Problems

63,351 problems across 17 subjects, plus 106.2 trillion generated practice problems · 0 solved

1,907 curated

Puzzles

#	Title	Difficulty	Source	Tags
16	The bat and the ballThink twice	Easy	Lemma	Setting up an equationReading carefully
17	The snail and the wellA gentle trap	Medium	Lemma	Careful countingWatch the last step
18	The lily padDoubling	Easy	Lemma	Exponential growthWorking backwards
19	Five machinesRates	Easy	Lemma	RatesProportional reasoning
20	The heavier coinWeighing	Medium	Lemma	Divide into threeInformation
21	There and backA sneaky average	Medium	Lemma	Average speedTotal distance over total time
22	At least one headProbability surprise	Medium	Lemma	Conditional probabilityListing cases
32	The Monty Hall problemProbability surprise	Medium	Lemma	Conditional probabilityCounting cases
33	The birthday paradoxA likely coincidence	Medium	Lemma	Complementary countingProbability
34	One hundred days onModular time	Easy	Lemma	RemaindersWorking mod 7
35	The coin rotation paradoxA rolling surprise	Medium	Lemma	Rolling without slippingCircumference
67	The twelve coinsWeighing	Deep	Lemma	InformationTernary outcomes
68	Wolf, goat, and cabbageLogic	Deep	Lemma	State searchCareful casework
69	Two eggs, a hundred floorsOptimization	Deep	Lemma	Worst-case balancingTriangular numbers
70	The hundred lockersA surprising pattern	Medium	Lemma	Divisor countingPerfect squares
638	We are both knavesLogic	Easy	Classical	Logic
639	Three wrong labelsLogic	Medium	Lemma	Logic
640	A hundred statementsLogic	Medium	Lemma	Logic
641	Same type as meLogic	Medium	Lemma	Logic
642	Day puzzleLogic	Easy	Lemma	Logic
643	Brothers and sisters have I noneLogic	Easy	Lemma	Logic
644	Overtaking in a raceLogic	Easy	Lemma	Logic
645	A slow clockLogic	Medium	Lemma	Logic
646	The striking clockLateral thinking	Easy	Lemma	Lateral thinking
647	When the hands meetLateral thinking	Medium	Lemma	Lateral thinking

Arithmetic

#	Title	Difficulty	Source	Tags
91	Times ninety-nineMental math	Easy	Lemma	Distributive lawCompensation
92	Squares ending in fiveMental math	Easy	Lemma	SquaresPatterns
93	Half of a third of a quarterFractions	Easy	Lemma	Multiplying fractions
94	A sum of unit fractionsFractions	Easy	Lemma	Common denominatorsEgyptian fractions
95	Up twenty, down twentyDecimals & percentages	Easy	Lemma	Percent changeMultipliers
96	A percentage swapDecimals & percentages	Easy	Lemma	Commutativity
97	Two pipes, one tankRatio & rate	Easy	Lemma	Work rates
98	There and back againRatio & rate	Medium	Lemma	Average speedHarmonic mean
99	The viral oneOrder of operations	Easy	Lemma	PrecedenceLeft-to-right
100	Which is bigger?Exponents & roots	Medium	Lemma	Laws of exponents
101	A root inside a rootExponents & roots	Easy	Lemma	Radicals
102	Gauss in the schoolroomMental math	Easy	Classical	PairingArithmetic series
232	Times twenty-fiveMental math	Easy	Lemma	Mental math
233	Almost a thousandMental math	Easy	Lemma	Mental math
234	Times one hundred and oneMental math	Easy	Lemma	Mental math
235	Forty-nine squaredMental math	Easy	Lemma	Mental math
236	Making changeMental math	Easy	Lemma	Mental math
237	One hundred twenty-fiveMental math	Easy	Lemma	Mental math
238	Around sixty-fiveMental math	Medium	Lemma	Mental math
239	Unlike denominatorsFractions	Easy	Lemma	Fractions
240	Dividing fractionsFractions	Easy	Lemma	Fractions
241	Which fraction is bigger?Fractions	Easy	Lemma	Fractions
242	Lowest termsFractions	Easy	Lemma	Fractions
243	A fraction towerFractions	Easy	Lemma	Fractions
244	Two and a half batchesFractions	Easy	Lemma	Fractions

