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. An Inequality with Constraint XX
  34. A Cyclic Inequality in Three Variables XX
  35. A Cyclic Inequality in Three Variables XXI
  36. A Cyclic Inequality in Three Variables XXII
  37. A Cyclic Inequality in Three Variables XXIII
  38. A Cyclic Inequality in Three Variables XXIV
  39. A Cyclic Inequality in Three Variables XXV
  40. A Cyclic Inequality in Three Variables And One More
  41. A Cyclic Inequality in Three Variables with Constraint
  42. A Cyclic Inequality in Three or More Variables
  43. A Cyclic Inequality of Degree Four
  44. A Cycling Inequality with Integrals
  45. A Cycling Inequality with Integrals II
  46. A Cyclic Inequality with Many Sums
  47. A Cyclic Inequality in Three Variables with a Variable Hierarchy
  48. A Diophantine Equation - Problem from 2000 Belarus MO
  49. A Diophantine Equation with Square Roots - Problem from the 2008 British MO
  50. A DIY Proof of the Irrationality of sqrt(2)
  51. A Formula for sin(3x)
  52. A Functional Equation from 1968 IMO
  53. A Functional Equation from the 2012 IMO
  54. A Generalization of an Inequality from a Romanian Olympiad
  55. A Limit of a Sequence
  56. A Limit of Another Sequence >li>A Limit with Fractions, Roots, Powers and Series
  57. A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
  58. A Mathematical Rabbit out of an Algebraic Hat
  59. A Middle School Problem from a Moscow Olympiad
  60. A New Proof That the Rationals Are Countable
  61. A Not Quite Cyclic Inequality
  62. A Not Symmetric Inequality [JavaScript]
  63. A Not So Simple Inequality in Three Variables
  64. A One-Line Proof of the Infinitude of Primes
  65. A Partly Cyclic Inequality in Four Variables
  66. A Problem for the 2015 New Year's Eve
  67. A Problem From a Mongolian Olympiad for Grade 11
  68. A Problem from Croatian TST
  69. A Problem from the Danubius Contest 2016
  70. A Problem from the Danubius-XI Contest
  71. A Problem From the Romanian Mathematical Competitions (2016)
  72. A Problem in Complex Numbers Sets 3 Aside
  73. A Problem with a Magical Solution from Secrets in Inequalities
  74. A Proof by Game for a Sum of a Convergent Series
  75. A Property of Number 4
  76. A Property of the fusc() Function
  77. A Refinement of Turkevich's Inequality
  78. A Representation of Rational Numbers
  79. A Quartic Equation
  80. A Series with k-Permutations
  81. A Simple Cyclic Inequality in Three Variables
  82. A Simple Inequality in Three Variables
  83. A simple integral
  84. A simple integral, III
  85. A Sum by Dorin Marghidanu
  86. A Sum of a Double Series
  87. A System of Equations Begging for Generalization
  88. A System of Equations in Determinants
  89. A System of Two Equations Replete with Squares
  90. A Telescoping Series
  91. A Tricky Integral Inequality
  92. A Trigonometric Inequality from the RMM
  93. A Triple Integral Inequality
  94. ab + bc + ca ≤ aa + bb + cc
  95. About numbers
  96. Absolute Value
  97. Absolute Value and 1998 integers
  98. Achieve the Equilibrium [Flash]
  99. Addition
  100. Addition and Subtraction Formulas for Sine and Cosine III
  101. Addition and Subtraction Formulas for Sine and Cosine IV
  102. Advancing a Millenium Problem [JavaScript]
  103. Affirmative Action Problem [Java]
  104. Algebraic-Geometric Inequality
  105. Algebraic Structure of Complex Numbers
  106. All Integers Are Equal to 1
  107. All Integers Are Even
  108. All Powers of 2 Are Equal to 1
  109. All Powers of x are Constant
  110. Almost every integer has the digit 3 in it
  111. Almost Integers - Gaussian Integers
  112. Amazing Mix of Triangular Numbers and Golden Ratio
  113. Ambassadors at a Round Table [Java]
  114. An Acyclic Inequality in Three Variables
  115. An Algebraic Lemma with Geometric Consequences [JavaScript]
  116. An Application of Quadratic Formula
  117. An Arctan Identity and its Consequence [JavaScript, GeoGebra]
  118. An Easy Inequality with Three Integrals
  119. An Elementary Inequality by Non-elementary Means
