# Algebra articles, problems, and puzzles.

### What Is Arithmetic?

What Is Algebra?

Arithmetic: Articles and Problems

- 17 rooks on an 8×8 chessboard
- 1962 IMO, Problem 2: an inequality with a twist
- 1=2 via Continued Fractions
- 3-Colors Tower of Hanoi [Java]
- 3 glass puzzle [JavaScript]
- 3 glasses puzzle, second, more general tool [JavaScript]
- 3 Glasses Problem in Barycentric Coordinates
- 3 Jugs Problem - A Water Doubling Variant
- Three Jugs Problem [Java]
- Two Pails Puzzle [Java]

- 3-Term Arithmetic Progression
- A 2-Variable Optimization From a China Competition
- A 4-variable Inequality from the RMM
- 4x4 coloring problem
- 6th European Mathematical Cup (2017), Junior Problem 4
- A 49
^{th}Degree Challenge - A Case for Determinants
- A Case of Divergence
- A Clique of Acquaintances
- A Curious Property of Numbers 3 and 5
- A Cyclic But Not Symmetric Inequality in Four Variables
- A Cyclic Inequality by Seyran Ibrahimov
- A Cyclic Inequality from Math Phenomenon
- A Cyclic Inequality in Three Variables
- A Cyclic Inequality in Three Variables II
- A Cyclic Inequality in Three Variables III
- A Cyclic Inequality in Three Variables IV
- A Cyclic Inequality in Three Variables V
- A Cyclic Inequality in Three Variables VI
- A Cyclic Inequality in Three Variables VII
- A Cyclic Inequality in Three Variables VIII
- A Cyclic Inequality in Three Variables IX
- A Cyclic Inequality in Three Variables X
- A Cyclic Inequality in Three Variables XI
- A Cyclic Inequality in Three Variables XII
- A Cyclic Inequality in Three Variables XIII
- A Cyclic Inequality in Three Variables XIV
- A Cyclic Inequality in Three Variables XV
- A Cyclic Inequality in Three Variables XVI
- A Cyclic Inequality in Three Variables XVII
- A Cyclic Inequality in Three Variables XVIII
- A Cyclic Inequality in Three Variables XIX
- An Estimate for the Roots of Quadratic Equation
- An Exercise in Complex Numbers: Problem 4217 from Crux
- An Inequality with Constraint XX
- A Cyclic Inequality in Three Variables XX
- A Cyclic Inequality in Three Variables XXI
- A Cyclic Inequality in Three Variables XXII
- A Cyclic Inequality in Three Variables XXIII
- A Cyclic Inequality in Three Variables XXIV
- A Cyclic Inequality in Three Variables XXV
- A Cyclic Inequality in Three Variables XXVI
- A Cyclic Inequality in Three Variables And One More
- >A Cyclic Inequality In Three Variables by Sorin Radulescu
- A Cyclic Inequality in Three Variables by Uche E. Okeke
- A Cyclic Inequality from India In Three Variables And More
- A Cyclic Inequality in Three Variables with Constraint
- A Cyclic Inequality in Three or More Variables
- A Cyclic Inequality of Degree Four
- A Cyclic Inequality With Constraint in Two Triples of Variables
- A Cycling Inequality with Integrals
- A Cycling Inequality with Integrals II
- A Cyclic Inequality with Many Sums
- A Cyclic Inequality with Powers 2 through 7
- A Cyclic Inequality in Three Variables with a Variable Hierarchy
- A Cyclic Inequality on [-1,1]
- A Diophantine Equation - Problem from 2000 Belarus MO
- A Diophantine Equation with Square Roots - Problem from the 2008 British MO
- A DIY Proof of the Irrationality of sqrt(2)
- A Fancy Sum from the APMO 2000, Problem 1
- A Few Variants of a Popular Inequality And a Generalization
- A Formula for sin(3x)
- A Functional Equation from 1968 IMO
- A Functional Equation from the 2012 IMO
- A Generalization of an Inequality from a Romanian Olympiad
- A Limit of a Sequence
- A Limit of Another Sequence
- A Limit of a Series of Determinants
- A Limit with Fractions, Roots, Powers and Series
- A Limit With Sums, Roots, And Binomial Coefficients
- A Little of Algebra for an Inequality, A Little of Calculus for a Generalization
- A Little More of Algebra for an Inequality, A Little Less of Calculus for a Generalization
- A Long Cyclic Inequality of Degree 4
- A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
- A Mathematical Rabbit out of an Algebraic Hat
- A Memorable Fact
- A Middle School Problem from a Moscow Olympiad
- A Moscow Olympiad Question with Two Inequalities
- A New Proof That the Rationals Are Countable
- A Not Quite Cyclic Inequality
- A Not Quite Cyclic Inequality from Tibet
- A Not Symmetric Inequality [JavaScript]
- A Not So Simple Inequality in Three Variables
- A One-Line Proof of the Infinitude of Primes
- A Partly Cyclic Inequality in Four Variables
- A Problem for the 2015 New Year's Eve
- A Problem From a Mongolian Olympiad for Grade 11
- A Problem from Croatian TST
- A Problem form the Short List of the 2018 JBMO
- A Problem from the Danubius Contest 2016
- A Problem from the Danubius-XI Contest
- A Problem From the Romanian Mathematical Competitions (2016)
- A Problem in Complex Numbers Sets 3 Aside
- A Problem in Division of Polynomials
- A Problem with a Magical Solution from Secrets in Inequalities
- A Proof by Game for a Sum of a Convergent Series
- A Property of Number 4
- A Property of Product of Special Matrices, Probably Folklore
- A Property of the fusc() Function
- A Refinement of Turkevich's Inequality
- A Representation of Rational Numbers
- A Quartic Equation
- A Series with k-Permutations
- A Simple Inequality with Many Variables
- A Simple Cyclic Inequality in Three Variables
- A Simple Inequality in Three Variables
- A simple integral
- A simple integral, III
- A Simple Integral of a Peculiar Function
- A Simplified Version of Leo Giugiuc's Inequality from the AMM
- A Sum by Dorin Marghidanu
- A Sum of a Double Series
- A System of Equations Begging for Generalization
- A System of Equations in Determinants
- A System of Two Equations Replete with Squares
- A Telescoping Series
- A Tricky Integral Inequality
- A Trigonometric Inequality from the RMM
- >A Trigonometric Observation in Right Triangle
- A Triple Integral Inequality
- A True Algebraic-Geometric Inequality
- A Warmup Inequality from Vasile Cirtoaje
- ab + bc + ca ≤ aa + bb + cc
- About numbers
- Absolute Value
- Absolute Value and 1998 integers
- Achieve the Equilibrium [Flash]
- Addition
- Addition and Subtraction Formulas for Sine and Cosine III
- Addition and Subtraction Formulas for Sine and Cosine IV
- Advancing a Millenium Problem [JavaScript]
- Affirmative Action Problem [Java]
- Algebraic-Geometric Inequality
- Algebraic Structure of Complex Numbers
- All Integers Are Equal to 1
- All Integers Are Even
- All Powers of 2 Are Equal to 1
- All Powers of x are Constant
- Almost every integer has the digit 3 in it
- Almost Integers - Gaussian Integers
- Amazing Mix of Triangular Numbers and Golden Ratio
- Ambassadors at a Round Table [Java]
- An Acyclic Inequality in Three Variables
- An Algebraic Lemma with Geometric Consequences [JavaScript]
- An Application of Quadratic Formula
- An Arctan Identity and its Consequence [JavaScript, GeoGebra]
- An Easy Cyclic Inequality And a Remark
- An Easy Inequality with Three Integrals
- An Elementary Inequality by Non-elementary Means
- An Elementary Proof for Euler's Series
- An Equation in Determinants
- An Equation in Factorials
- An Equation in Matrices
- An Equation in Radicals
- An Equation in Reciprocals
- An Equation in RxR from Romania
- An Equation in Squares [JavaScript]
- An Equation Involving Four Collinear Points
- An Exercise without Calculating Aids
- An Identity between the Roots of a Cubic Polynomial - Problem from the 2007 Irish MO
- An Identity From the 2010 China Mathematical Competition
- An Identity Involving the Integer Part - Problem from the 1981 Canadian MO
- An Incidental Sum of 24
- An Increasing Sequence from 2014 IMO
- An Inequality and Its Modifications
- An Inequality between AM, QM and GM
- An Inequality by Uncommon Induction
- An Inequality for Grade 8