Algebra

851

#	Title	Difficulty	Tags
6	The mean of twoInequalities	Easy	AM–GMSquares are nonnegative
7	A sum that collapsesSums	Easy	TelescopingPartial fractions
12	Integer echoesRecurrences	Medium	InductionRecurrence+1
31	What comes next?Patterns	Easy	Perfect squaresSequences
45	A third at a timeSums	Medium	Geometric series
46	A difference of squaresFactoring	Medium	Difference of squaresFactoring
47	Three fractions, at least threeInequalities	Deep	AM–GMProducts telescoping
48	Roots without solvingPolynomials	Medium	Vieta's formulasSymmetric functions
49	The lowest pointQuadratics	Medium	Completing the square
50	A dot product can't be too bigInequalities	Deep	Cauchy–SchwarzSquares are nonnegative
51	A radical that goes on foreverFixed points	Medium	Self-referenceQuadratics
52	Two equations from oneFunctional equations	Deep	SubstitutionSolving a system
103	One step at a timeLinear equations	Easy	Distributive lawBalancing
104	Three pairwise sumsSystems	Easy	Adding equationsSymmetry
105	Factor itQuadratics	Easy	FactoringZero product
106	A remainder without dividingPolynomials	Medium	Remainder theorem
107	Run it backwardsFunctions	Easy	Inverse functions
108	Where exponentials crossExponentials & logarithms	Medium	Logarithms
109	Doubling, added upSequences & series	Easy	Geometric series
740	Clear the fractionsLinear equations	Easy	Linear equations
741	A decimal coefficientLinear equations	Easy	Linear equations
742	A line through two pointsLinear equations	Easy	Linear equations
743	Brackets on both sidesLinear equations	Easy	Linear equations
744	Sum and differenceSystems	Easy	Systems
745	SubstituteSystems	Easy	Systems

Geometry

#	Title	Difficulty	Tags
5	Six equal piecesEuclidean geometry	Medium	AreaMedians+1
8	The angle in a semicircleCircles	Easy	Isosceles trianglesInscribed angle+1
9	Viviani's constant sumArea	Medium	Area decompositionEquilateral triangle
14	Pythagoras, without algebraTransformations	Medium	DissectionArea+1
26	Hands on the clockAngles	Easy	AnglesFractions of a circle
27	Across the rectanglePythagoras	Easy	Pythagorean theorem
28	Every square on the boardCounting shapes	Medium	CountingSum of squares
29	Three straight cutsThe lazy caterer	Medium	Adding intersectionsPatterns
39	Diagonals of a hexagonCounting shapes	Medium	CountingCombinations
40	Volume of a sphereSolids	Medium	Volume formula
41	Angles of a hexagonAngles	Easy	Angle sumTriangles
63	Crossing chordsCircles	Medium	Power of a pointSimilar triangles
64	The 13-14-15 triangleArea	Deep	Heron's formula
65	The fourth cornerDistances	Deep	British flag theoremPythagoras
66	Across the boxSolids	Medium	Pythagoras in 3D
110	Angles of a decagonAngles	Easy	Polygon angle sum
111	The circle inside 3-4-5Triangles	Medium	InradiusArea = rs
112	A chord from its distanceCircles	Easy	Perpendicular bisectorPythagoras
113	Area of an equilateral triangleArea & perimeter	Easy	AltitudePythagoras
114	Distance to a lineCoordinate geometry	Easy	Point–line distance
115	Equidistant from two pointsCoordinate geometry	Medium	MidpointPerpendicular slopes
116	An ice-cream coneSolid geometry	Easy	Volume of a cone
117	Through the mirrorTransformations	Easy	Reflections
771	A hexagon's cornerAngles	Easy	Angles
772	A supplementary angleAngles	Easy	Angles

