Algebra articles, problems, and puzzles.

What Is Arithmetic?
What Is Algebra?
Arithmetic: Articles and Problems

  1. 17 rooks on an 8×8 chessboard
  2. 1962 IMO, Problem 2: an inequality with a twist
  3. 1=2 via Continued Fractions
  4. 3-Colors Tower of Hanoi [Java]
  5. 3 glass puzzle [JavaScript]
  6. 3-Term Arithmetic Progression
  7. A 4-variable Inequality from the RMM
  8. 4x4 coloring problem
  9. A 49th Degree Challenge
  10. A Case for Determinants
  11. A Case of Divergence
  12. A Clique of Acquaintances
  13. A Cyclic But Not Symmetric Inequality in Four Variables
  14. A Cyclic Inequality in Three Variables
  15. A Cyclic Inequality in Three Variables II
  16. A Cyclic Inequality in Three Variables III
  17. A Cyclic Inequality in Three Variables IV
  18. A Cyclic Inequality in Three Variables V
  19. A Cyclic Inequality in Three Variables VI
  20. A Cyclic Inequality in Three Variables VII
  21. A Cyclic Inequality in Three Variables VIII
  22. A Cyclic Inequality in Three Variables IX
  23. A Cyclic Inequality in Three Variables X
  24. A Cyclic Inequality in Three Variables XI
  25. A Cyclic Inequality in Three Variables XII
  26. A Cyclic Inequality in Three Variables XIII
  27. A Cyclic Inequality in Three Variables XIV
  28. A Cyclic Inequality in Three Variables XV
  29. A Cyclic Inequality in Three Variables XVI
  30. A Cyclic Inequality in Three Variables XVII
  31. A Cyclic Inequality in Three Variables XVIII
  32. A Cyclic Inequality in Three Variables XIX
  33. A Cyclic Inequality in Three Variables XX
  34. A Cyclic Inequality in Three Variables XXI
  35. A Cyclic Inequality in Three Variables XXII
  36. A Cyclic Inequality in Three Variables XXIII
  37. A Cyclic Inequality in Three Variables XXIV
  38. A Cyclic Inequality in Three Variables XXV
  39. A Cyclic Inequality in Three Variables And One More
  40. A Cyclic Inequality in Three Variables with Constraint
  41. A Cyclic Inequality in Three or More Variables
  42. A Cyclic Inequality of Degree Four
  43. A Cycling Inequality with Integrals
  44. A Cycling Inequality with Integrals II
  45. A Cyclic Inequality with Many Sums
  46. A Cyclic Inequality in Three Variables with a Variable Hierarchy
  47. A Diophantine Equation - Problem from 2000 Belarus MO
  48. A Diophantine Equation with Square Roots - Problem from the 2008 British MO
  49. A DIY Proof of the Irrationality of sqrt(2)
  50. A Formula for sin(3x)
  51. A Functional Equation from 1968 IMO
  52. A Functional Equation from the 2012 IMO
  53. A Generalization of an Inequality from a Romanian Olympiad
  54. A Limit of a Sequence
  55. A Limit of Another Sequence >li>A Limit with Fractions, Roots, Powers and Series
  56. A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
  57. A Mathematical Rabbit out of an Algebraic Hat
  58. A Middle School Problem from a Moscow Olympiad
  59. A New Proof That the Rationals Are Countable
  60. A Not Quite Cyclic Inequality
  61. A Not Symmetric Inequality [JavaScript]
  62. A Not So Simple Inequality in Three Variables
  63. A One-Line Proof of the Infinitude of Primes
  64. A Partly Cyclic Inequality in Four Variables
  65. A Problem for the 2015 New Year's Eve
  66. A Problem From a Mongolian Olympiad for Grade 11
  67. A Problem from Croatian TST
  68. A Problem from the Danubius Contest 2016
  69. A Problem from the Danubius-XI Contest
  70. A Problem in Complex Numbers Sets 3 Aside
  71. A Problem with a Magical Solution from Secrets in Inequalities
  72. A Proof by Game for a Sum of a Convergent Series
  73. A Property of Number 4
  74. A Property of the fusc() Function
  75. A Representation of Rational Numbers
  76. A Quartic Equation
  77. A Series with k-Permutations
  78. A Simple Cyclic Inequality in Three Variables
  79. A Simple Inequality in Three Variables
  80. A simple integral
  81. A simple integral, III
  82. A Sum by Dorin Marghidanu
  83. A Sum of a Double Series
  84. A System of Equations Begging for Generalization
  85. A System of Equations in Determinants
  86. A System of Two Equations Replete with Squares
  87. A Telescoping Series
  88. A Tricky Integral Inequality
  89. A Trigonometric Inequality from the RMM
  90. A Triple Integral Inequality
  91. ab + bc + ca ≤ aa + bb + cc
  92. About numbers
  93. Absolute Value
  94. Absolute Value and 1998 integers
  95. Achieve the Equilibrium [Flash]
  96. Addition
  97. Addition and Subtraction Formulas for Sine and Cosine III
  98. Addition and Subtraction Formulas for Sine and Cosine IV
  99. Advancing a Millenium Problem [JavaScript]
  100. Affirmative Action Problem [Java]
  101. Algebraic-Geometric Inequality
  102. Algebraic Structure of Complex Numbers
  103. All Integers Are Equal to 1
  104. All Integers Are Even
  105. All Powers of 2 Are Equal to 1
  106. All Powers of x are Constant
  107. Almost every integer has the digit 3 in it
  108. Almost Integers - Gaussian Integers
  109. Amazing Mix of Triangular Numbers and Golden Ratio
  110. Ambassadors at a Round Table [Java]
  111. An Acyclic Inequality in Three Variables
  112. An Algebraic Lemma with Geometric Consequences [JavaScript]