- An Inequality from a Mongolian Exam
- An Inequality from the 1967 IMO Shortlist
- An inequality from 1971 Canadian Mathematical Olympiad
- An Inequality from Marocco [JavaScript]
- An Inequality from Tibet
- An inequality in 2+2 variables from SSMA magazine
- Inequality in Convex Quadrilateral
- An Inequality in Determinants
- An Inequality in Four Weighted Variables
- An Inequality in Fractions with Absolute Values
- An Inequality in Integers [JavaScript]
- An Inequality in Integers II
- An Inequality in Parallelogram of Unit Area
- An Inequality in Reciprocals
- An Inequality in Reciprocals II An Inequality in Reciprocals III
- An Inequality in Three (Or Is It Two) Variables
- An Inequality in Triangle and In General
- An Inequality In Triangle and Without
- An Inequality In Triangle and Without II
- An Inequality in Two Or More Variables
- An Inequality in Two Or More Variables II
- An Inequality Involving Arithmetic And Geometric Means
- An Inequality Solved by Changing Appearances
- An Inequality with Angles and Integers
- An Inequality with Arbitrary Roots
- An Inequality with Areas, Norms, and Complex Numbers
- An Inequality with Complex Numbers of Unit Length
- An Inequality with Complex Numbers of Unit Length II
- An Inequality with Cubic Roots
- An Inequality with Cyclic Sums And Products
- An Inequality with Cyclic Sums on Both Sides
- An Inequality with Cyclic Sums on Both Sides II
- An Inequality with Cyclic Sums on Both Sides III
- An Inequality with Exponents
- An Inequality with Five Variables, Only Three Cyclic
- An Inequality with Fractions
- An Inequality with Integrals
- An Inequality with Integrals and Radicals
- An Inequality with Just Two Variable And an Integer
- An Inequality with Sines But Not in a Triangle
- An Inequality with One Tangent and Six Sines
- An Inequality with Powers And Logarithm
- An Inequality With Six Variables and Constraints
- An Inequality Where One Term Is More Equal Than Others
- An Inequality with a Parameter and a Constraint
- An Inequality with Absolute Values
- An Inequality with an Infinite Series
- An Inequality with Constraint
- An Inequality with Constraints II
- An Inequality with Constraint III
- An Inequality with Constraint IV
- An Inequality with Constraint V
- An Inequality with Constraint VI
- An Inequality with Constraint VII
- An Inequality with Constraint VIII [JavaScript]
- An Inequality with Constraint IX [JavaScript]
- An Inequality with Constraint X [JavaScript]
- An Inequality with Constraint XI [JavaScript]
- An Inequality with Constraint XII
- An Inequality with Constraint XIII
- An Inequality with Constraint XIV
- Inequalities with Constraint XV and XVI
- An Inequality with Constraint XVII
- An Inequality with Constraint XVIII
- An Inequality with Constraint XIX
- An Inequality with Constraint in Four Variables
- An Inequality with Constraint in Four Variables II
- An Inequality with Constraint in Four Variables III
- An Inequality with Constraint in Four Variables IV
- An Inequality with Constraint in Four Variables V
- An Inequality with Constraint in Four Variables VI
- An Inequality with Cot, Cos, and Sin
- An Inequality with Cycling Sums
- An Inequality with Determinants [JavaScript]
- An Inequality with Determinants II
- An Inequality with Determinants III
- An Inequality with Determinants IV
- An Inequality with Determinants V
- An Inequality with Determinants VI
- An Inequality with Determinants VII
- An Inequality with Exponents [JavaScript]
- An Inequality with a Generic Proof [JavaScript]
- An Inequality with Nested Radicals
- An Inequality via Young's Theorem
- An Inequality with Just Two Variable
- An Inequality with Just Two Variable II
- An Inequality with Just Two Variable III
- An Inequality with Just Two Variables IV
- An Inequality with Just Two Variables V
- An Inequality with Just Two Variables VI
- An Inequality with Just Two Variable VII
- An Inequality with Just Two Variable VIII
- An Inequality with Permutations, I [JavaScript]
- An Inequality with Permutations, II [JavaScript]
- An Inequality with Two Cyclic Sums
- An Inequality with Two Pairs of Triplets An Inequality with Two Triples of Variables
- An Inequality with Two Triples of Variables II
- Inequality with Two Variables? Think Again
- An Inequality with Two Variables from Awesome Math
- An Infinite Product from Recursion
- An Integer Root of a Polynomial with Integer Coefficients
- An Integral Estimate
- An Integral Identity with Two Functions
- An Integral from the RMM
- An Integral Inequality from the RMM
- An Old Inequality
- An Unsolvable Riccati Equation
- An Unusual Quadratic Equation
- Another Face and Proof of a Trigonometric Identity
- Another Inequality with Logarithms, But Not Really
- Another Integral Inequality from the RMM
- Another Problem from the 2016 Danubius Contest
- Another Property of Number 4
- Another simple integral
- An Unusual Problem by Leo Giugiuc
- Any Integer with Three 2s
- Application of Cauchy-Schwarz Inequality
- Applications of Schweitzer's Inequality
- Approximation Of Roots
- Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
- Areas on the Graphs of Power Functions
- Arithmetic-Geometric Means Inequality
- Arithmetic of Pie Sharing
- Ascending Bases and Exponents in Pascal's Triangle
- Assigning Numbers to Points in the Plane
- Associativity from 1977 IMO
- Averages, Arithmetic and Harmonic Means [Java]
- Averages in a sequence
- Ballot Lemma [Java]
- Banzhaf Power Index Calculator [Java]
- Barcode Encoding
- Barcode Magic [Java]
- Base Converter [JavaScript]
- Bergström's inequality
- Birth of an Inequality
- Bayes' Theorem
- Beatty Sequences
- Beauty and the Beast - in Trigonometry
- Bertrand's Postulate
- Bicubal Domino [JavaScript]
- Bilateral Diagrams [Java]
- Binet's Formula via Generating Functions
- Binet's Formula with Cosines
- Binomial Distribution [Java]
- Binomial Theorem
- Bisector of an imaginary angle may be real
- Brahmagupta-Fibonacci Identity [JavaScript]
- Byzantine Basketball
- Calculation of the Digits of pi by the Spigot Algorithm [Java]
- Calculus Proof of the Pythagorean Theorem
- Candido's Identity
- Cantor-Bernstein-Schroeder theorem
- Cantor's First Uncountability Proof
- Cardinal Numbers
- Cartesian Coordinate System [Java]
- Cassini's Identity
- Catalan Numbers in Pascal's Triangle
- Cavalieri's Principle [Java]
- Central and Inscribed Angles in Complex Numbers
- Chebyshev Inequality
- Chebyshev Holds a Key
- Check for Negative Roots
- Chess tournament with 1.5 Points Winners
- Chinese Remainder Theorem
- Clifford Algebras
- Clubs in a Vector Space
- Collatz Conjecture [Java]
- Coloring a Graph [Java]
- Coloring Squares, Adding up Numbers
- Combinatorial Proofs
- Combo Playground: Calculating the Number of Paths on a Square Grid
- Committee Chairs
- Comparing complex numbers
- Completing a Square and Quadratic Formula
- Completing the Square Where It Matters
- Complex Numbers: A Dynamic Tool [Java]
- Complex number to a complex power may be real
- Complicated Constraint - Simple Inequality
- Composition of Functions, an Exercise Constrained Inequality with Reciprocals
- Constrained Intermarriages [JavaScript]
- Constructible Numbers
- Construction of Pascal's Triangle [Java]
- Constructive Existence of the de Bruijn Cycles
- Continued fractions
- cos/sin inequality
- cos(π/7)-cos(2π/7)+cos(3π/7) = 1/2
- Cospherical Points
- Countability of Rational Numbers
- Countability Principle
- Counting Chips On a Circle [Java]
- Counting Fat Sets
- Counting Squares in a Square [Java]
- Counting Triangles [Java]
- Counting Triangles II [Java]
- Covering a Chessboard with a Hole with L-Trominoes [Java]
- Covering A Chessboard With Domino
- Dominoes on a Chessboard [Java]

