

Extracting quantitative information from bar graphs, line charts, pie charts, and scatter plots; includes identifying trends, reading scales, and comparing categories.

Mean of squared deviations from the mean: Σ(x_i − x̄)^2 / n (or /(n−1) for sample variance); quantifies dispersion.

Mean where each value is multiplied by a weight: (Σ w_i x_i) / (Σ w_i); used in mixture problems, grade computation, and expected value.

Starting from the desired end state and reversing operations to recover the initial state; effective when the forward path has many branches but the target is fixed.

Performing addition, subtraction, multiplication, and division in non-decimal bases, with carrying/borrowing adapted to the base.

Computing days-of-week, leap-year counts, and recurring date offsets using modular arithmetic (mostly mod 7) and careful handling of month lengths.

If moduli are pairwise coprime, a system of simultaneous congruences has a unique solution modulo the product; used to reconstruct integers from residues.

Quick tests for divisibility by 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13 based on digit patterns and alternating sums; derived from modular arithmetic.

Generalization of Fermat: if gcd(a,n) = 1, then a^φ(n) ≡ 1 (mod n), where φ is Euler's totient function; central to RSA and large-exponent reductions.

If p is prime and gcd(a,p) = 1, then a^(p−1) ≡ 1 (mod p); used for exponent reduction and primality testing.

The largest positive integer dividing two integers; computed efficiently via the Euclidean algorithm, with gcd(a,b)·lcm(a,b) = |ab|.

The smallest positive integer divisible by both a and b; equals ab / gcd(a,b) and is computed by taking max exponents across prime factorizations.

The exponent of prime p in n! equals Σ⌊n/p^k⌋; paired with v_p(n), the p-adic valuation, for counting factors in products, factorials, and binomial coefficients.

Integer solutions to ax + by = c: solvable iff gcd(a,b) | c; general solution parameterized by one integer k using the Extended Euclidean algorithm.

Arithmetic on residue classes mod n; includes properties of addition, multiplication, modular inverses, and solving linear congruences.

Converting integers (and fractions) between bases (binary, octal, decimal, hexadecimal, etc.); includes repeated-division and positional-value methods.

If n = p1^a1·…·pk^ak, then d(n) = (a1+1)(a2+1)…(ak+1); counts positive divisors of n from its prime factorization.

Unique representation of a positive integer as a product of prime powers (Fundamental Theorem of Arithmetic); the entry point for nearly all number-theoretic computations.

σ(n) = Π (p_i^(a_i+1)−1)/(p_i−1); multiplicative formula giving the sum of all positive divisors of n from its prime factorization.