  113. An Application of Quadratic Formula
  114. An Arctan Identity and its Consequence [JavaScript, GeoGebra]
  115. An Easy Inequality with Three Integrals
  116. An Elementary Inequality by Non-elementary Means
  117. An Elementary Proof for Euler's Series
  118. An Equation in Determinants
  119. An Equation in Factorials
  120. An Equation in Matrices [JavaScript]
  121. An Equation in Radicals
  122. An Equation in Reciprocals
  123. An Equation in Squares [JavaScript]
  124. An Equation Involving Four Collinear Points
  125. An Exercise without Calculating Aids
  126. An Identity between the Roots of a Cubic Polynomial - Problem from the 2007 Irish MO
  127. An Identity From the 2010 China Mathematical Competition
  128. An Identity Involving the Integer Part - Problem from the 1981 Canadian MO
  129. An Increasing Sequence from 2014 IMO
  130. An Inequality and Its Modifications
  131. An Inequality by Uncommon Induction
  132. An Inequality for Grade 8
  133. An inequality from 1971 Canadian Mathematical Olympiad
  134. An Inequality from Marocco [JavaScript]
  135. An Inequality from Tibet
  136. An Inequality in Determinants
  137. An Inequality in Four Weighted Variables
  138. An Inequality in Fractions with Absolute Values
  139. An Inequality in Integers [JavaScript]
  140. An Inequality in Integers II
  141. An Inequality in Parallelogram of Unit Area
  142. An Inequality in Reciprocals
  143. An Inequality in Reciprocals II An Inequality in Reciprocals III
  144. An Inequality in Three (Or Is It Two) Variables
  145. An Inequality in Triangle and In General
  146. An Inequality In Triangle and Without
  147. An Inequality In Triangle and Without II
  148. An Inequality in Two Or More Variables
  149. An Inequality in Two Or More Variables II
  150. An Inequality Solved by Changing Appearances
  151. An Inequality with Angles and Integers
  152. An Inequality with Arbitrary Roots
  153. An Inequality with Areas, Norms, and Complex Numbers
  154. An Inequality with Complex Numbers of Unit Length
  155. An Inequality with Complex Numbers of Unit Length II
  156. An Inequality with Cyclic Sums And Products
  157. An Inequality with Cyclic Sums on Both Sides
  158. An Inequality with Cyclic Sums on Both Sides II
  159. An Inequality with Cyclic Sums on Both Sides III
  160. An Inequality with Fractions
  161. An Inequality with Integrals and Radicals
  162. An Inequality with Sines But Not in a Triangle
  163. An Inequality with One Tangent and Six Sines
  164. An Inequality with Powers And Logarithm
  165. An Inequality With Six Variables and Constraints
  166. An Inequality Where One Term Is More Equal Than Others
  167. An Inequality with a Parameter and a Constraint
  168. An Inequality with Absolute Values
  169. An Inequality with an Infinite Series
  170. An Inequality with Constraint
  171. An Inequality with Constraints II
  172. An Inequality with Constraint III
  173. An Inequality with Constraint IV
  174. An Inequality with Constraint V
  175. An Inequality with Constraint VI
  176. An Inequality with Constraint VII
  177. An Inequality with Constraint VIII [JavaScript]
  178. An Inequality with Constraint IX [JavaScript]
  179. An Inequality with Constraint X [JavaScript]
  180. An Inequality with Constraint XI [JavaScript]
  181. An Inequality with Constraint XII
  182. An Inequality with Constraint XIII
  183. An Inequality with Constraint XIV
  184. Inequalities with Constraint XV and XVI
  185. An Inequality with Constraint XVII
  186. An Inequality with Constraint in Four Variables
  187. An Inequality with Constraint in Four Variables II
  188. An Inequality with Constraint in Four Variables III
  189. An Inequality with Constraint in Four Variables IV
  190. An Inequality with Constraint in Four Variables V
  191. An Inequality with Cot, Cos, and Sin
  192. An Inequality with Cycling Sums
  193. An Inequality with Determinants [JavaScript]
  194. An Inequality with Determinants II
  195. An Inequality with Determinants III
  196. An Inequality with Determinants IV
  197. An Inequality with Determinants V
  198. An Inequality with Determinants VI
  199. An Inequality with Exponents [JavaScript]
  200. An Inequality with a Generic Proof [JavaScript]
  201. An Inequality with Nested Radicals
  202. An Inequality via Young's Theorem
  203. An Inequality with Just Two Variable
  204. An Inequality with Just Two Variable II
  205. An Inequality with Just Two Variable III
  206. An Inequality with Just Two Variables IV
  207. An Inequality with Just Two Variables V
  208. An Inequality with Just Two Variables VI
  209. An Inequality with Just Two Variable VII
  210. An Inequality with Just Two Variable VIII
  211. An Inequality with Permutations, I [JavaScript]
  212. An Inequality with Permutations, II [JavaScript]
  213. An Inequality with Two Cyclic Sums
  214. Inequality with Two Variables? Think Again
  215. An Infinite Product from Recursion
  216. An Integral Inequality from the RMM
  217. An Old Inequality
  218. An Unsolvable Riccati Equation
  219. An Unusual Quadratic Equation