  120. An Elementary Proof for Euler's Series
  121. An Equation in Determinants
  122. An Equation in Factorials
  123. An Equation in Matrices [JavaScript]
  124. An Equation in Radicals
  125. An Equation in Reciprocals
  126. An Equation in RxR from Romania
  127. An Equation in Squares [JavaScript]
  128. An Equation Involving Four Collinear Points
  129. An Exercise without Calculating Aids
  130. An Identity between the Roots of a Cubic Polynomial - Problem from the 2007 Irish MO
  131. An Identity From the 2010 China Mathematical Competition
  132. An Identity Involving the Integer Part - Problem from the 1981 Canadian MO
  133. An Increasing Sequence from 2014 IMO
  134. An Inequality and Its Modifications
  135. An Inequality by Uncommon Induction
  136. An Inequality for Grade 8
  137. An Inequality from the 1967 IMO Shortlist
  138. An inequality from 1971 Canadian Mathematical Olympiad
  139. An Inequality from Marocco [JavaScript]
  140. An Inequality from Tibet
  141. An Inequality in Determinants
  142. An Inequality in Four Weighted Variables
  143. An Inequality in Fractions with Absolute Values
  144. An Inequality in Integers [JavaScript]
  145. An Inequality in Integers II
  146. An Inequality in Parallelogram of Unit Area
  147. An Inequality in Reciprocals
  148. An Inequality in Reciprocals II An Inequality in Reciprocals III
  149. An Inequality in Three (Or Is It Two) Variables
  150. An Inequality in Triangle and In General
  151. An Inequality In Triangle and Without
  152. An Inequality In Triangle and Without II
  153. An Inequality in Two Or More Variables
  154. An Inequality in Two Or More Variables II
  155. An Inequality Solved by Changing Appearances
  156. An Inequality with Angles and Integers
  157. An Inequality with Arbitrary Roots
  158. An Inequality with Areas, Norms, and Complex Numbers
  159. An Inequality with Complex Numbers of Unit Length
  160. An Inequality with Complex Numbers of Unit Length II
  161. An Inequality with Cyclic Sums And Products
  162. An Inequality with Cyclic Sums on Both Sides
  163. An Inequality with Cyclic Sums on Both Sides II
  164. An Inequality with Cyclic Sums on Both Sides III
  165. An Inequality with Fractions
  166. An Inequality with Integrals and Radicals
  167. An Inequality with Just Two Variable And an Integer
  168. An Inequality with Sines But Not in a Triangle
  169. An Inequality with One Tangent and Six Sines
  170. An Inequality with Powers And Logarithm
  171. An Inequality With Six Variables and Constraints
  172. An Inequality Where One Term Is More Equal Than Others
  173. An Inequality with a Parameter and a Constraint
  174. An Inequality with Absolute Values
  175. An Inequality with an Infinite Series
  176. An Inequality with Constraint
  177. An Inequality with Constraints II
  178. An Inequality with Constraint III
  179. An Inequality with Constraint IV
  180. An Inequality with Constraint V
  181. An Inequality with Constraint VI
  182. An Inequality with Constraint VII
  183. An Inequality with Constraint VIII [JavaScript]
  184. An Inequality with Constraint IX [JavaScript]
  185. An Inequality with Constraint X [JavaScript]
  186. An Inequality with Constraint XI [JavaScript]
  187. An Inequality with Constraint XII
  188. An Inequality with Constraint XIII
  189. An Inequality with Constraint XIV
  190. Inequalities with Constraint XV and XVI
  191. An Inequality with Constraint XVII
  192. An Inequality with Constraint XVIII
  193. An Inequality with Constraint XIX
  194. An Inequality with Constraint in Four Variables
  195. An Inequality with Constraint in Four Variables II
  196. An Inequality with Constraint in Four Variables III
  197. An Inequality with Constraint in Four Variables IV
  198. An Inequality with Constraint in Four Variables V
  199. An Inequality with Cot, Cos, and Sin
  200. An Inequality with Cycling Sums
  201. An Inequality with Determinants [JavaScript]
  202. An Inequality with Determinants II