Trigonometry

#	Title	Difficulty	Tags
71	The exact sine of fifteen degreesExact values	Easy	Angle subtractionExact values
72	One identity, many disguisesIdentities	Easy	Pythagorean identity
73	The leaning ladderRight triangles	Easy	Sine and cosine in right triangles
74	The triple angleMultiple angles	Medium	Angle additionDouble angle
75	A product of three sinesProduct identities	Medium	Product to sumTriple angle
76	The pentagon's cosineThe golden ratio	Medium	Double angleQuadratic in cosine
77	The sine rule sees the circleTriangles and circles	Medium	Inscribed angle theoremCircumcircle
78	Twenty, forty, eightyProduct of sines	Medium	Product identityTriple angle
79	Morrie's lawA telescoping product	Deep	Double angleTelescoping
80	The sum of a triangle's sinesInequalities	Deep	ConcavityJensen's inequality
118	A thirty-degree rampRight triangles	Easy	Sine ratio
119	Around the unit circleUnit circle	Easy	Reference angles
120	A tangent in disguiseIdentities	Medium	Double-angle formulas
121	Two solutions in a turnEquations	Easy	Symmetry of sine
122	The third sideLaws of sines & cosines	Easy	Law of cosines
123	Sine of an arccosineInverse functions	Easy	Inverse trigRight triangles
673	From tangent to sineRight triangles	Easy	Right triangles
674	A ladder against a wallRight triangles	Easy	Right triangles
675	A forty-five degree look-upRight triangles	Easy	Right triangles
676	A tree and its shadowRight triangles	Easy	Right triangles
677	Complementary valuesRight triangles	Easy	Right triangles
678	A 30-60-90 triangleRight triangles	Easy	Right triangles
679	Sine in the third quadrantUnit circle	Easy	Unit circle
680	Tangent in the second quadrantUnit circle	Easy	Unit circle
681	Two quarter turnsUnit circle	Easy	Unit circle

Complex Numbers

#	Title	Difficulty	Source	Tags
124	The cycle of iArithmetic of ℂ	Easy	Lemma	Powers of iPeriodicity
125	Dividing by a complex numberArithmetic of ℂ	Easy	Lemma	Conjugate
126	A modulus without multiplyingArithmetic of ℂ	Easy	Lemma	ModulusMultiplicativity
127	Into polar formPolar form	Easy	Lemma	ArgumentModulus
128	A big power, quicklyDe Moivre	Medium	Lemma	De Moivre's theorem
129	The most beautiful equationPolar form	Easy	Classical	Euler's formula
130	Roots that cancelRoots of unity	Easy	Lemma	Roots of unityFactoring
131	A product of rootsRoots of unity	Medium	Lemma	Cyclotomic polynomialFactorisation
132	Square roots of iPolar form	Medium	Lemma	Square roots in ℂ
133	A root and its twinPolynomials over ℂ	Medium	Lemma	Conjugate root theoremVieta
134	Counting rootsPolynomials over ℂ	Easy	Lemma	Roots of unityVieta
135	Rotate a squarePolar form	Medium	Lemma	Multiplication by i is rotation
269	Adding complex numbersArithmetic of ℂ	Easy	Lemma	Arithmetic of ℂ
270	Multiplying complex numbersArithmetic of ℂ	Easy	Lemma	Arithmetic of ℂ
271	Squaring a complex numberArithmetic of ℂ	Easy	Lemma	Arithmetic of ℂ
272	A quotientArithmetic of ℂ	Easy	Lemma	Arithmetic of ℂ
273	A Pythagorean modulusArithmetic of ℂ	Easy	Lemma	Arithmetic of ℂ
274	One over iArithmetic of ℂ	Easy	Lemma	Arithmetic of ℂ
275	An equation with a conjugateArithmetic of ℂ	Medium	Lemma	Arithmetic of ℂ
276	A hundred powers of iArithmetic of ℂ	Easy	Lemma	Arithmetic of ℂ
277	An argument in the third quadrantPolar form	Easy	Lemma	Polar form
278	Back to rectangularPolar form	Easy	Lemma	Polar form
279	Multiply in polar formPolar form	Easy	Lemma	Polar form
280	Divide in polar formPolar form	Easy	Lemma	Polar form
281	Argument of a productPolar form	Easy	Lemma	Polar form