  220. Another Face and Proof of a Trigonometric Identity
  221. Another Inequality with Logarithms, But Not Really
  222. Another Integral Inequality from the RMM
  223. Another Problem from the 2016 Danubius Contest
  224. Another simple integral
  225. Any Integer with Three 2s
  226. Application of Cauchy-Schwarz Inequality
  227. Applications of Schweitzer's Inequality
  228. Approximation Of Roots
  229. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
  230. Areas on the Graphs of Power Functions
  231. Arithmetic-Geometric Means Inequality
  232. Another Property of Number 4
  233. Ascending Bases and Exponents in Pascal's Triangle
  234. Assigning Numbers to Points in the Plane
  235. Associativity from 1977 IMO
  236. Averages, Arithmetic and Harmonic Means [Java]
  237. Averages in a sequence
  238. Ballot Lemma [Java]
  239. Banzhaf Power Index Calculator [Java]
  240. Barcode Encoding
  241. Barcode Magic [Java]
  242. Base Converter [JavaScript]
  243. Bergström's inequality
  244. Birth of an Inequality
  245. Bayes' Theorem
  246. Beatty Sequences
  247. Beauty and the Beast - in Trigonometry
  248. Bertrand's Postulate
  249. Bicubal Domino [JavaScript]
  250. Bilateral Diagrams [Java]
  251. Binet's Formula via Generating Functions
  252. Binet's Formula with Cosines
  253. Binomial Distribution [Java]
  254. Binomial Theorem
  255. Bisector of an imaginary angle may be real
  256. Brahmagupta-Fibonacci Identity [JavaScript]
  257. Byzantine Basketball
  258. Calculation of the Digits of pi by the Spigot Algorithm [Java]
  259. Calculus Proof of the Pythagorean Theorem
  260. Candido's Identity
  261. Cantor-Bernstein-Schroeder theorem
  262. Cantor's First Uncountability Proof
  263. Cardinal Numbers
  264. Cartesian Coordinate System [Java]
  265. Cassini's Identity
  266. Catalan Numbers in Pascal's Triangle
  267. Cavalieri's Principle [Java]
  268. Central and Inscribed Angles in Complex Numbers
  269. Chebyshev Inequality
  270. Chebyshev Holds a Key
  271. Chinese Remainder Theorem
  272. Clifford Algebras
  273. Clubs in a Vector Space
  274. Collatz Conjecture [Java]
  275. Coloring a Graph [Java]
  276. Combinatorial Proofs
  277. Combo Playground: Calculating the Number of Paths on a Square Grid
  278. Committee Chairs
  279. Comparing complex numbers
  280. Completing a Square and Quadratic Formula
  281. Completing the Square Where It Matters
  282. Complex Numbers: A Dynamic Tool [Java]
  283. Complex number to a complex power may be real
  284. Complicated Constraint - Simple Inequality
  285. Composition of Functions, an Exercise Constrained Inequality with Reciprocals
  286. Constrained Intermarriages [JavaScript]
  287. Constructible Numbers
  288. Construction of Pascal's Triangle [Java]
  289. Constructive Existence of the de Bruijn Cycles
  290. Continued fractions
  291. cos/sin inequality
  292. cos(π/7)-cos(2π/7)+cos(3π/7) = 1/2
  293. Cospherical Points
  294. Countability of Rational Numbers
  295. Countability Principle
  296. Counting Chips On a Circle [Java]
  297. Counting Fat Sets
  298. Counting Squares in a Square [Java]
  299. Counting Triangles [Java]
  300. Counting Triangles II [Java]
  301. Covering a Chessboard with a Hole with L-Trominoes [Java]
  302. Covering A Chessboard With Domino
  303. Cubes Constrained
  304. Cubes in Pascal's Triangle
  305. Cubic Equation for the Sides of a Triangle
  306. Cubic Formula [JavaScript]
  307. Cubics and Bézout's Theorem
  308. Cubic Optimization with Linear Constraints
  309. Cubic Optimization with Partly Linear Constraints
  310. Curious Identities In Pythagorean Triangles
  311. Curious Identities Involving Integer Products
  312. Cyclic Inequality with Logarithms
  313. Cyclic Inequality with Square Roots
  314. Cyclic Inequality with Square Roots And Absolute Values
  315. Dan Sitaru's Amazing, Never Ending Inequality
  316. Dan Sitaru's Cyclic Inequality In One Variable
  317. Dan Sitaru's Cyclic Inequality In Many Variables
  318. Dan Sitaru's Cyclic Inequality In Three Variables
  319. Dan Sitaru's Cyclic Inequality in Three Variables II
  320. Dan Sitaru's Cyclic Inequality in Three Variables III
  321. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints
  322. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints IV
  323. Dan Sitaru's Inequality: From Three Variables to Many in Two Ways
  324. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints II
  325. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints III
  326. Dan Sitaru's Inequality by Induction
  327. Dan Sitaru's Inequality with Roots and Powers
  328. Dan Sitaru's Inequality with Tangents
  329. Dan Sitaru's Inequality with Three Related Integrals and Derivatives
  330. Dan Sitaru's Integral Inequality with Powers of a Function
  331. Daniel Dan's Optimization in Three Variables
  332. Darij Grinberg's Inequality In Three Variables
  333. Deceptive Appearances
  334. Decimal Sums of Successive Integers
  335. Derivatives without Limits
  336. Descartes' Rule of Signs
  337. Determinant and Divisibility
  338. Determinant on a Circle