  203. An Inequality with Determinants III
  204. An Inequality with Determinants IV
  205. An Inequality with Determinants V
  206. An Inequality with Determinants VI
  207. An Inequality with Determinants VII
  208. An Inequality with Exponents [JavaScript]
  209. An Inequality with a Generic Proof [JavaScript]
  210. An Inequality with Nested Radicals
  211. An Inequality via Young's Theorem
  212. An Inequality with Just Two Variable
  213. An Inequality with Just Two Variable II
  214. An Inequality with Just Two Variable III
  215. An Inequality with Just Two Variables IV
  216. An Inequality with Just Two Variables V
  217. An Inequality with Just Two Variables VI
  218. An Inequality with Just Two Variable VII
  219. An Inequality with Just Two Variable VIII
  220. An Inequality with Permutations, I [JavaScript]
  221. An Inequality with Permutations, II [JavaScript]
  222. An Inequality with Two Cyclic Sums
  223. An Inequality with Two Pairs of Triplets
  224. Inequality with Two Variables? Think Again
  225. An Infinite Product from Recursion
  226. An Integral from the RMM
  227. An Integral Inequality from the RMM
  228. An Old Inequality
  229. An Unsolvable Riccati Equation
  230. An Unusual Quadratic Equation
  231. Another Face and Proof of a Trigonometric Identity
  232. Another Inequality with Logarithms, But Not Really
  233. Another Integral Inequality from the RMM
  234. Another Problem from the 2016 Danubius Contest
  235. Another simple integral
  236. Any Integer with Three 2s
  237. Application of Cauchy-Schwarz Inequality
  238. Applications of Schweitzer's Inequality
  239. Approximation Of Roots
  240. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
  241. Areas on the Graphs of Power Functions
  242. Arithmetic-Geometric Means Inequality
  243. Another Property of Number 4
  244. Ascending Bases and Exponents in Pascal's Triangle
  245. Assigning Numbers to Points in the Plane
  246. Associativity from 1977 IMO
  247. Averages, Arithmetic and Harmonic Means [Java]
  248. Averages in a sequence
  249. Ballot Lemma [Java]
  250. Banzhaf Power Index Calculator [Java]
  251. Barcode Encoding
  252. Barcode Magic [Java]
  253. Base Converter [JavaScript]
  254. Bergström's inequality
  255. Birth of an Inequality
  256. Bayes' Theorem
  257. Beatty Sequences
  258. Beauty and the Beast - in Trigonometry
  259. Bertrand's Postulate
  260. Bicubal Domino [JavaScript]
  261. Bilateral Diagrams [Java]
  262. Binet's Formula via Generating Functions
  263. Binet's Formula with Cosines
  264. Binomial Distribution [Java]
  265. Binomial Theorem
  266. Bisector of an imaginary angle may be real
  267. Brahmagupta-Fibonacci Identity [JavaScript]
  268. Byzantine Basketball
  269. Calculation of the Digits of pi by the Spigot Algorithm [Java]
  270. Calculus Proof of the Pythagorean Theorem
  271. Candido's Identity
  272. Cantor-Bernstein-Schroeder theorem
  273. Cantor's First Uncountability Proof
  274. Cardinal Numbers
  275. Cartesian Coordinate System [Java]
  276. Cassini's Identity
  277. Catalan Numbers in Pascal's Triangle
  278. Cavalieri's Principle [Java]
  279. Central and Inscribed Angles in Complex Numbers
  280. Chebyshev Inequality
  281. Chebyshev Holds a Key
  282. Chinese Remainder Theorem
  283. Clifford Algebras
  284. Clubs in a Vector Space
  285. Collatz Conjecture [Java]
  286. Coloring a Graph [Java]
  287. Combinatorial Proofs
  288. Combo Playground: Calculating the Number of Paths on a Square Grid
  289. Committee Chairs
  290. Comparing complex numbers
  291. Completing a Square and Quadratic Formula
  292. Completing the Square Where It Matters
  293. Complex Numbers: A Dynamic Tool [Java]
  294. Complex number to a complex power may be real
  295. Complicated Constraint - Simple Inequality
  296. Composition of Functions, an Exercise Constrained Inequality with Reciprocals
  297. Constrained Intermarriages [JavaScript]
  298. Constructible Numbers
  299. Construction of Pascal's Triangle [Java]