Calculus

#	Title	Difficulty	Tags
81	The derivative from first principlesLimits & derivatives	Easy	Limit definition of the derivative
82	The open boxOptimization	Medium	OptimizationCritical points
83	Where a cubic turnsExtrema	Easy	Critical pointsSecond-derivative test
84	The sliding ladderRelated rates	Medium	Related ratesImplicit differentiation
85	The cheapest canOptimization	Medium	OptimizationConstraint substitution
86	The area under a parabolaThe definite integral	Medium	Riemann sumsSum of squares
87	An area equal to a quarter-turnDefinite integrals	Medium	AntiderivativesInverse trigonometric functions
88	The volume of a sphere, by slicingSolids of revolution	Medium	Disk methodDefinite integral
89	Parts, applied onceTechniques of integration	Medium	Integration by parts
90	Gabriel's hornImproper integrals	Deep	Improper integralsComparison test+1
136	A limit worth rememberingLimits	Medium	Standard limitsConjugates
137	The product ruleDerivatives	Easy	Product rule
138	A tangent lineApplications of derivatives	Easy	Tangent lines
139	Under one arch of sineIntegrals	Easy	Fundamental theorem of calculus
140	Spot the inner derivativeTechniques of integration	Easy	Substitution
141	Between a line and a parabolaApplications of integrals	Easy	Area between curves
142	A Taylor coefficientSeries	Easy	Maclaurin series
707	Cancel the holeLimits	Easy	Limits
708	A limit at infinityLimits	Easy	Limits
709	Tangent over xLimits	Easy	Limits
710	RationaliseLimits	Easy	Limits
711	Exponential beats linearLimits	Easy	Limits
712	Chain ruleDerivatives	Easy	Derivatives
713	Derivative of ln cos xDerivatives	Easy	Derivatives
714	Quotient ruleDerivatives	Easy	Derivatives

Multivariable Calculus

#	Title	Difficulty	Source	Tags
143	The angle between two vectorsVectors & geometry	Easy	Lemma	Dot product
144	A vector perpendicular to bothVectors & geometry	Easy	Lemma	Cross product
145	A plane through three pointsVectors & geometry	Medium	Lemma	Intercept formNormal vector
146	A first partial derivativePartial derivatives	Easy	Lemma	Partial differentiationChain rule
147	The direction of steepest ascentGradient & optimization	Easy	Lemma	Gradient
148	Walking downhillGradient & optimization	Medium	Lemma	Directional derivativeUnit vector
149	Saddle or valley?Gradient & optimization	Medium	Lemma	Critical pointsSecond-derivative test
150	Biggest rectangle in a circleGradient & optimization	Medium	Lemma	Lagrange multipliersAM–GM
151	Volume over a rectangleMultiple integrals	Easy	Lemma	Iterated integralsFubini
152	The Gaussian integralMultiple integrals	Deep	Classical	Polar coordinatesSquaring trick
153	Work along a pathVector calculus	Medium	Lemma	Conservative fieldsPotential function
154	Area from the boundaryVector calculus	Medium	Classical	Green's theoremParametrisation
307	Length of a vectorVectors & geometry	Easy	Lemma	Vectors & geometry
308	A dot productVectors & geometry	Easy	Lemma	Vectors & geometry
309	A unit vectorVectors & geometry	Easy	Lemma	Vectors & geometry
310	A triangle in spaceVectors & geometry	Medium	Lemma	Vectors & geometry
311	Distance to a planeVectors & geometry	Easy	Lemma	Vectors & geometry
312	A point on a lineVectors & geometry	Easy	Lemma	Vectors & geometry
313	Volume of a parallelepipedVectors & geometry	Medium	Lemma	Vectors & geometry
314	Perpendicular?Vectors & geometry	Easy	Lemma	Vectors & geometry
315	A partial in yPartial derivatives	Easy	Lemma	Partial derivatives
316	A mixed partialPartial derivatives	Medium	Lemma	Partial derivatives
317	A logarithmic partialPartial derivatives	Easy	Lemma	Partial derivatives
318	Going round in a circlePartial derivatives	Easy	Lemma	Partial derivatives
319	Mixed partials agreePartial derivatives	Medium	Lemma	Partial derivatives