  339. Diagonal Argument
  340. Diophantine Equations
  341. Distance between the Orthocenter and Circumcenter
  342. Distance Between Strings [Java]
  343. Distance From a Point to a Straight Line [Java]
  344. Distance Inequality
  345. Distances to Three Points on a Circle
  346. Distinct Multisets with the Same Pairwise Sum
  347. Divergence of {cos(n)} [JavaScript]
  348. Divergence of the Harmonic Series (Details)
  349. Divide And Conquer in Cyclic Sums
  350. Dividing Loot Fairly
  351. Divisibility and Modulo Arithmetic at IMO 2007
  352. Divisibility by Powers of 2
  353. Dollars in Pockets
  354. Doodling on April 1st
  355. Dorin Marghidanu's Believe It Or Not
  356. Dorin Marghidanu's Cyclic Inequality in Three Variables
  357. Dorin Marghidanu's Cyclic Inequality in Three Variables II
  358. Dorin Marghidanu's Cyclic Inequality with Constraint
  359. Dorin Marghidanu's Inequality in Complex Plane
  360. Dorin Marghidanu's Inequality in Integer Variables
  361. Dorin Marghidanu's Inequality with Constraint
  362. Dorin Marghidanu's Inequality with Generalization
  363. Dorin Marghidanu's Light Elegance in Four Variables
  364. Dorin Marghidanu's Spanish Problem
  365. Dorin Marghidanu's Tribute to Descartes
  366. Dot Patterns and Sierpinski Gasket [Java]
  367. Dots and fractions [Java]
  368. Double and Half Angle Formulas
  369. Ducci Sequences [Java]
  370. e in the Pascal Triangle
  371. EAN-13 Encoding [Java]
  372. Eigenvectors by Inspection
  373. Elves and Gnomes [JavaScript]
  374. Entry 1.414... in Ramanujan's Notebooks
  375. Equation in Permutations
  376. Equation in Radicals as a System of Equations
  377. Equations of the Staight Line [Java]
  378. Equivalence Relations
  379. Escape of the Clones
  380. Euclid's Algorithm
  381. Euclid's Algorithm: An Interactive Illustration [Java]
  382. Euler Cycles in Digraph
  383. Euler's formula for e
  384. Euler's Function and Congruence Theorem
  385. Evolution of Algebraic Symbolism
  386. Exercise with Square Spiral [GeoGebra]
  387. Existence of the de Bruijn Cycles via Graphs
  388. Exponential Inequalities for Means
  389. Extension of Euclid's Algorithm
  390. Extrema of x^3+y^3+z^3-xyz on the unit sphere
  391. Factorial Products
  392. Factorials and division
  393. Fair Division: Method of Sealed Bids II [Java]
  394. Falsity implies anything
  395. Farey Series
  396. Fermat's Like Equation
  397. Fermat's Little Theorem
  398. Fermat's Only Published Proof
  399. Fibonacci Idendtities with Matrices
  400. Fibonacci's Quickies
  401. Fibonacci Tilings
  402. Filling a Grid with Good Neighbors [Java]
  403. Filling an Array with 0s and 1s - and Counting [Java]
  404. Filling Pool with Fluids
  405. Find Minimum without Calculus
  406. Finding Hamilton Paths and Circuits [Java]
  407. Finite Sums of Terms 2^(n-i) i²
  408. First 1008 Digits
  409. First Geometric Interpretation of Complex Numbers [Java]
  410. Five Methods of Apportionment [Java]
  411. Five Lattice Points [Java]
  412. Five Squares in Complex Numbers [GeoGebra]
  413. Fleury's Algorithm and Euler's Paths and Cycles [Java]
  414. Flipping Items Simultaneously [Java]
  415. Flipping Items Simultaneously II [Java]
  416. Flipping Items Simultaneously III [Java]
  417. Flipping pancakes [Java]
  418. Floor Function
  419. Formal Systems - an Example
  420. Formula for Primes
  421. Four Hinged Squares: using complex numbers
  422. Four Integrals in One Inequality
  423. Four Numbers, Six Differences, GCD of the Products
  424. Four Pegs That Form a Square
  425. Four Voting Methods [Java]
  426. From The Sum of Angles To The Sum of Sines
  427. Function
  428. Function in the Plane That Vanishes
  429. Fundamental Theorem of Algebra
  430. Futurama Theorem and Puzzle [Java]
  431. Gasoline Stations on a Circular Trek
  432. Gauss and Euler Integrals
  433. Gauss' Lemma
  434. Gauss' theorem on constructibility of regular polygons
  435. GCD of Fibonacci Numbers
  436. GCD(M, N) × LCM(M, N) = M × N [Java]
  437. Generalization That Solves the Problem
  438. Generating Functions
  439. Geometric Construction of Roots of Quadratic Equation [Java]
  440. Getting a Scout out of Desert [Java]
  441. Getting Your Rightful Share Back [JavaScript]
  442. Gheorghe Stoica's Problem from Gazeta Matematica Gireaux's Theorem
  443. Gödel's Incompleteness Theorem
  444. Golden Ratio is Irrational
  445. Graph and Roots of a Third Degree Polynomial [Java]
  446. Graph and Roots of Quadratic Polynomial [Java]
  447. Graph of a Polynomial [Java]
  448. Graph of a Polynomial Defined by Its Roots [Java]
  449. Graph with Nodes of Even Degree
  450. Graph without 3-Cycles
  451. Graphing Equations Is Useful
  452. Graphing Equations Is Useful, II [Java]
  453. Graphing Equations Is Useful, III
  454. Graphing Equations Is Useful, IV
  455. Graphing Equations Is Useful, V
  456. Greatest Difference in Arithmetic Progression
  457. Group multiplication of permutations [Java]
  458. 1/2 = 1
  459. Harmonic Serires And Its Parts
  460. 2006 Hellenic Mathematical Olympiad, Problem 4 [JavaScript]