  300. Constructive Existence of the de Bruijn Cycles
  301. Continued fractions
  302. cos/sin inequality
  303. cos(π/7)-cos(2π/7)+cos(3π/7) = 1/2
  304. Cospherical Points
  305. Countability of Rational Numbers
  306. Countability Principle
  307. Counting Chips On a Circle [Java]
  308. Counting Fat Sets
  309. Counting Squares in a Square [Java]
  310. Counting Triangles [Java]
  311. Counting Triangles II [Java]
  312. Covering a Chessboard with a Hole with L-Trominoes [Java]
  313. Covering A Chessboard With Domino
  314. Cubes Constrained
  315. Cubes in Pascal's Triangle
  316. Cubic Equation for the Sides of a Triangle
  317. Cubic Formula [JavaScript]
  318. Cubics and Bézout's Theorem
  319. Cubic Optimization with Linear Constraints
  320. Cubic Optimization with Partly Linear Constraints
  321. Curious Identities In Pythagorean Triangles
  322. Curious Identities Involving Integer Products
  323. Cyclic Inequality in Three Variables by Marian Cucoanes
  324. Cyclic Inequality with Arctangents
  325. Cyclic Inequality with Logarithms
  326. Cyclic Inequality with Square Roots
  327. Cyclic Inequality with Square Roots And Absolute Values
  328. Dan Sitaru's Amazing, Never Ending Inequality
  329. Dan Sitaru's Cyclic Inequality In One Variable
  330. Dan Sitaru's Cyclic Inequality In Many Variables
  331. Dan Sitaru's Cyclic Inequality In Three Variables
  332. Dan Sitaru's Cyclic Inequality in Three Variables II
  333. Dan Sitaru's Cyclic Inequality in Three Variables III
  334. Dan Sitaru's Cyclic Inequality in Three Variables IV
  335. Dan Sitaru's Cyclic Inequality in Three Variables V
  336. Dan Sitaru's Cyclic Inequality in Three Variables VI
  337. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints
  338. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints IV
  339. Dan Sitaru's Exercise with Pi and Ln
  340. Dan Sitaru's Inequality: From Three Variables to Many in Two Ways
  341. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints II
  342. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints III
  343. Dan Sitaru's Inequality by Induction
  344. Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots
  345. Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots II
  346. Dan Sitaru's Inequality with Roots and Powers
  347. Dan Sitaru's Inequality with Tangents
  348. Dan Sitaru's Inequality with Three Related Integrals and Derivatives
  349. Dan Sitaru's Integral Inequality with Powers of a Function
  350. Daniel Dan's Optimization in Three Variables
  351. Darij Grinberg's Inequality In Three Variables
  352. Deceptive Appearances
  353. Decimal Sums of Successive Integers
  354. Derivatives without Limits
  355. Descartes' Rule of Signs
  356. Determinant and Divisibility
  357. Determinant on a Circle
  358. Diagonal Argument
  359. Diophantine Equations
  360. Distance between the Orthocenter and Circumcenter
  361. Distance Between Strings [Java]
  362. Distance From a Point to a Straight Line [Java]
  363. Distance Inequality
  364. Distances to Three Points on a Circle
  365. Distinct Multisets with the Same Pairwise Sum
  366. Divergence of {cos(n)} [JavaScript]
  367. Divergence of the Harmonic Series (Details)
  368. Divide And Conquer in Cyclic Sums
  369. Dividing Loot Fairly
  370. Divisibility and Modulo Arithmetic at IMO 2007
  371. Divisibility by Powers of 2
  372. Dollars in Pockets
  373. Doodling on April 1st
  374. Dorin Marghidanu's Believe It Or Not
  375. Dorin Marghidanu's Cyclic Inequality in Three Variables
  376. Dorin Marghidanu's Cyclic Inequality in Three Variables II
  377. Dorin Marghidanu's Cyclic Inequality with Constraint
  378. Dorin Marghidanu's Inequality in Complex Plane
  379. Dorin Marghidanu's Inequality in Integer Variables
  380. Dorin Marghidanu's Inequality in Many Variables
  381. Dorin Marghidanu's Inequality in Many Variables Plus Two More
  382. Dorin Marghidanu's Inequality with Constraint
  383. Dorin Marghidanu's Inequality with Generalization
  384. Dorin Marghidanu's Inequality with Radicals