- Cubes Constrained
- Cubes in Pascal's Triangle
- Cubic Equation for the Sides of a Triangle
- Cubic Formula [JavaScript]
- Cubics and Bézout's Theorem
- Cubic Optimization with Linear Constraints
- Cubic Optimization with Partly Linear Constraints
- Curious Identities In Pythagorean Triangles
- Curious Identities Involving Integer Products
- Cyclic Inequality in Four Variables
- Cyclic Inequality in Four Variables By D. Sitaru
- Cyclic Inequality in Three Variables by Marian Cucoanes
- Cyclic Inequality In Three Variables from the 2018 Romanian Olympiad, Grade 9
- Cyclic Inequality In Three Variables From Vietnam by Rearrangement
- Cyclic Inequality with Arctangents
- Cyclic Inequality with Logarithms
- Cyclic Inequality with Square Roots
- Cyclic Inequality with Square Roots And Absolute Values
- Dan Sitaru's Amazing, Never Ending Inequality
- Dan Sitaru's Cyclic Inequality in Four Variables
- Dan Sitaru's Cyclic Inequality In One Variable
- Dan Sitaru's Cyclic Inequality In Many Variables
- Dan Sitaru's Cyclic Inequality In Three Variables
- Dan Sitaru's Cyclic Inequality in Three Variables II
- Dan Sitaru's Cyclic Inequality in Three Variables III
- Dan Sitaru's Cyclic Inequality in Three Variables IV
- Dan Sitaru's Cyclic Inequality in Three Variables V
- Dan Sitaru's Cyclic Inequality in Three Variables VI
- Dan Sitaru's Cyclic Inequality in Three Variables VII
- Dan Sitaru's Cyclic Inequality in Three Variables VIII
- Cyclic Inequality In Three Variables From Kvant
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints IV
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints V
- Dan Sitaru's Exercise with Pi and Ln
- Dan Sitaru's Inequality: From Three Variables to Many in Two Ways
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints II
- Dan Sitaru's Cyclic Inequality In Three Variables with Constraints III
- Dan Sitaru's Cyclic Inequality in Three Variables IX
- Dan Sitaru's Inequality by Induction
- Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots
- Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots II
- Dan Sitaru's Inequality with a Double Integral
- Dan Sitaru's Inequality with Roots and Powers
- Dan Sitaru's Inequality with Tangents
- Dan Sitaru's Inequality with Three Related Integrals and Derivatives
- Dan Sitaru's Integral Inequality with Powers of a Function
- Dan Sitaru's Sum of a Series
- Daniel Dan's Optimization in Three Variables
- Darij Grinberg's Inequality In Three Variables
- Deceptive Appearances
- Decimal Sums of Successive Integers
- Derivatives without Limits
- Descartes' Rule of Signs
- Determinant and Divisibility
- Determinant on a Circle
- Determinants in Pascal's Triangle
- Diagonal Argument
- Diophantine Equations
- Distance between the Orthocenter and Circumcenter
- Distance Between Strings [Java]
- Distance From a Point to a Straight Line [Java]
- Distance Inequality
- Distances to Three Points on a Circle
- Distinct Multisets with the Same Pairwise Sum
- Divergence of {cos(n)} [JavaScript]
- Divergence of the Harmonic Series (Details)
- Divide And Conquer in Cyclic Sums
- Dividing Loot Fairly
- Divisibility and Modulo Arithmetic at IMO 2007
- Divisibility by Powers of 2
- Dollars in Pockets
- Doodling on April 1st
- Dorin Marghidanu's Believe It Or Not
- Dorin Marghidanu's Calculus Lemma
- Dorin Marghidanu's Cyclic Inequality in Three Variables
- Dorin Marghidanu's Cyclic Inequality in Three Variables II
- Dorin Marghidanu's Cyclic Inequality in Three Variables III
- Dorin Marghidanu's Cyclic Inequality with Constraint
- Dorin Marghidanu's Inequality in Complex Plane
- Dorin Marghidanu's Functional Equation
- Dorin Marghidanu's Inequality in Integer Variables
- Dorin Marghidanu's Inequality in Many Variables
- Dorin Marghidanu's Inequality in Many Variables Plus Two More
- Dorin Marghidanu's Inequality with Constraint
- Dorin Marghidanu's Inequality with Generalization
- Dorin Marghidanu's Inequality with Powers and Reciprocals
- Dorin Marghidanu's Inequality with Radicals
- Dorin Marghidanu's Light Elegance in Four Variables
- Dorin Marghidanu's Permuted Inequality
- Dorin Marghidanu's Spanish Problem Dorin Marghidanu's Sums and Products
- Dorin Marghidanu's Tribute to Descartes
- Dorin Marghidanu's Two-Sided Inequality
- Dot Patterns and Sierpinski Gasket [Java]
- Arithmetic in Disguise [Java]