  461. Hlawka's Inequality
  462. Horner's Method [Java]
  463. How Jost Burgi Computed the Sines of All Integer Angles in 1586
  464. How to Ask an Embarrassing Question
  465. How to define a permutation? [Java]
  466. How to write an equation of the union of two sets
  467. However You Solve It ... A Wonderful Equation
  468. Hung Viet's Inequality
  469. Hung Viet's Inequality II
  470. Hung Viet's Inequality III
  471. Hyperintegers and Hyperreal Numbers
  472. Identities with Triangular Numbers
  473. Imad Zak's Cyclic Inequality in Three Variables
  474. Imad Zak's Cyclic Inequality in Three Variables II
  475. Implementation of Base Conversion
  476. Importance of the Absolute Value
  477. Improving an Inequality [JavaScript]
  478. Improving on an Escalator [JavaScript]
  479. In Calculus too 1 = 0
  480. Inclusion-Exclusion Principle
  481. Inclusion-Exclusion Principle: an Example
  482. Inequalities and Quadratic Forms
  483. An Inequality for Mixed Means
  484. An Inequality from RMM with a Generic 5
  485. An Inequality from RMM with Powers of 2
  486. Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
  487. Inequality (1 + 1-3)(1 + 2-3)(1 + 2-3)...(1 + n-3) < 3
  488. Inequality 1 + 2-2 + 3-2 + 4-2 + ... + n-2 < 2.
  489. Inequality by Calculus
  490. Inequality for Integer Sequences
  491. Inequality for Integer Sequences II
  492. Inequality for Mixed Means
  493. An Inequality from Gazeta Matematica, March 2016
  494. An Inequality from Gazeta Matematica, March 2016 II
  495. An Inequality from Gazeta Matematica, March 2016 III
  496. An Inequality from Gazeta Matematica, March 2016 IV
  497. Inequalities with Double And Triple Integrals
  498. Inequality from Math Magazine 2009
  499. Inequality from Math Phenomenon
  500. Inequality in Acute Triangle
  501. Inequality of a Series of Square Roots
  502. Inequality with Constraint from Dan Sitaru's Math Phenomenon
  503. Inequality with Csc And Sin
  504. An Inequality with Exponents [JavaScript]
  505. An Inequality with Finite Sums
  506. An Inequality with Integrals and Rearrangement
  507. Inequality with Logarithms
  508. Inequality with Powers And Radicals
  509. Inequality with Three Linear Constraints
  510. Inequality with Three Numbers, Not All Zero
  511. Inequality with Two Minima
  512. Infinite Discent for x4 + y4 = z4
  513. Infinitesimals. Non-standard Analysis
  514. Infinitude of Primes
  515. Infinity As a Limit
  516. Inflection Points of Fourth Degree Polynomials [Java]
  517. An innovative recurrence
  518. Integer Iterations on a Circle [Java]
  519. Candy Game (Change Discharged) [Java]
  520. Integer Iterations on a Circle III [Java]
  521. Integer roots of integers are irrational
  522. Integer Triangle That Sums to Cube
  523. Integers and Rectangles [Java]
  524. Integers and Rectangles: a Proof by Induction [Java]
  525. Integers and Rectangles: Two Simple Proofs [Java]
  526. Integers With Digits 1 and 2 and Residues Modulo Powers of 2
  527. Integral in Normal Distribution
  528. Integral Is Area
  529. Integrals of a Function And Its Inverse
  530. Interest Calculations [Java]
  531. Integral of a Piece-wise Function
  532. Irrationality of k-th roots
  533. Irrationality of sqrt(2) via a Difference Equation
  534. Is GLOBALHELLFRY a Prime? [JavaScript]
  535. Is the Triangle Inequality Necessary?
  536. Is There a Limit?
  537. ISBN Encoding [Java]
  538. Irrational number to an irrational power may be rational
  539. Is 0.999... = 1?
  540. It's All About Powers of 10
  541. Iterations on Monotone Functions
  542. Jensen's Inequality [JavaScript]
  543. Joachimsthal's Notations
  544. Jumping Isn't Everything [Java]
  545. Kaprekar's Iterations and Numbers [Java]
  546. Kulikowski's Theorem
  547. Lagrange's identity, Proof
  548. Lagrange interpolation
  549. Lamé's Theorem - First Application of Fibonacci Numbers
  550. Langford and Skolem Sequences [Java]
  551. Latin Squares [Java]
  552. Le Khanh Sy's Problem
  553. Leech Trees and Taylor's Criterion
  554. Legendre's Formula
  555. Legendre's Theorem - The Prime Factorization of Factorials
  556. Leibniz and Pascal Triangles [Java]