  385. Dorin Marghidanu's Light Elegance in Four Variables
  386. Dorin Marghidanu's Spanish Problem
  387. Dorin Marghidanu's Tribute to Descartes
  388. Dot Patterns and Sierpinski Gasket [Java]
  389. Dots and fractions [Java]
  390. Double and Half Angle Formulas
  391. Ducci Sequences [Java]
  392. e in the Pascal Triangle
  393. EAN-13 Encoding [Java]
  394. Eigenvectors by Inspection
  395. Elves and Gnomes [JavaScript]
  396. Entry 1.414... in Ramanujan's Notebooks
  397. Equation in Permutations
  398. Equation in Radicals as a System of Equations
  399. Equations of the Staight Line [Java]
  400. Equivalence Relations
  401. Escape of the Clones
  402. Euclid's Algorithm
  403. Euclid's Algorithm: An Interactive Illustration [Java]
  404. Euler Cycles in Digraph
  405. Euler's formula for e
  406. Euler's Function and Congruence Theorem
  407. Evolution of Algebraic Symbolism
  408. Exercise with Square Spiral [GeoGebra]
  409. Existence of the de Bruijn Cycles via Graphs
  410. Exponential Inequalities for Means
  411. Extension of Euclid's Algorithm
  412. Extrema of x^3+y^3+z^3-xyz on the unit sphere
  413. Factorial Products
  414. Factorials and division
  415. Fair Division: Method of Sealed Bids II [Java]
  416. Falsity implies anything
  417. Farey Series
  418. Fermat's Like Equation
  419. Fermat's Little Theorem
  420. Fermat's Only Published Proof
  421. Fibonacci Idendtities with Matrices
  422. Fibonacci's Quickies
  423. Fibonacci Tilings
  424. Filling a Grid with Good Neighbors [Java]
  425. Filling an Array with 0s and 1s - and Counting [Java]
  426. Filling Pool with Fluids
  427. Find Minimum without Calculus
  428. Finding Hamilton Paths and Circuits [Java]
  429. Finite Sums of Terms 2^(n-i) i²
  430. First 1008 Digits
  431. First Geometric Interpretation of Complex Numbers [Java]
  432. Five Methods of Apportionment [Java]
  433. Five Lattice Points [Java]
  434. Five Squares in Complex Numbers [GeoGebra]
  435. Fleury's Algorithm and Euler's Paths and Cycles [Java]
  436. Flipping Items Simultaneously [Java]
  437. Flipping Items Simultaneously II [Java]
  438. Flipping Items Simultaneously III [Java]
  439. Flipping pancakes [Java]
  440. Floor Function
  441. Formal Systems - an Example
  442. Formula for Primes
  443. Four Hinged Squares: using complex numbers
  444. Four Integrals in One Inequality
  445. Four Numbers, Six Differences, GCD of the Products
  446. Four Pegs That Form a Square
  447. Four Voting Methods [Java]
  448. From The Sum of Angles To The Sum of Sines
  449. Function
  450. Function in the Plane That Vanishes
  451. Fundamental Theorem of Algebra
  452. Futurama Theorem and Puzzle [Java]
  453. Gasoline Stations on a Circular Trek
  454. Gauss and Euler Integrals
  455. Gauss' Lemma
  456. Gauss' theorem on constructibility of regular polygons
  457. GCD of Fibonacci Numbers
  458. GCD(M, N) × LCM(M, N) = M × N [Java]
  459. Generalization That Solves the Problem
  460. Generating Functions
  461. Geometric Construction of Roots of Quadratic Equation [Java]
  462. Getting a Scout out of Desert [Java]
  463. Getting Your Rightful Share Back [JavaScript]
  464. Gheorghe Stoica's Problem from Gazeta Matematica Gireaux's Theorem
  465. Gödel's Incompleteness Theorem
  466. Golden Ratio is Irrational
  467. Graph and Roots of a Third Degree Polynomial [Java]
  468. Graph and Roots of Quadratic Polynomial [Java]
  469. Graph of a Polynomial [Java]
  470. Graph of a Polynomial Defined by Its Roots [Java]
  471. Graph with Nodes of Even Degree
  472. Graph without 3-Cycles
  473. Graphing Equations Is Useful
  474. Graphing Equations Is Useful, II [Java]
  475. Graphing Equations Is Useful, III
  476. Graphing Equations Is Useful, IV
  477. Graphing Equations Is Useful, V
  478. Greatest Difference in Arithmetic Progression
  479. Group multiplication of permutations [Java]
  480. 1/2 = 1
  481. Harmonic Serires And Its Parts
  482. 2006 Hellenic Mathematical Olympiad, Problem 4 [JavaScript]