Linear Algebra

#	Title	Difficulty	Source	Tags
155	Two lines, one pointLinear systems	Easy	Lemma	Elimination
156	Multiply two matricesMatrices	Easy	Lemma	Matrix multiplicationPermutation matrix
157	Undo a matrixMatrices	Easy	Lemma	InverseAdjugate
158	A three-by-three determinantDeterminants	Easy	Lemma	Cofactor expansion
159	Scaling volumeDeterminants	Easy	Lemma	Determinant as volumeMultiplicativity
160	How many independent rows?Vector spaces	Easy	Lemma	RankRow reduction
161	Rank–nullityVector spaces	Easy	Lemma	Rank–nullity theorem
162	Find the eigenvaluesEigenvalues	Easy	Lemma	Characteristic polynomialTrace & determinant
163	A direction that doesn't turnEigenvalues	Easy	Lemma	EigenvectorsNull space
164	A hundredth powerEigenvalues	Medium	Lemma	DiagonalisationPattern
165	A shadow on a lineOrthogonality	Easy	Lemma	Projection
166	Best-fit line through three pointsOrthogonality	Medium	Classical	Normal equationsLeast squares
345	Three equations, three unknownsLinear systems	Medium	Lemma	Linear systems
346	Parallel linesLinear systems	Easy	Lemma	Linear systems
347	The same line twiceLinear systems	Easy	Lemma	Linear systems
348	When does it fail?Linear systems	Medium	Lemma	Linear systems
349	A clean pairLinear systems	Easy	Lemma	Linear systems
350	How many free variables?Linear systems	Easy	Lemma	Linear systems
351	Solve with an inverseLinear systems	Medium	Lemma	Linear systems
352	Transpose of a productMatrices	Easy	Lemma	Matrices
353	A matrix that squares to zeroMatrices	Easy	Lemma	Matrices
354	The traceMatrices	Easy	Lemma	Matrices
355	Matrix times vectorMatrices	Easy	Lemma	Matrices
356	Rotate by ninety degreesMatrices	Easy	Lemma	Matrices
357	A shear, five timesMatrices	Easy	Lemma	Matrices

Differential Equations

#	Title	Difficulty	Source	Tags
167	Separate the variablesFirst-order	Easy	Lemma	Separable equations
168	An integrating factorFirst-order	Medium	Lemma	Linear first-orderIntegrating factor
169	Two exponentialsSecond-order linear	Medium	Lemma	Characteristic equation
170	A spring, releasedSecond-order linear	Easy	Lemma	Simple harmonic motion
171	A repeated rootSecond-order linear	Medium	Lemma	Critical dampingRepeated roots
172	Guess a polynomialSecond-order linear	Medium	Lemma	Undetermined coefficients
173	Two quantities feeding each otherSystems	Medium	Lemma	Linear systemsEigenvalues
174	Growth with a ceilingApplications	Medium	Classical	Logistic equationPartial fractions
175	A cooling cup of coffeeApplications	Medium	Lemma	Newton's law of coolingExponential decay
176	Carbon datingApplications	Easy	Lemma	Half-lifeExponential decay
177	Salt in a tankApplications	Medium	Lemma	Rate in − rate outLinear first-order
178	Doubling timeFirst-order	Easy	Lemma	Exponential growth
383	Growth proportional to sizeFirst-order	Easy	Lemma	First-order
384	Just integrateFirst-order	Easy	Lemma	First-order
385	A solution that blows upFirst-order	Medium	Lemma	First-order
386	A bell curve from an ODEFirst-order	Easy	Lemma	First-order
387	Approach to equilibriumFirst-order	Easy	Lemma	First-order
388	Where tangent comes fromFirst-order	Medium	Lemma	First-order
389	A power-law solutionFirst-order	Easy	Lemma	First-order
390	A linear equationFirst-order	Medium	Lemma	First-order
391	Find the equilibriaFirst-order	Easy	Lemma	First-order
392	Stable or unstable?First-order	Easy	Lemma	First-order
393	An integrating factor of xFirst-order	Medium	Lemma	First-order
394	Two steps of Euler's methodFirst-order	Easy	Lemma	First-order
395	Characteristic rootsSecond-order linear	Easy	Lemma	Second-order linear