  557. Leo Giugiuc's Cyclic Inequality in Three Variables
  558. Leo Giugiuc's Exercise
  559. Leo Giugiuc's Inequality with Constraint
  560. Leo Giugiuc's Optimization with Constraint
  561. Leo Giugiuc's Trigonometric Lemma
  562. Less than, Equal to, Greater Than Symbols
  563. Lewis Carroll's Game of Logic [Java]
  564. Light Bulbs in a Circle [Java]
  565. Lights Out [Java]
  566. Limit of a Recursive Sequence
  567. Linear Algebra Tools for Proving Inequalities
  568. Linear Algebra Tools for Proving Inequalities: Cauchy-Binet Formula
  569. Linear Function with Coefficients in Arithmetic Progression [GeoGebra]
  570. Linear functions
  571. Linear Programming
  572. Linear System with Parameter
  573. Listing All Combinations [Java]
  574. Listing all permutations [Java]
  575. Listing All Permutations: Johnson-Trotter Method [Java]
  576. Logistic Model applet [Java]
  577. Long Runs of Composite Numbers
  578. Long Up Short Down
  579. Looking your best
  580. Loop or Halt [Java]
  581. Lucas' Theorem
  582. Lucas' Theorem II
  583. Lucky Numbers [Java]
  584. Marian Dinca's Refinement of Nesbitt's Inequality
  585. Marriage Theorem
  586. Marriage Problem, simulation [Java]
  587. Math Lotto
  588. Mathematical Induction
  589. Mathematical Proof
  590. Mathematicians Doze Off
  591. Matrices and Determinants as Optimization Tools: an Example
  592. Matrix Groups
  593. Maximum Volume of a Cut Off Box [Java]
  594. Means [Java]
  595. Mediant Fractions
  596. Method of Markers II [Java]
  597. Michael Rozenberg's Inequality in Three Variables with Constraints
  598. Michael Rozenberg's Inequality in Two Variables
  599. Minesweeper Theorem
  600. Mini Lights Out [Java]
  601. Minimax Principle [Java]
  602. Minimax Theorem [Java]
  603. Minimum of a Cyclic Sum with Logarithms
  604. Minumum Spanning Trees and Kruskal's Algorithm [Java]
  605. Minimum under Two Constraints
  606. Minus One But What a Difference
  607. Modular Arithmetic [Java]
  608. Monotone Convergence Theorem
  609. Monotone Number Sequences [Java]
  610. Monthly Problem 11199
  611. Much from Little
  612. Muirhead's Inequality
  613. Multiplication
  614. Multiplication of Points on a Circle [JavaScript, GeoGebra]
  615. Multiplication of Points on an Ellipse [JavaScript, GeoGebra]
  616. Napoleon's Relatives [Java]
  617. Negative on the left, positive on the right: how come?
  618. Nested Radicals
  619. Nested Subsets
  620. New Proof of Euclid's Theorem
  621. Newton's Method [Java]
  622. Newton's and Maclaurin's Inequalities [JavaScript]
  623. Nine Squares [Java]
  624. No Flipping without End [Java]
  625. NonSquare Matrix as a Tool for Proving an Inequality
  626. Number 8 Is Interesting
  627. Number Comparison - from the 1969 Canadian MO
  628. Number sequences and Pigeonhole [Java]
  629. Number System With Base 36
  630. Modern Vietnamese Classic
  631. On Dedekind Cuts
  632. On the Games of Hex and Y
  633. On the Intersection of kx and |sin(x)| [JavaScript]
  634. On the Intersection of 1+x and e^x and Trigonometric Consequences [JavaScript]
  635. One Dimensional Ants
  636. One Inequality - Two Domains
  637. One Trigonometric Formula and Its Consequences
  638. Optimization in Four Variables with Two Constraints
  639. Optimization with Many Variables
  640. Order. Well-ordered sets.
  641. Order and Chaos in Multiple Pendulums
  642. Ordinal Numbers
  643. Orthocenter in Complex Plane
  644. Outline Mathematics
  645. p-adic Expansions
  646. p-adic Numbers
  647. Pairing Monotone and Convex Functions
  648. Parabola and Abutting Equilateral Triangles
  649. Parabolic Sieve of Prime Numbers
  650. Parity counting [Java]
  651. Parrondo Paradox Via Simpson Paradox [Java]
  652. Partial But General Case Of Fermat's Last Theorem
  653. Party Acquaintances [Java]
  654. Patterns in Pascal's Triangle
  655. Peano axioms
  656. Pennies in Boxes [Java]
  657. Percent
  658. Permutations
  659. Perrin Sequence
  660. Pi in Pascal's Triangle
  661. Pi in Pascal's Triangle via Triangular Numbers
  662. pi^e vs e^pi [JavaScript]
  663. Pigeonhole Principle
  664. Pinocchio as Epimenides
  665. Place Value [Java]