  483. Hlawka's Inequality
  484. Horner's Method [Java]
  485. How Jost Burgi Computed the Sines of All Integer Angles in 1586
  486. How to Ask an Embarrassing Question
  487. How to define a permutation? [Java]
  488. How to write an equation of the union of two sets
  489. However You Solve It ... A Wonderful Equation
  490. Hung Viet's Inequality
  491. Hung Viet's Inequality II
  492. Hung Viet's Inequality III
  493. Hung Viet's Inequality IV
  494. Hyperintegers and Hyperreal Numbers
  495. Identities with Triangular Numbers
  496. Imad Zak's Cyclic Inequality in Three Variables
  497. Imad Zak's Cyclic Inequality in Three Variables II
  498. Implementation of Base Conversion
  499. Importance of the Absolute Value
  500. Improving an Inequality [JavaScript]
  501. Improving on an Escalator [JavaScript]
  502. In Calculus too 1 = 0
  503. Inclusion-Exclusion Principle
  504. Inclusion-Exclusion Principle: an Example
  505. Inequalities and Quadratic Forms
  506. An Inequality for Mixed Means
  507. An Inequality from RMM with a Generic 5
  508. An Inequality from RMM with Powers of 2
  509. Inequality 101 from the Cyclic Inequalities Marathon
  510. Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
  511. Inequality (1 + 1-3)(1 + 2-3)(1 + 2-3)...(1 + n-3) < 3
  512. Inequality 1 + 2-2 + 3-2 + 4-2 + ... + n-2 < 2.
  513. Inequality by Calculus
  514. Inequality for Integer Sequences
  515. Inequality for Integer Sequences II
  516. Inequality for Mixed Means
  517. An Inequality from Gazeta Matematica, March 2016
  518. An Inequality from Gazeta Matematica, March 2016 II
  519. An Inequality from Gazeta Matematica, March 2016 III
  520. An Inequality from Gazeta Matematica, March 2016 IV
  521. Inequalities with Double And Triple Integrals
  522. Inequality from Math Magazine 2009
  523. Inequality from Math Phenomenon
  524. Inequality in Acute Triangle
  525. Inequality of a Series of Square Roots
  526. Inequality with Constraint from Dan Sitaru's Math Phenomenon
  527. Inequality with Csc And Sin
  528. An Inequality with Exponents [JavaScript]
  529. An Inequality with Finite Sums
  530. An Inequality with Integrals and Rearrangement
  531. Inequality with Logarithms
  532. Inequality with Powers And Radicals
  533. Inequality with Three Linear Constraints
  534. Inequality with Three Numbers, Not All Zero
  535. Inequality with Two Minima
  536. Infinite Discent for x4 + y4 = z4
  537. Infinitesimals. Non-standard Analysis
  538. Infinitude of Primes
  539. Infinity As a Limit
  540. Inflection Points of Fourth Degree Polynomials [Java]
  541. An innovative recurrence
  542. Integer Iterations on a Circle [Java]
  543. Candy Game (Change Discharged) [Java]
  544. Integer Iterations on a Circle III [Java]
  545. Integer roots of integers are irrational
  546. Integer Triangle That Sums to Cube
  547. Integers and Rectangles [Java]
  548. Integers and Rectangles: a Proof by Induction [Java]
  549. Integers and Rectangles: Two Simple Proofs [Java]
  550. Integers With Digits 1 and 2 and Residues Modulo Powers of 2
  551. Integral in Normal Distribution
  552. Integral Is Area
  553. Integrals of a Function And Its Inverse
  554. Interest Calculations [Java]
  555. Integral of a Piece-wise Function
  556. Irrationality of k-th roots
  557. Irrationality of sqrt(2) via a Difference Equation
  558. Is GLOBALHELLFRY a Prime? [JavaScript]
  559. Is the Triangle Inequality Necessary?
  560. Is There a Limit?
  561. ISBN Encoding [Java]
  562. Irrational number to an irrational power may be rational
  563. Is 0.999... = 1?
  564. It's All About Powers of 10
  565. Iterations on Monotone Functions
  566. Jensen's Inequality [JavaScript]
  567. Joachimsthal's Notations
  568. Jumping Isn't Everything [Java]
  569. Kaprekar's Iterations and Numbers [Java]
  570. Kulikowski's Theorem
  571. Lagrange's identity, Proof
  572. Lagrange interpolation
  573. Lamé's Theorem - First Application of Fibonacci Numbers
  574. Langford and Skolem Sequences [Java]
  575. Latin Squares [Java]
  576. Le Khanh Sy's Problem
  577. Leech Trees and Taylor's Criterion
  578. Legendre's Formula
  579. Legendre's Theorem - The Prime Factorization of Factorials
  580. Leibniz and Pascal Triangles [Java]
  581. Leo Giugiuc's Cyclic Inequality in Three Variables
  582. