AMC & AIME Problems by Concept
Every real AMC 8, AMC 10, AMC 12, and AIME problem in our archive is tagged by the math concepts its solution uses. Pick a concept to practice every problem that uses it, each with a professionally curated solution, many with a video solution too, from LIVE by Po-Shen Loh. You can also browse the full past-contest archive.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
arithmetic
Clock (11) · Date and time (32) · Decimal (24) · Distance rate and time (89) · Distributive property (10) · Estimation (59) · Exponent (84) · Factorial (44) · Fraction (162) · Money (58) · Number base (51) · Order of operations (35) · Percentage (164) · Place value (74) · Rate (78) · Ratio and proportion (168) · Relative speed (17) · Repeating decimal (15) · Unit conversion (66) · Whole number operations (12)
algebra
Absolute value (53) · Ages (20) · Algebraic manipulation (99) · AM-GM Inequality (9) · Arithmetic sequence (125) · Binomial theorem (19) · Calculus (7) · Cauchy-Schwarz Inequality (1) · Completing the square (20) · Complex number (90) · Custom operation (40) · De Moivre’s Theorem (9) · Determinant (3) · Difference of squares (61) · Factoring (87) · Fibonacci (12) · Floor and ceiling functions (58) · Function (34) · Functional equation (14) · Geometric sequence (85) · Inequality (92) · Linear equation (154) · Logarithm (99) · Matrix (1) · Mixture (11) · Newton’s Sums (3) · Partial fractions (4) · Polynomial (93) · Quadratic (105) · Radical (33) · Rational equation (10) · Rationalizing denominator (10) · Recursion (88) · Roots of unity (35) · Simon’s Favorite Factoring Trick (18) · Slope (34) · Substitution (63) · Sum and difference of cubes (11) · Sum of first n cubes (1) · Sum of first n odd numbers (5) · Sum of first n squares (10) · Summation (68) · Symmetry (algebra) (22) · System of equations (169) · Telescoping (43) · Vieta’s Formulas (85)
number theory
Chicken McNugget Theorem (2) · Chinese Remainder Theorem (22) · Continued fraction (6) · Digits (171) · Diophantine Equation (99) · Divisibility (174) · Euler’s Totient Function (7) · Factor (36) · Factor counting (53) · Fermat’s Little Theorem (1) · Greatest common divisor (49) · Least common multiple (53) · Legendre’s Formula (11) · Modular arithmetic (193) · Modular exponentiation (15) · Multiple (16) · Multiplicative order (15) · Palindrome (18) · Parity (121) · Perfect power (10) · Perfect square (72) · Power of 2 (17) · Prime (57) · Prime factorization (136) · Sum of factors (9) · Trailing zeros (7) · Triangular number (26) · Units digit (26)
geometry
3D geometry (74) · Altitude (21) · Angle bisector (18) · Angle bisector theorem (24) · Angle chasing (50) · Angle sum (27) · Annulus (8) · Arc (23) · Area (107) · Area decomposition (125) · Area ratio (100) · Brahmagupta’s Formula (2) · Centroid (18) · Chord (27) · Circle (52) · Circle area (79) · Circumcircle, circumcenter, and circumradius (28) · Circumference (20) · Cone (26) · Congruence (geometry) (15) · Coordinate geometry (205) · Cube geometry (65) · Cyclic quadrilateral (26) · Cylinder (19) · Diagonal (8) · Distance formula (41) · Ellipse (10) · Equiangular polygon (12) · Equilateral triangle (92) · Euler’s Polyhedron Formula (2) · Heron’s Formula (19) · Homothety (9) · Hyperbola (4) · Incircle, incenter, and inradius (36) · Inscribed angle (31) · Isosceles triangle (54) · Kite (6) · Lattice point (41) · Law of cosines (56) · Law of sines (21) · Mass points (2) · Median (geometry) (21) · Midpoint (26) · Net (3D geometry) (18) · Paper folding (20) · Parabola (26) · Parallel lines (20) · Parallelogram (22) · Perimeter (43) · Perpendicular bisector (16) · Pick’s Theorem (1) · Polyhedron (9) · Power of a point (22) · Power scaling of length, area, and volume (24) · Ptolemy’s Theorem (6) · Pyramid (18) · Pythagorean Theorem (159) · Pythagorean Triple (19) · Radical axis (9) · Rectangle (50) · Rectangular prism (15) · Reflection (geometry) (8) · Regular polygon (65) · Rhombus (21) · Right triangle (43) · Sector (43) · Shoelace formula (22) · Similarity (121) · Special right triangle (44) · Sphere (33) · Square (geometry) (63) · Stewart’s Theorem (1) · Surface area (28) · Tangent circles (67) · Tangent line (49) · Transformation (60) · Trapezoid (53) · Triangle area (126) · Triangle inequality (40) · Trigonometric identity (73) · Trigonometry (67) · Vector (30) · Viviani’s Theorem (1) · Volume (88)
combinatorics
Arrangements with restrictions (72) · Basic counting (44) · Bijection (25) · Burnside’s Lemma (8) · Catalan Number (1) · Circular arrangements (13) · Combinations (108) · Combinatorial game (8) · Complementary counting (57) · Counting integers in a range (69) · Counting intersections (20) · Counting pairs (26) · Counting regions (2) · Counting shapes in figures (15) · Derangements (3) · Double counting (26) · Fencepost counting (6) · Generating functions (3) · Graph theory (42) · Inclusion-exclusion (60) · Lattice paths (12) · Multiplication principle (75) · Multiset permutations (22) · Partitions and compositions (16) · Pascal’s Triangle (6) · Permutations (46) · Pigeonhole principle (12) · Recursive counting (20) · Stars and bars (21) · Subsets (35) · Tiling (13) · Tree diagram (4) · Vandermonde’s Convolution (4) · Venn Diagram (10)
probability and statistics
Basic probability (165) · Bayes’ Theorem (3) · Binomial probability (12) · Complementary probability (19) · Conditional probability (27) · Data and graph interpretation (47) · Dice (probability) (44) · Expected value (15) · Geometric distribution (6) · Geometric probability (62) · Harmonic mean (10) · Independent events (24) · Mean (152) · Median (data) (53) · Mode (19) · Random walk (19) · Range (5) · Recursive probability (19) · Sampling without replacement (12) · Weighted mean (4)
logic
Counterexample (14) · Cryptarithm (11) · Logical deduction (70) · Magic square (7) · Truth-tellers and liars (4)
problem-solving techniques
Bounding to limit cases (98) · Casework (415) · Extremal argument (51) · Induction (8) · Invariant (27) · Optimization (202) · Pairing and grouping (43) · Pattern recognition (77) · Process simulation (20) · Small cases (7) · Symmetry (136) · Systematic listing (58) · Work backwards (40)