- Dots and fractions [Java]
- Double and Half Angle Formulas
- Double Sums by Dorin Marghidanu
- Ducci Sequences [Java]
- e in the Pascal Triangle
- EAN-13 Encoding [Java]
- Eigenvectors by Inspection
- Elves and Gnomes [JavaScript]
- Entry 1.414... in Ramanujan's Notebooks
- Equation in Permutations
- Equation in Radicals as a System of Equations
- Equations of the Staight Line [Java]
- Equivalence Relations
- Escape of the Clones
- Euclid's Algorithm
- Euclid's Algorithm: An Interactive Illustration [Java]
- Euler Cycles in Digraph
- Euler's formula for
*e*^{iθ} - Euler's Function and Congruence Theorem
- Evolution of Algebraic Symbolism
- Exercise with Square Spiral [GeoGebra]
- Existence of the de Bruijn Cycles via Graphs
- Exponential Inequalities for Means
- Extension of Euclid's Algorithm
- Extrema of x^3+y^3+z^3-xyz on the unit sphere
- Factorial Products
- Factorials and division
- Fair Division: Method of Sealed Bids II [Java]
- Falsity implies anything
- Farey Series
- Fermat's Like Equation
- Fermat's Little Theorem
- Fermat's Only Published Proof
- Fibonacci Idendtities with Matrices
- Fibonacci's Quickies
- Fibonacci Tilings
- Filling a Grid with Good Neighbors [Java]
- Filling an Array with 0s and 1s - and Counting [Java]
- Filling Pool with Fluids
- Find Minimum without Calculus
- Find the Maximum and Minimum of a Function
- Finding Hamilton Paths and Circuits [Java]
- Finite Sums of Terms 2^(n-i) i²
- First 1008 Digits
- First Geometric Interpretation of Complex Numbers [Java]
- Five Methods of Apportionment [Java]
- Five Lattice Points [Java]
- Five Squares in Complex Numbers [GeoGebra]
- Fixed Point of a Family of Parabolas
- Fleury's Algorithm and Euler's Paths and Cycles [Java]
- Flipping Items Simultaneously [Java]
- Flipping Items Simultaneously II [Java]
- Flipping Items Simultaneously III [Java]
- Flipping pancakes [Java]
- Floor Function
- Formal Systems - an Example
- Formula for Primes
- Four Hinged Squares: using complex numbers
- Four Integrals in One Inequality
- Four Numbers, Six Differences, GCD of the Products
- Four Pegs That Form a Square
- Four Pegs That Form a Square [Java]

- Four Voting Methods [Java]
- From The Sum of Angles To The Sum of Sines
- Function
- Function in the Plane That Vanishes
- Functional Inequality from Vietnam
- Fundamental Theorem of Algebra
- Futurama Theorem and Puzzle [Java]
- Gasoline Stations on a Circular Trek
- Gauss and Euler Integrals
- Gauss' Lemma
- Gauss' theorem on constructibility of regular polygons
- GCD of Fibonacci Numbers
GCD(M, N) × LCM(M, N) = M × N [Java]- Generalization That Solves the Problem
- Generating Functions
- Generating Functions [Java]
- Coin Change [Java]
- Fast Power indices [Java]
- Weighted Voting and Power Indices [Java]
- Generating Functions from Recurrences