  666. Powers and Fractions Inequality
  667. Presence of the Antipodes on a Circle
  668. Prime Sums in Pairs
  669. Probability of Divisibility
  670. Problem 1 from the IMO 2009
  671. Problem 11804 from the AMM
  672. Problem A1, 69th Putnam Competition
  673. Problem A2, 69th Putnam Competition
  674. Problem of Divisibility by 5n
  675. Product of Diagonals in Regular N-gon
  676. Product Of Four Consecutive Integers
  677. Product of Simple Fractions [Java]
  678. Product of permutations [Java]
  679. Proizvolov's Identity [Java]
  680. Proofs Without Words [Java]
  681. Property of an
  682. Property of Quadratic Polynomials
  683. Property of the Powers of 2 [Java]
  684. Puzzlist vs. Mathematician: A Practical Arithmetic Problem
  685. Pythagorean Triples [Java]
  686. Quadratic Function for Solving Inequalities
  687. An Inequality Where One Term Is More Equal Than Others
  688. Quasigroups and block designs from triangular grid [Java]
  689. Rabbits Reproduce; Integers Don't
  690. Radicals, Radicals, And More Radicals in an Inequality
  691. Radon's Inequality and Applications
  692. Ramsey's Number R(3, 3, 3)
  693. Ramsey Number R(4, 3)
  694. Ramsey Number R(5, 3)
  695. Ramsey Number R(4, 4)
  696. Ramsey's Theorem
  697. Random Walks [Java]
  698. Rascal Triangle
  699. Rational Solutions to xy = yx
  700. Rearrangement Inequality [JavaScript]
  701. Recombining Permutations [Java]
  702. Rectangle on a Chessboard [Java]
  703. Rectilinear Paradox
  704. Reflection Lemma [Java]
  705. Riemann Rearrangement Theorem
  706. Riemann Sphere and Möbius Transformation
  707. Riemann Sums - Function Integration [Java]
  708. Right Replacement [Java]
  709. Rolle's and Mean Value Theorems [Java]
  710. Rolle's Theorem in Action
  711. RomanoNorwegian Inequality
  712. Rooks on a Chessboard
  713. Roots and Tangents [GeoGebra]
  714. Round Robin Tournament
  715. Russell's Paradox
  716. Solve A Fourth Degree Equation
  717. Sophie Germain's Identity
  718. Square Roots and Triangle Inequality
  719. Squares in Pascal's Triangle
  720. Squeeze Theorem
  721. Sum of Absolute Values [JavaScript]
  722. Sum of All Natural Numbers
  723. Sums of Binomial Reciprocals in Pascal's Triangle
  724. Squeeze (Sandwich, Pinching ...) Theorem (Lemma): an Example
  725. Scalar Product Optimization [JavaScript]
  726. Scheduling and Critical Path Algorithm [Java]
  727. Schur's Inequality
  728. Scrub Tile Puzzle [Java]
  729. Self-descriptive Strings [Java]
  730. Self-documenting Iterations [JavaScript]
  731. Self-reference
  732. Seven integers under 127 and their Ratios
  733. Seven Terms Periodic Sequence
  734. Shapley-Shubik Power Index [Java]
  735. Shapley-Shubik Power Index Calculator [Java]
  736. Shifting Digits and a Point of View
  737. Ship-Dock Theorem
  738. Shuttle Puzzle Practice II [Java]
  739. Sicherman Dice
  740. Sierpinski Gasket and Tower of Hanoi
  741. Sierpinski's Gasket and Dihedral Symmetry [Java]
  742. Sieve of Squares [Java]
  743. Σ2-n = Σn·2-n
  744. Signs and Sums in a Sequence [Java]
  745. Signs in a Matrix [Java]
  746. Simple But Uncommon Equation from 1968 IMO
  747. Simple Cellular Automaton [Java]
  748. Simple Graph Practice [Java]
  749. Simple Inequality with a Variety of Solutions
  750. Simple Inequality with Many Faces And Variables
  751. Simple Somos Sequence - a Russian Olympiad problem
  752. Simple Somos Sequence Calculator [Java]
  753. Simpson Paradox [Java]
  754. $\sin 1^{\circ}+\sin {2^\circ}+\sin 3^{\circ}+\cdots+\sin 180^{\circ}=\tan 89.5^{\circ}$
  755. Sine And Cosine Are Continuous Functions
  756. Sine of a Sum Formula, Proof Without Words
  757. Single formula to express any integer with digits 0-9 each used just once
  758. Sitaru-Schweitzer Inequality
  759. Six Balls, Two Weighings
  760. Size of a Class: Two Viewpoints
  761. Sladjan Stankovik's Inequality In Four Variables
  762. Sladjan Stankovik's Inequality With Constraint
  763. Sladjan Stankovik's Inequality With Constraint II
  764. Slider Puzzles [Java]
  765. Solitaire in Two Colors [Java]
  766. Solutions of F(x)=x and F(F(x))=x [JavaScript]
  767. Solve - an Olympiad Problem - in Your Mind
  768. Solve xx3 = 3