Probability

#	Title	Difficulty	Source	Tags
179	Lucky sevenCounting & chance	Easy	Lemma	Equally likely outcomes
180	The Chevalier's wagerCounting & chance	Easy	Classical	Complement ruleIndependence
181	At least one boyConditional probability	Medium	Classical	ConditioningSample space
182	A positive testConditional probability	Medium	Lemma	Bayes' theoremBase rates
183	The average rollExpectation	Easy	Lemma	Expected value
184	Waiting for a sixExpectation	Easy	Lemma	Geometric distributionFirst-step analysis
185	See every faceExpectation	Medium	Lemma	Coupon collectorLinearity of expectation
186	The hat-check problemExpectation	Medium	Lemma	Indicator variablesLinearity of expectation
187	Three heads in fiveDistributions	Easy	Lemma	Binomial distribution
188	Meeting for lunchDistributions	Medium	Lemma	Geometric probability
189	A quiet hourDistributions	Easy	Lemma	Poisson distribution
190	Bertrand's boxesParadoxes	Medium	Classical	Conditional probabilityCounting coins, not boxes
421	A heart or a kingCounting & chance	Easy	Lemma	Counting & chance
422	Exactly two headsCounting & chance	Easy	Lemma	Counting & chance
423	Ten or moreCounting & chance	Easy	Lemma	Counting & chance
424	Two aces in a rowCounting & chance	Easy	Lemma	Counting & chance
425	On the committeeCounting & chance	Easy	Lemma	Counting & chance
426	A PIN with no repeatsCounting & chance	Easy	Lemma	Counting & chance
427	Who comes first?Counting & chance	Easy	Lemma	Counting & chance
428	A flushCounting & chance	Medium	Lemma	Counting & chance
429	Given the sumConditional probability	Easy	Lemma	Conditional probability
430	Drawing without replacementConditional probability	Easy	Lemma	Conditional probability
431	Either oneConditional probability	Easy	Lemma	Conditional probability
432	A suspicious coinConditional probability	Medium	Lemma	Conditional probability
433	Defects from two machinesConditional probability	Easy	Lemma	Conditional probability

Number Theory

148

#	Title	Difficulty	Tags
2	The primes never endPrime numbers	Medium	Proof by contradictionDivisibility+1
3	A sum that's never wholeDivisibility	Deep	2-adic valuationParity+1
10	A desert of primesPrime numbers	Medium	FactorialsDivisibility
13	Squares leave a traceDivisibility	Easy	ParityModular arithmetic
23	Adding 1 to 100A famous trick	Easy	PairingArithmetic series
24	The nine testDivisibility	Easy	Digit sumsDivisibility by 9
25	The last digit of a giantPatterns	Medium	CyclesUnits digit
36	Every square is a sum of oddsOdd numbers	Easy	Square numbersPairing
37	Zeros at the end of 100!Factorials	Deep	Factors of 5Legendre's formula
38	The next perfect numberPerfect numbers	Medium	DivisorsPerfect numbers
59	A giant power, tamedModular arithmetic	Deep	Fermat's little theorem
60	A hidden common factorDivisibility	Medium	Prime factorisationEuclid's algorithm
61	One more than a common multipleRemainders	Medium	Least common multipleRemainders
62	The last two digits of a giantPatterns	Deep	Cycles mod 100Powers
567	A greatest common divisorDivisibility	Easy	Divisibility
568	A least common multipleDivisibility	Easy	Divisibility
569	Always divisibleDivisibility	Medium	Divisibility
570	A remainderDivisibility	Easy	Divisibility
571	Odd divisorsDivisibility	Easy	Divisibility
572	Factor 1001Divisibility	Easy	Divisibility
573	Count the primesPrime numbers	Easy	Prime numbers
574	Factor this yearPrime numbers	Medium	Prime numbers
575	Three primes, two apartPrime numbers	Medium	Prime numbers
576	Primes in a windowPrime numbers	Easy	Prime numbers
577	PrimorialPrime numbers	Easy	Prime numbers