- Geometric Construction of Roots of Quadratic Equation [Java]
- Getting a Scout out of Desert [Java]
- Getting Your Rightful Share Back [JavaScript]
- Gheorghe Stoica's Problem from Gazeta Matematica Gireaux's Theorem
- Gödel's Incompleteness Theorem
- Golden Ratio is Irrational
- Graph and Roots of a Third Degree Polynomial [Java]
- Graph and Roots of Quadratic Polynomial [Java]
- Graph of a Polynomial [Java]
- Graph of a Polynomial Defined by Its Roots [Java]
- Graph with Nodes of Even Degree
- Graph without 3-Cycles
- Graphing Equations Is Useful
- Graphing Equations Is Useful, II [Java]
- Graphing Equations Is Useful, III
- Graphing Equations Is Useful, IV
- Graphing Equations Is Useful, V
- Greatest Difference in Arithmetic Progression
- Group multiplication of permutations [Java]
- 1/2 = 1
- Hadamard's Determinant Inequalities and Applications I
- Hadamard's Determinant Inequalities and Applications II
- Harmonic Serires And Its Parts
- 2006 Hellenic Mathematical Olympiad, Problem 4 [JavaScript]
- Hlawka's Inequality
- Horner's Method [Java]
- How Jost Burgi Computed the Sines of All Integer Angles in 1586
- How to Ask an Embarrassing Question
- How to define a permutation? [Java]
- How to write an equation of the union of two sets
- However You Solve It ... A Wonderful Equation
- Hung Viet's Inequality
- Hung Viet's Inequality II
- Hung Viet's Inequality III
- Hung Viet's Inequality IV
- Hyperintegers and Hyperreal Numbers
- Identities with Triangular Numbers
- Imad Zak's Cyclic Inequality in Three Variables
- Imad Zak's Cyclic Inequality in Three Variables II
- Implementation of Base Conversion
- Importance of the Absolute Value
- Improving an Inequality [JavaScript]
- Improving on an Escalator [JavaScript]
- In Calculus too 1 = 0
- Inclusion-Exclusion Principle
- Inclusion-Exclusion Principle: an Example
- Inequalities and Quadratic Forms
- An Inequality for Mixed Means
- An Inequality from RMM with a Generic 5
- An Inequality from RMM with Powers of 2
- Inequality 101 from the Cyclic Inequalities Marathon
- Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
- Inequality
(1 + 1 ^{-3})(1 + 2^{-3})(1 + 2^{-3})...(1 + n^{-3}) < 3 - Inequality
1 + 2 ^{-2}+ 3^{-2}+ 4^{-2}+ ... + n^{-2}< 2. - Inequality by Calculus
- Inequality for Integer Sequences
- Inequality for Integer Sequences II
- Inequality for Mixed Means
- An Inequality from Gazeta Matematica, March 2016
- An Inequality from Gazeta Matematica, March 2016 II
- An Inequality from Gazeta Matematica, March 2016 III
- An Inequality from Gazeta Matematica, March 2016 IV
- Inequalities with Double And Triple Integrals
- Inequality from Math Magazine 2009
- Inequality from Math Phenomenon
- Inequality in Acute Triangle
- Inequality of a Series of Square Roots
- Inequality with Constraint from Dan Sitaru's Math Phenomenon
- Inequality with Csc And Sin
- An Inequality with Exponents [JavaScript]
- An Inequality with Finite Sums
- An Inequality with Integrals and Rearrangement
- Inequality with Logarithms
- Inequality with Powers And Radicals
- Inequality with Three Linear Constraints
- Inequality with Three Numbers, Not All Zero
- Inequality with Two Minima
- Infinite Discent for x
^{4}+ y^{4}= z^{4} - Infinitesimals. Non-standard Analysis
- Infinitude of Primes
- Infinitude of Primes - A Topological Proof
- Infinitude of Primes - A Topological Proof without Topology
- Infinitude of Primes Via *-Sets
- Infinitude of Primes Via Coprime Pairs
- Infinitude of Primes Via Fermat Numbers
- Infinitude of Primes Via Harmonic Series
- Infinitude of Primes Via Lower Bounds
- Infinitude Of Primes From Legendre's Formula
- Infinitude of Primes - via Fibonacci Numbers
- Infinitude of Primes - via Bertrand's Postulate
- Infinitude of Primes - An Impossible Injection
- Infinitude of Primes - from Infinitude of Mutual Primes
- Infinitude of Primes Via Euler's Product Formula
- Infinitude of Primes Via Euler's Product Formula for Pi
- Infinitude of Primes - Golomb's Topological Proof
- Infinitude of Primes Via Powers of 2
- Infinitude of Primes As a Quickie
- Infinitude of Primes - A. Thue's Proof

