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
algebra
- Absolute value83
- Ages26
- Algebraic manipulation288
- AM-GM Inequality18
- Arithmetic sequence187
- Arithmetico-geometric series2
- Binomial theorem42
- Calculus14
- Cauchy-Schwarz Inequality5
- Completing the square37
- Complex number126
- Custom operation49
- De Moivre’s Theorem10
- Determinant8
- Difference of squares97
- Factoring163
- Fibonacci15
- Floor and ceiling functions72
- Function67
- Functional equation23
- Geometric sequence125
- Inequality189
- Linear equation236
- Logarithm170
- Matrix1
- Mixture17
- Newton’s Sums4
- Partial fractions7
- Polynomial150
- Quadratic240
- Radical100
- Rational equation61
- Rationalizing denominator22
- Recursion115
- Roots of unity47
- Simon’s Favorite Factoring Trick18
- Slope55
- Substitution156
- Sum and difference of cubes21
- Sum of first n cubes1
- Sum of first n odd numbers8
- Sum of first n squares12
- Summation120
- Symmetry (algebra)41
- System of equations264
- Telescoping52
- Vieta’s Formulas128
- Zero product property16
number theory
- Chicken McNugget Theorem2
- Chinese Remainder Theorem23
- Continued fraction7
- Digits209
- Diophantine Equation126
- Divisibility239
- Euler’s Totient Function9
- Factor43
- Factor counting62
- Fermat’s Little Theorem2
- Greatest common divisor67
- Least common multiple63
- Legendre’s Formula13
- Modular arithmetic243
- Modular exponentiation20
- Multiple16
- Multiplicative order15
- Palindrome19
- Parity145
- Perfect power17
- Perfect square102
- Power of 224
- Prime67
- Prime factorization165
- Quadratic residue2
- Sum of factors9
- Trailing zeros11
- Triangular number31
- Units digit32
geometry
- 3D geometry85
- Altitude43
- Angle bisector31
- Angle bisector theorem33
- Angle chasing84
- Angle sum57
- Annulus10
- Arc41
- Area158
- Area decomposition143
- Area ratio135
- Brahmagupta’s Formula2
- British Flag Theorem1
- Centroid31
- Ceva’s Theorem1
- Chord59
- Circle132
- Circle area102
- Circumcircle, circumcenter, and circumradius38
- Circumference35
- Cone28
- Congruence (geometry)20
- Coordinate geometry312
- Cube geometry75
- Cyclic quadrilateral31
- Cylinder24
- Diagonal22
- Distance formula72
- Ellipse13
- Equiangular polygon14
- Equilateral triangle133
- Euler’s Polyhedron Formula3
- Heron’s Formula23
- Homothety12
- Hyperbola6
- Incircle, incenter, and inradius53
- Inscribed angle47
- Isosceles triangle77
- Kite8
- Lattice point48
- Law of cosines73
- Law of sines29
- Mass points8
- Median (geometry)45
- Midpoint47
- Net (3D geometry)19
- Paper folding23
- Parabola37
- Parallel lines31
- Parallelogram34
- Perimeter66
- Perpendicular bisector27
- Pick’s Theorem1
- Polyhedron12
- Power of a point33
- Power scaling of length, area, and volume34
- Ptolemy’s Theorem7
- Pyramid21
- Pythagorean Theorem197
- Pythagorean Triple24
- Radical axis9
- Rectangle78
- Rectangular prism25
- Reflection (geometry)18
- Regular polygon81
- Rhombus27
- Right triangle92
- Sector48
- Shoelace formula24
- Similarity171
- Special right triangle59
- Sphere37
- Square (geometry)94
- Stewart’s Theorem5
- Surface area39
- Tangent circles86
- Tangent line76
- Transformation72
- Trapezoid65
- Triangle area180
- Triangle inequality58
- Trigonometric identity105
- Trigonometry100
- Vector56
- Viviani’s Theorem2
- Volume106
combinatorics
- Arrangements with restrictions77
- Basic counting63
- Bijection27
- Burnside’s Lemma10
- Catalan Number2
- Circular arrangements14
- Combinations138
- Combinatorial game8
- Complementary counting65
- Counting integers in a range93
- Counting intersections33
- Counting pairs34
- Counting regions5
- Counting shapes in figures16
- Derangements3
- Double counting33
- Fencepost counting7
- Generating functions4
- Graph theory48
- Inclusion-exclusion71
- Lattice paths15
- Multiplication principle85
- Multiset permutations23
- Partitions and compositions17
- Pascal’s Triangle8
- Permutations54
- Pigeonhole principle16
- Recursive counting21
- Stars and bars23
- Subsets48
- Tiling14
- Tree diagram4
- Vandermonde’s Convolution4
- Venn Diagram13
probability and statistics
- Basic probability183
- Bayes’ Theorem3
- Binomial probability16
- Complementary probability24
- Conditional probability33
- Data and graph interpretation53
- Dice (probability)47
- Expected value17
- Geometric distribution7
- Geometric probability65
- Harmonic mean17
- Independent events25
- Mean171
- Median (data)53
- Mode22
- Random walk21
- Range7
- Recursive probability22
- Sampling with replacement1
- Sampling without replacement19
- Weighted mean9