  769. Solve cos²x + cos²2x + cos²3x = 1
  770. Solving an Equation with Inverse Functions
  771. Soriteses
  772. Sorting Algorithms
  773. Sperner's Theorem
  774. Splitting piles
  775. Square Root of 2 as a Continued Fraction
  776. Square root of 2 is irrational
  777. Star of David Theorems in Pascal Triangle
  778. Steinhaus' Problem [Java]
  779. Step into the Elliptic Realm
  780. Stern-Brocot Tree
  781. Straight Tromino on a Chessboard [Java]
  782. Structure of Hyperreal Numbers
  783. Subsets and Intersections
  784. Subtraction Game [Java]
  785. Sums of the Elements of Three Element Subsets [JavaScript]
  786. Sum of Three Squares [JavaScript]
  787. Sums and Products, a Generalization
  788. Sums, Products, and 1-1 Functions
  789. Sums of Geometric Series - Proofs Without Words [Java]
  790. Sums of Power Series
  791. Sums Of Successive Squares
  792. Sum without Adding
  793. Sums and Products [Java]
  794. Sums of powers of digits [Java]
  795. Superposition Principle
  796. Surreal Numbers
  797. Swapping Rows and Columns [Java]
  798. Sylvester's Problem
  799. Taking Games Seriously [Java]
  800. Tangential Chaos
  801. Telescoping Cubes
  802. Telescoping Squares
  803. Telescoping Sums and Series [JavaScript]
  804. Telescoping Tangents
  805. The ass and the mule [Java]
  806. The Basic Rules of Counting
  807. The Local-Global Principle
  808. The Regula Falsi - Iterated
  809. The Roads We Take
  810. The Size is in the Eyes of the Beholder
  811. Three Dimensional Cross
  812. Three Junior Problems from Vietnam
  813. Tool for Constructing Orthogonal Latin Squares [Java]
  814. Tower of Hanoi, the Hard Way [Java]
  815. Train on Bridge [JavaScript]
  816. Tran Hoang Nam's Cyclic Inequality in Three Variables
  817. Transpositions
  818. Traveling Salesman Problem [Java]
  819. Treatise on Arithmetical Triangle
  820. Triangular numbers that are also square
  821. Trilateral Diagrams [Java]
  822. Trinary Tree(s) underlying Primitive Pythagorean Triples
  823. Tromino Puzzle: Deficient Squares [Java]
  824. Twenty five boys and twenty five girls around a table.
  825. Twin Inequalities in Four Variables: Twin 1
  826. Twin Inequalities in Four Variables: Twin 2
  827. Two Circles and a Limit [Java]
  828. Two Colors in Two Rows [Java]
  829. Two Cyclic Inequalities [JavaScript]
  830. Two Equations in Three Variables [JavaScript]
  831. Two Men of Tibet [Java]
  832. Two-Parameter Solutions to Three Almost Fermat Equations
  833. Two Problems from a Serbian 7 Grade Math Competition
  834. Two Products: Constraint and Inequality
  835. Two Sequences Defined by Recurrence
  836. Two-Sided Inequality - One Provenance
  837. Uncountability of the Reals - via a Game
  838. Undefined vs Indeterminate in Mathematics
  839. (Un)fair Division by Consensus
  840. Universal Coloring [Java]
  841. Universal Product Code [Java]
  842. Unproperty of the Powers of 2
  843. Unsolved Problem from Crux Solved
  844. USA 1975 Problem 3
  845. Vandermonde's Convolution Formula
  846. Vandermonde Determinant
  847. Venn Diagrams
  848. Viète's Formulas
  849. Voronoi Diagram [JavaSCript]
  850. Ways To Count
  851. Weierstrass Product Inequality
  852. Weierstrass Substitution
  853. What, how, and the Web [Java]
  854. What 1, 6, and 14 Have to Do with 5?
  855. What Is Algorithm?
  856. What Is Determinant?
  857. What Is Finite?
  858. What Is Infinite Descent?
  859. What Is Infinity?
  860. What Is Limit?
  861. What is so natural about "natural logarithms"?
  862. What Is Space Completion?
  863. WHEN THE COUNTING GETS TOUGH, THE TOUGH COUNT ON MATHEMATICS
  864. When Two Segments Intersect? [JavaScript]
  865. Where To Start? [Java]
  866. White and Black Balls in Urn. 1 in, 2 out. What Color Remains?
  867. Why 17?
  868. Why sin(x)/x tends to 1
  869. Why The Number of Primes Could Not Be Finite?
  870. Wilson's Theorem
  871. Wonderful Inequality on Unit Circle
  872. Word Problems [Java]
  873. Wu's Inequality
  874. $x_n=\sin 1+\sin 3+\sin 5+\cdots+\sin (2n-1)$
  875. Yet Another Puzzle with Hats
  876. Zeros and Nines [Java]

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