- Infinity As a Limit
- Inflection Points of Fourth Degree Polynomials [Java]
- Inheriting Inequality
- An innovative recurrence
- Integer Iterations on a Circle [Java]
- Candy Game (Change Discharged) [Java]
- Integer Iterations on a Circle III [Java]
- Integer roots of integers are irrational
- Integer Triangle That Sums to Cube
- Integers and Rectangles [Java]
- Integers and Rectangles: a Proof by Induction [Java]
- Integers and Rectangles: Two Simple Proofs [Java]
- Integers With Digits 1 and 2 and Residues Modulo Powers of 2
- Integral in Normal Distribution
- Integral Is Area
- Integrals of a Function And Its Inverse
- Interest Calculations [Java]
- Integral of a Piece-wise Function
- Irrationality of k-th roots
- Irrationality of sqrt(2) via a Difference Equation
- Is GLOBALHELLFRY a Prime? [JavaScript]
- Is the Triangle Inequality Necessary?
- Is There a Limit?
- ISBN Encoding [Java]
- Irrational number to an irrational power may be rational
- Is 0.999... = 1?
- It's All About Powers of 10
- Iterations on Monotone Functions
- Jensen's Inequality [JavaScript]
- Ji Chen's Inequality
- Joachimsthal's Notations
- Jumping Isn't Everything [Java]
- Kaprekar's Iterations and Numbers [Java]
- Kulikowski's Theorem
- Kunihiko Chikaya's Inequality
- Kunihiko Chikaya's Inequality with a Constraint
- Kunihiko Chikaya's Inequality with Parameter
- Labeling Subsets
- Lagrange's identity, Proof
- Lagrange interpolation
- Lamé's Theorem - First Application of Fibonacci Numbers
- Langford and Skolem Sequences [Java]
- Latin Squares [Java]
- Le Khanh Sy's Problem
- Leech Trees and Taylor's Criterion
- Legendre's Formula
- Legendre's Theorem - The Prime Factorization of Factorials
- Leibniz and Pascal Triangles [Java]
- Leo Giugiuc and Vasile Cirtoaje's Cyclic Inequality
- Leo Giugiuc's Cyclic Inequality in Square Roots
- Leo Giugiuc's Cyclic Inequality in Three Variables
- Leo Giugiuc's Cyclic Quickie in Four Variables
- Leo Giugiuc's Cyclic Inequality with a Constraint
- Leo Giugiuc's Exercise
- Leo Giugiuc's Inequality with Constraint
- Leo Giugiuc's Optimization with Constraint
- Leo Giugiuc's Trigonometric Lemma
- Less than, Equal to, Greater Than Symbols
- Lewis Carroll's Game of Logic [Java]
- Light Bulbs in a Circle [Java]
- Lights Out [Java]
- Limit of a Recursive Sequence
- Linear Algebra Tools for Proving Inequalities
- Linear Algebra Tools for Proving Inequalities: Cauchy-Binet Formula
- Linear Function with Coefficients in Arithmetic Progression [GeoGebra]
- Linear functions
- Linear Programming
- Linear System with Parameter
- Listing All Combinations [Java]
- Listing all permutations [Java]
- Listing All Permutations: Johnson-Trotter Method [Java]
- Logistic Model applet [Java]
- Long Huynh Huu's Inequality and Solution
- Long Runs of Composite Numbers
- Long Up Short Down
- Looking your best
- Loop or Halt [Java]
- Lorian Saceanu's Cyclic Inequalities with Three Variables
- Lorian Saceanu's Inequality with Many Variables
- Lucas' Theorem
- Lucas' Theorem II
- Lucky Numbers [Java]
- Marian Dinca's Refinement of Nesbitt's Inequality
- Marriage Theorem
- Marriage Problem, simulation [Java]
- Math Lotto
- Mathematical Induction
- Mathematical Proof
- Mathematicians Doze Off
- Matrices and Determinants as Optimization Tools: an Example
- Matrix Groups
- Maximum Volume of a Cut Off Box [Java]
- Means [Java]
- Mediant Fractions
- Method of Markers II [Java]
- Michael Rozenberg's Inequality in Three Variables with Constraints
- Michael Rozenberg's Inequality in Two Variables
- Minesweeper Theorem
- Mini Lights Out [Java]
- Minimax Principle [Java]
- Minimax Theorem [Java]
- Minimum of a Cyclic Sum with Logarithms
- Minimum of Cotangents from Saint Petersburg
- Minumum Spanning Trees and Kruskal's Algorithm [Java]
- Minimum under Two Constraints
- Minus One But What a Difference
- Modular Arithmetic [Java]
- Monotone Convergence Theorem
- Monotone Number Sequences [Java]
- Monthly Problem 11199
- Much from Little
- Muirhead's Inequality
- Multiplication
- Multiplication of Points on a Circle [JavaScript, GeoGebra]
- Multiplication of Points on an Ellipse [JavaScript, GeoGebra]
- Napoleon's Relatives [Java]
- Negative on the left, positive on the right: how come?
- Nested Radicals
- Nested Subsets
- New Proof of Euclid's Theorem
- Newton's Method [Java]
- Newton's and Maclaurin's Inequalities [JavaScript]
- Hung Nguyen Viet's Inequality with a Constraint
- Nine Squares [Java]
- No Flipping without End [Java]
- NonSquare Matrix as a Tool for Proving an Inequality
- Number 8 Is Interesting
- Number Comparison - from the 1969 Canadian MO
- Number sequences and Pigeonhole [Java]
- Number System With Base 36
- Modern Vietnamese Classic
- On Dedekind Cuts
- On the Games of Hex and Y
- On the Intersection of kx and |sin(x)| [JavaScript]
- On the Intersection of 1+x and e^x and Trigonometric Consequences [JavaScript]
- One Dimensional Ants
- One Inequality - Two Domains
- One Trigonometric Formula and Its Consequences
- Optimization in Four Variables with Two Constraints
- Optimization with Many Variables
- Order. Well-ordered sets.
- Order and Chaos in Multiple Pendulums
- Ordinal Numbers
- Orthocenter in Complex Plane
- Outline Mathematics
- Billy is twice as old as Sally [Java]
- Coin Counting Word Problem [Java]
- Crab's Weight [Java]
- "Math trick" with two dice [Java]
- Primes as differences of squares [Java]
- Problem 1 from the 2017 Canada MO
- Problem 2 from the 2017 Canada MO
- Problem #24 from the Rhind Papyrus [Java]
- Robbery #1 [Java]
- Robbery #2 [Java]
- Robbery #3 [Java]
- Robbery #4 [Java]

