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 (19) · Date and time (35) · Decimal (30) · Distance rate and time (119) · Distributive property (21) · Estimation (79) · Exponent (186) · Factorial (54) · Fraction (203) · Money (87) · Number base (72) · Order of operations (45) · Percentage (208) · Place value (104) · Rate (106) · Ratio and proportion (262) · Relative speed (36) · Repeating decimal (22) · Unit conversion (81) · Whole number operations (22)
algebra
Absolute value (83) · Ages (26) · Algebraic manipulation (288) · AM-GM Inequality (18) · Arithmetic sequence (187) · Arithmetico-geometric series (2) · Binomial theorem (42) · Calculus (14) · Cauchy-Schwarz Inequality (5) · Completing the square (37) · Complex number (126) · Custom operation (49) · De Moivre’s Theorem (10) · Determinant (8) · Difference of squares (97) · Factoring (163) · Fibonacci (15) · Floor and ceiling functions (72) · Function (67) · Functional equation (23) · Geometric sequence (125) · Inequality (189) · Linear equation (236) · Logarithm (170) · Matrix (1) · Mixture (17) · Newton’s Sums (4) · Partial fractions (7) · Polynomial (150) · Quadratic (240) · Radical (100) · Rational equation (61) · Rationalizing denominator (22) · Recursion (115) · Roots of unity (47) · Simon’s Favorite Factoring Trick (18) · Slope (55) · Substitution (156) · Sum and difference of cubes (21) · Sum of first n cubes (1) · Sum of first n odd numbers (8) · Sum of first n squares (12) · Summation (120) · Symmetry (algebra) (41) · System of equations (264) · Telescoping (52) · Vieta’s Formulas (128) · Zero product property (16)
number theory
Chicken McNugget Theorem (2) · Chinese Remainder Theorem (23) · Continued fraction (7) · Digits (209) · Diophantine Equation (126) · Divisibility (239) · Euler’s Totient Function (9) · Factor (43) · Factor counting (62) · Fermat’s Little Theorem (2) · Greatest common divisor (67) · Least common multiple (63) · Legendre’s Formula (13) · Modular arithmetic (243) · Modular exponentiation (20) · Multiple (16) · Multiplicative order (15) · Palindrome (19) · Parity (145) · Perfect power (17) · Perfect square (102) · Power of 2 (24) · Prime (67) · Prime factorization (165) · Quadratic residue (2) · Sum of factors (9) · Trailing zeros (11) · Triangular number (31) · Units digit (32)
geometry
3D geometry (85) · Altitude (43) · Angle bisector (31) · Angle bisector theorem (33) · Angle chasing (84) · Angle sum (57) · Annulus (10) · Arc (41) · Area (158) · Area decomposition (143) · Area ratio (135) · Brahmagupta’s Formula (2) · British Flag Theorem (1) · Centroid (31) · Ceva’s Theorem (1) · Chord (59) · Circle (132) · Circle area (102) · Circumcircle, circumcenter, and circumradius (38) · Circumference (35) · Cone (28) · Congruence (geometry) (20) · Coordinate geometry (312) · Cube geometry (75) · Cyclic quadrilateral (31) · Cylinder (24) · Diagonal (22) · Distance formula (72) · Ellipse (13) · Equiangular polygon (14) · Equilateral triangle (133) · Euler’s Polyhedron Formula (3) · Heron’s Formula (23) · Homothety (12) · Hyperbola (6) · Incircle, incenter, and inradius (53) · Inscribed angle (47) · Isosceles triangle (77) · Kite (8) · Lattice point (48) · Law of cosines (73) · Law of sines (29) · Mass points (8) · Median (geometry) (45) · Midpoint (47) · Net (3D geometry) (19) · Paper folding (23) · Parabola (37) · Parallel lines (31) · Parallelogram (34) · Perimeter (66) · Perpendicular bisector (27) · Pick’s Theorem (1) · Polyhedron (12) · Power of a point (33) · Power scaling of length, area, and volume (34) · Ptolemy’s Theorem (7) · Pyramid (21) · Pythagorean Theorem (197) · Pythagorean Triple (24) · Radical axis (9) · Rectangle (78) · Rectangular prism (25) · Reflection (geometry) (18) · Regular polygon (81) · Rhombus (27) · Right triangle (92) · Sector (48) · Shoelace formula (24) · Similarity (171) · Special right triangle (59) · Sphere (37) · Square (geometry) (94) · Stewart’s Theorem (5) · Surface area (39) · Tangent circles (86) · Tangent line (76) · Transformation (72) · Trapezoid (65) · Triangle area (180) · Triangle inequality (58) · Trigonometric identity (105) · Trigonometry (100) · Vector (56) · Viviani’s Theorem (2) · Volume (106)
combinatorics
Arrangements with restrictions (77) · Basic counting (63) · Bijection (27) · Burnside’s Lemma (10) · Catalan Number (2) · Circular arrangements (14) · Combinations (138) · Combinatorial game (8) · Complementary counting (65) · Counting integers in a range (93) · Counting intersections (33) · Counting pairs (34) · Counting regions (5) · Counting shapes in figures (16) · Derangements (3) · Double counting (33) · Fencepost counting (7) · Generating functions (4) · Graph theory (48) · Inclusion-exclusion (71) · Lattice paths (15) · Multiplication principle (85) · Multiset permutations (23) · Partitions and compositions (17) · Pascal’s Triangle (8) · Permutations (54) · Pigeonhole principle (16) · Recursive counting (21) · Stars and bars (23) · Subsets (48) · Tiling (14) · Tree diagram (4) · Vandermonde’s Convolution (4) · Venn Diagram (13)
probability and statistics
Basic probability (183) · Bayes’ Theorem (3) · Binomial probability (16) · Complementary probability (24) · Conditional probability (33) · Data and graph interpretation (53) · Dice (probability) (47) · Expected value (17) · Geometric distribution (7) · Geometric probability (65) · Harmonic mean (17) · Independent events (25) · Mean (171) · Median (data) (53) · Mode (22) · Random walk (21) · Range (7) · Recursive probability (22) · Sampling with replacement (1) · Sampling without replacement (19) · Weighted mean (9)
logic
Counterexample (38) · Cryptarithm (12) · Logical deduction (95) · Magic square (8) · Truth-tellers and liars (5)
problem-solving techniques
Bounding to limit cases (116) · Casework (451) · Extremal argument (68) · Induction (10) · Invariant (31) · Optimization (244) · Pairing and grouping (51) · Pattern recognition (99) · Process simulation (29) · Small cases (7) · Symmetry (152) · Systematic listing (73) · Work backwards (42)