Combinatorics

143

#	Title	Difficulty	Tags
1	A party of sixGraph theory	Medium	PigeonholeRamsey R(3,3)+1
4	The mutilated chessboardInvariants	Medium	Colouring argumentInvariant+1
11	The odd shakers are evenGraph theory	Easy	Handshake lemmaSum of degrees+1
15	Two subsets, one sumCounting	Deep	PigeonholeCounting subsets+1
30	Everyone shakes handsCounting	Easy	Counting pairsAvoiding double counting
42	Arranging MATHArrangements	Easy	FactorialsCounting choices
43	Every subsetCounting	Easy	Powers of twoIndependent choices
44	Choosing a pairCounting	Easy	Combinationsn choose 2
53	Corner to cornerCounting paths	Medium	CombinationsArranging steps
54	Balls into boxesCounting	Medium	Stars and bars
55	Divisible by 2 or 3Inclusion–exclusion	Medium	Inclusion–exclusionCounting multiples
56	Five points in a squarePigeonhole	Deep	PigeonholeDistances in a square
57	Every letter in the wrong envelopeDerangements	Deep	DerangementsInclusion–exclusion
58	Balanced parenthesesCatalan numbers	Deep	Catalan numbersCareful listing
603	Four different digitsCounting	Easy	Counting
604	Licence platesCounting	Easy	Counting
605	Around a tableCounting	Easy	Counting
606	Anagrams of BANANACounting	Easy	Counting
607	President and vice-presidentCounting	Easy	Counting
608	Sharing ten sweetsCounting	Medium	Counting
609	Walking on a gridCounting	Easy	Counting
610	Diagonals of a decagonCounting	Easy	Counting
611	Ten choose threeBinomial coefficients	Easy	Binomial coefficients
612	A binomial coefficientBinomial coefficients	Easy	Binomial coefficients
613	With a coefficient insideBinomial coefficients	Easy	Binomial coefficients

Real Analysis

118

#	Title	Difficulty	Source	Tags
191	Supremum and infimumThe real numbers	Easy	Lemma	SupremumInfimum
192	A rational in every gapThe real numbers	Medium	Lemma	Archimedean propertyDensity
193	The number e as a limitSequences	Easy	Lemma	Euler's number
194	The n-th root of nSequences	Medium	Lemma	LogarithmsGrowth rates
195	An oscillating sequenceSequences	Medium	Lemma	limsupliminf+1
196	Convergent sequences are boundedSequences	Easy	Lemma	ε–N definition
197	A geometric seriesSeries	Easy	Lemma	Geometric series
198	Odd squares from BaselSeries	Medium	Classical	Splitting a seriesBasel problem
199	Radius of convergenceSeries	Deep	Lemma	Ratio testLimits involving e
200	Barely divergentSeries	Medium	Lemma	Integral test
201	Squeezed to zeroContinuity	Easy	Lemma	Squeeze theorem
202	Fill the holeContinuity	Easy	Lemma	Removable discontinuity
203	Differentiable, but wildDifferentiation	Medium	Lemma	Definition of the derivativeSqueeze theorem
204	A sum that is secretly an integralIntegration	Medium	Lemma	Riemann sums
495	The least upper bound of a setThe real numbers	Easy	Lemma	The real numbers
496	An infimum never reachedThe real numbers	Easy	Lemma	The real numbers
497	The square root of two is irrationalThe real numbers	Easy	Classical	The real numbers
498	Minimise a sum of distancesThe real numbers	Easy	Lemma	The real numbers
499	Integers in an intervalThe real numbers	Easy	Lemma	The real numbers
500	How large must n be?The real numbers	Easy	Lemma	The real numbers
501	Leading terms winSequences	Easy	Lemma	Sequences
502	A difference of rootsSequences	Medium	Lemma	Sequences
503	A decaying compoundSequences	Easy	Lemma	Sequences
504	A recursive sequenceSequences	Medium	Lemma	Sequences
505	n times sin(1/n)Sequences	Easy	Lemma	Sequences

Complex Analysis

#	Title	Difficulty	Source	Tags
205	A harmonic conjugateAnalytic functions	Easy	Lemma	Cauchy–Riemann equations
206	Recover f from its real partAnalytic functions	Medium	Lemma	Cauchy–RiemannMilne-Thomson
207	i to the power iAnalytic functions	Medium	Classical	Complex exponentPrincipal logarithm
208	Once around the originContour integrals	Easy	Lemma	ParametrisationWinding
209	Integrating the conjugateContour integrals	Medium	Lemma	Non-analytic integrands
210	Cauchy's integral formulaCauchy's theorem	Easy	Classical	Cauchy integral formula
211	Bounded and entireCauchy's theorem	Deep	Classical	Cauchy estimates
212	A simple residueResidues	Easy	Lemma	Simple poles
213	A real integral via the complex planeResidues	Medium	Lemma	Semicircular contourResidue theorem
214	A trigonometric integralResidues	Deep	Lemma	Unit-circle substitutionResidue theorem
215	An essential singularitySeries & singularities	Medium	Lemma	Laurent seriesEssential singularities
216	How bad is the pole?Series & singularities	Easy	Lemma	Laurent seriesOrder of a pole
217	Why the real series stops at 1Series & singularities	Medium	Lemma	Radius of convergenceComplex singularities
459	Is the conjugate analytic?Analytic functions	Easy	Lemma	Analytic functions
460	A complex derivativeAnalytic functions	Easy	Lemma	Analytic functions
461	The conjugate of eˣ cos yAnalytic functions	Medium	Lemma	Analytic functions
462	Size of e^{iz}Analytic functions	Easy	Lemma	Analytic functions
463	Logarithm of −1Analytic functions	Easy	Lemma	Analytic functions
464	Cosine of iAnalytic functions	Easy	Lemma	Analytic functions
465	Differentiable at one pointAnalytic functions	Medium	Lemma	Analytic functions
466	Harmonic or not?Analytic functions	Easy	Lemma	Analytic functions
467	Along a segmentContour integrals	Easy	Lemma	Contour integrals
468	A closed loopContour integrals	Easy	Lemma	Contour integrals
469	One over z squaredContour integrals	Medium	Lemma	Contour integrals
470	A pole insideContour integrals	Easy	Lemma	Contour integrals

Abstract Algebra

#	Title	Difficulty	Source	Tags
218	Symmetries of a squareGroups	Easy	Lemma	Dihedral groups
219	Order of an elementGroups	Easy	Lemma	OrderCyclic groups
220	Order of a permutationGroups	Easy	Lemma	Cycle decompositionlcm
221	Elements of order twoGroups	Medium	Lemma	Cycle typesCounting
222	What Lagrange allowsSubgroups & cosets	Easy	Classical	Lagrange's theorem
223	Groups of prime orderSubgroups & cosets	Easy	Lemma	Lagrange's theoremCyclic subgroups
224	Generators of a cyclic groupSubgroups & cosets	Easy	Lemma	Euler's totientGenerators
225	Counting homomorphismsHomomorphisms	Medium	Lemma	Homomorphisms from cyclic groups
226	Which products are cyclic?Homomorphisms	Easy	Lemma	Chinese remainder theoremIsomorphism
227	Units modulo fifteenRings	Easy	Lemma	UnitsEuler's totient
228	An inverse mod 26Rings	Easy	Lemma	Extended Euclidean algorithm
229	Units of the Gaussian integersRings	Medium	Lemma	NormUnits
230	Division in a finite fieldFields	Easy	Lemma	Finite fieldsInverses
231	A degree-four extensionFields	Medium	Lemma	Tower lawField extensions
531	How big is S₄?Groups	Easy	Lemma	Groups
532	An element of order twoGroups	Easy	Lemma	Groups
533	Order of a 4-cycleGroups	Easy	Lemma	Groups
534	Elements of order fourGroups	Easy	Lemma	Groups
535	Is subtraction a group operation?Groups	Easy	Lemma	Groups
536	The centre of S₃Groups	Medium	Lemma	Groups
537	Even or odd?Groups	Easy	Lemma	Groups
538	Flips of a squareGroups	Medium	Lemma	Groups
539	An indexSubgroups & cosets	Easy	Lemma	Subgroups & cosets
540	All subgroups of ℤ₁₂Subgroups & cosets	Easy	Lemma	Subgroups & cosets
541	Cosets of A₄Subgroups & cosets	Easy	Lemma	Subgroups & cosets

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TrigonometryOne, two, three — a perfect angleArctangent identities · by SouvikMedium CalculusSine squared over a quarter turnSymmetry in integrals · by SouvikMedium CombinatoricsLines cutting the planeCounting regions · by SouvikDeep GeometryThe inradius of a right triangleIncircle · by SouvikMedium AlgebraA product that collapsesTelescoping · by SouvikMedium CombinatoricsSocks in the darkPigeonhole · by SouvikEasy Number TheoryA square is never two more than a multiple of threeModular arithmetic · by SouvikEasy