- Problem 1 From the 2016 Pan-African Math Olympiad
- Problem 11867 from the American Mathematical Monthly
- Problem 2, the 36th IMO (1995)
- Problem 2 from the 2017 IMO
- Problem 3824 from Crux Mathematicorum
- Problem 3980 from Crux Mathematicorum
- Problem 4002 from Crux Mathematicorum
- Problem 4010 from Crux Mathematicorum
- Problem 4020 from Crux Mathematicorum
- Problem 4033 from Crux Mathematicorum
- Problem 4117 from Crux Mathematicorum
- Problem 4142 From Crux Mathematicorum
- Problem 4165 from Crux Mathematicorum
- Problem 4186 from Crux Mathematicorum
- Problem 6 from Dan Sitaru's Algebraic Phenomenon
- Problem 790 from Pentagon: an Inequality in Two Variables
- Problem From the 2016 IMO Shortlist
- Problem M317 from Crux Mathematicorum
- Proofs of Schweitzer's inequality
- Prove 2[sqrt(n) - sqrt(n-1)] > 1/sqrt(n)
- Sam Walters' Four Questions
- Same Integral, Three Intervals
- Samuel G. Walters' Congruence
- Simple Nameless Inequality
- Small Change Makes Big Difference
- Smallest multiple of 9 with no odd digits [Java]
- Snake Permutations And Their Number
- Solving a Peculiar Third Degree Equation
- The Basics of Infinite Descent
- The Joy of Homogeneity
- The Lucky Find [Java]
- The Mean Property of the Mean
- The Number of Judges and the Number of Votes [JavaScript]
- The power of substitution: solving an equation in a right triangle
- The power of substitution II: proving an inequality with three variables
- The power of substitution III: proving an inequality with two variables
- The Very First Functional Equation at IMO
- The Very Last, for 2014, proof of the Irrationality of sqrt(2)
- Three Colors on a Chessboard
- Three digit twister [Java]
- Who Has Beard? [Java]
- Outline Mathematics: Diophantine Equation I [JavaScript]
- p-adic Expansions
- p-adic Numbers
- Pairing Monotone and Convex Functions
- Parabola and Abutting Equilateral Triangles
- Parabolic Sieve of Prime Numbers
- Parity counting [Java]
- Parrondo Paradox Via Simpson Paradox [Java]
- Partial But General Case Of Fermat's Last Theorem
- Party Acquaintances [Java]
- Patterns in Pascal's Triangle
- Peano axioms
- Pennies in Boxes [Java]
- Percent
- Permutations
- Perrin Sequence
- Pi in Pascal's Triangle
- Pi in Pascal's Triangle via Triangular Numbers
- pi^e vs e^pi [JavaScript]
- Pigeonhole Principle
- Pinocchio as Epimenides
- Place Value [Java]
- Letter Count in a Sentence [Java]

- Powers and Fractions Inequality
- Presence of the Antipodes on a Circle
- Prime Sums in Pairs
- Probability of Divisibility
- Problem 1 from the IMO 2009
- Problem 11804 from the AMM
- Problem A1, 69th Putnam Competition
- Problem A2, 69th Putnam Competition
- Problem of Divisibility by 5
^{n} - Product of Diagonals in Regular N-gon
- Product Of Four Consecutive Integers
- Product of Integers that Add Up to 2018
- Product of Simple Fractions [Java]
- Product of permutations [Java]
- Proizvolov's Identity [Java]
- Proofs Without Words [Java]
- Property of a
^{n} - Property of Quadratic Polynomials
- Property of the Powers of 2 [Java]
- Puzzlist vs. Mathematician: A Practical Arithmetic Problem
- Pythagorean Triples [Java]
- Quadratic Function for Solving Inequalities
- Quadratic Inequalities on [-1,1]
- An Inequality Where One Term Is More Equal Than Others
- Quasigroups and block designs from triangular grid [Java]
- Rabbits Reproduce; Integers Don't
- Radicals, Radicals, And More Radicals in an Inequality
- Radon's Inequality and Applications
- Ramsey's Number R(3, 3, 3)
- Ramsey Number R(4, 3)
- Ramsey Number R(5, 3)
- Ramsey Number R(4, 4)
- Ramsey's Theorem
- Random Walks [Java]
- Rascal Triangle
- Rational Solutions to x
^{y}= y^{x} - Rearrangement Inequality [JavaScript]
- Recombining Permutations [Java]
- Rectangle on a Chessboard [Java]
- Rectilinear Paradox
- Reflection Lemma [Java]
- Riemann Rearrangement Theorem
- Riemann Sphere and Möbius Transformation
- Riemann Sums - Function Integration [Java]
- Right Replacement [Java]
- Rolle's and Mean Value Theorems [Java]
- Rolle's Theorem in Action
- RomanoNorwegian Inequality
- Rooks on a Chessboard
- Roots and Tangents [GeoGebra]
- Round Robin Tournament
- Russell's Paradox
- Solve A Fourth Degree Equation
- A Followup on Solving A Fourth Degree Equation [JavaScript]

- Sophie Germain's Identity
- Square Root of 2 is Irrational, Proof 29
- Square Roots and Triangle Inequality
- Squares in Pascal's Triangle
- Squeeze Theorem
- Sum of Absolute Values [JavaScript]
- Sum of All Natural Numbers
- Sums of Binomial Reciprocals in Pascal's Triangle
- Squeeze (Sandwich, Pinching ...) Theorem (Lemma): an Example
- Scalar Product Optimization [JavaScript]
- Scheduling and Critical Path Algorithm [Java]
- Schur's Inequality
- Scrub Tile Puzzle [Java]
- Second Pair of Twin Inequalities: Twin 1
- Second Pair of Twin Inequalities: Twin 2
- Self-descriptive Strings [Java]
- Self-documenting Iterations [JavaScript]
- Self-reference
- Seven integers under 127 and their Ratios
- Seven Terms Periodic Sequence
- Shapley-Shubik Power Index [Java]
- Shapley-Shubik Power Index Calculator [Java]
- Shifting Digits and a Point of View
- Ship-Dock Theorem
- Shuttle Puzzle Practice II [Java]
- Sicherman Dice
- Sierpinski Gasket and Tower of Hanoi
- Sierpinski's Gasket and Dihedral Symmetry [Java]
- Sieve of Squares [Java]
- Σ2
^{-n}= Σn·2^{-n} - Signs and Sums in a Sequence [Java]
- Signs in a Matrix [Java]
- Simple But Uncommon Equation from 1968 IMO
- Simple Cellular Automaton [Java]
- Simple Graph Practice [Java]
- Simple Inequality with a Variety of Solutions
- Simple Inequality with Many Faces And Variables
- Simple Somos Sequence - a Russian Olympiad problem
- Simple Somos Sequence Calculator [Java]
- Simple Yet Uncommon Inequalities with Absolute Value
- Simpson Paradox [Java]
- $\sin 1^{\circ}+\sin {2^\circ}+\sin 3^{\circ}+\cdots+\sin 180^{\circ}=\tan 89.5^{\circ}$
- Sine And Cosine Are Continuous Functions
- Sine of a Sum Formula, Proof Without Words
- Single formula to express any integer with digits 0-9 each used just once
- Sitaru-Schweitzer Inequality
- Six Balls, Two Weighings
- Size of a Class: Two Viewpoints
- Sladjan Stankovik's Inequality In Four Variables
- Sladjan Stankovik's Inequality With Constraint
- Sladjan Stankovik's Inequality With Constraint II
- Slider Puzzles [Java]
- Solitaire in Two Colors [Java]
- Solutions of F(x)=x and F(F(x))=x [JavaScript]
- Solve - an Olympiad Problem - in Your Mind
- Solve x
^{x3}= 3 - Solve
cos²x + cos²2x + cos²3x = 1 - Solving an Equation with Inverse Functions
- Soriteses
- Sorting Algorithms
- Sperner's Theorem
- Splitting piles
- Square Root of 2 as a Continued Fraction
- Square root of 2 is irrational
- Star of David Theorems in Pascal Triangle
- Stars in Array
- Steinhaus' Problem [Java]
- Step into the Elliptic Realm
- Stern-Brocot Tree
- Straight Tromino on a Chessboard [Java]
- Structure of Hyperreal Numbers
- Subsets and Intersections
- Subtraction Game [Java]
- Sums of the Elements of Three Element Subsets [JavaScript]
- Sum of Three Squares [JavaScript]
- Sums and Products, a Generalization
- Sums, Products, and 1-1 Functions
- Sums of Geometric Series - Proofs Without Words [Java]
- Sums of Power Series
- Sums Of Successive Squares
- Sum without Adding
- Sums and Products [Java]
- Sums of powers of digits [Java]
- Superposition Principle
- Surreal Numbers
- Swapping Rows and Columns [Java]
- Sylvester's Problem
- Taking Games Seriously [Java]
- Tangential Chaos
- Telescoping Cubes
- Telescoping Squares
- Telescoping Sums and Series [JavaScript]
- Telescoping Tangents
- The ass and the mule [Java]
- The Averages of Sides and Angles in a Triangle
- The Basic Rules of Counting
- The Local-Global Principle
- The Regula Falsi - Iterated
- The Roads We Take
- The Size is in the Eyes of the Beholder
- Three Complex Numbers Satisfy Fermat's Identity For Prime Powers
- Three Dimensional Cross
- Three Junior Problems from Vietnam
- Three Variables, Three Constraints, Two Inequalities (Only One to Prove) - by Leo Giugiuc
- Tool for Constructing Orthogonal Latin Squares [Java]
- Tower of Hanoi, the Hard Way [Java]
- Train on Bridge [JavaScript]
- Tran Hoang Nam's Cyclic Inequality in Three Variables
- Transpositions
- Traveling Salesman Problem [Java]
- Treatise on Arithmetical Triangle
- Triangular numbers that are also square
- Trigonometric Identity for the Pi Day
- Trigonometric Inequality with Integrals
- A Trigonometric Inequality with Ordered Triple of Variables
- Trilateral Diagrams [Java]
- Trinary Tree(s) underlying Primitive Pythagorean Triples
- Tromino Puzzle: Deficient Squares [Java]
- Twenty five boys and twenty five girls around a table.
- Twin Inequalities in Four Variables: Twin 1
- Twin Inequalities in Four Variables: Twin 2
- Two Constraints, One Inequality by Qing Song
- Two Circles and a Limit [Java]
- Two Colors in Two Rows [Java]
- Two Colors in Two Rows II [Java]

- Two Cyclic Inequalities [JavaScript]
- Two Equations in Three Variables [JavaScript]
- Two Men of Tibet [Java]
- Two-Parameter Solutions to Three Almost Fermat Equations
- Two-Sided Inequality by Dorin Marghidanu
- Two Problems by Kunihiko Chikaya
- Two Problems from a Serbian 7 Grade Math Competition
- Two Products: Constraint and Inequality
- Two Sequences Defined by Recurrence
- Two-Sided Inequality - One Provenance
- Two Solutions: One Correct, One Illuminating. An Example
- Uncountability of the Reals - via a Game
- Undefined vs Indeterminate in Mathematics
- (Un)fair Division by Consensus
- Universal Coloring [Java]
- Universal Product Code [Java]
- Unproperty of the Powers of 2
- Unsolved Problem from Crux Solved
- USA 1975 Problem 3
- Vasile Cirtoaje's Cyclic Inequality with Three Variables
- Vandermonde's Convolution Formula
- Vandermonde Determinant
- Venn Diagrams
- Viète's Formulas
- Venn Diagrams I [Java]
- Venn Diagrams II [Java]

- Voronoi Diagram [JavaSCript]
- Ways To Count
- Weierstrass Product Inequality
- Weierstrass Substitution
- What, how, and the Web [Java]
- What 1, 6, and 14 Have to Do with 5?
- What Is Algorithm?
- What Is Determinant?
- What Is Finite?
- What Is Infinite Descent?
- What Is Infinity?
- What Is Limit?
- What is so natural about "natural logarithms"?
- What Is Space Completion?
- WHEN THE COUNTING GETS TOUGH, THE TOUGH COUNT ON MATHEMATICS
- When Two Segments Intersect? [JavaScript]
- Where To Start? [Java]
- White and Black Balls in Urn. 1 in, 2 out. What Color Remains?
- Why 17?
- Why Can't They Be All Different?
- Why sin(x)/x tends to 1
- Why The Number of Primes Could Not Be Finite?
- Wilson's Theorem
- Without Definitions: Why $\displaystyle\frac{1}{3}+\frac{2}{5}=\frac{11}{15}?$
- Wonderful Inequality on Unit Circle
- Word Problems [Java]
- Wu's Inequality
- $x_n=\sin 1+\sin 3+\sin 5+\cdots+\sin (2n-1)$
- Yet Another Puzzle with Hats
- Yet Another Simple Integral
- Zeros and Nines [Java]

|Contact| |Front page| |Contents|

Copyright © 1996-2018 Alexander Bogomolny

68646192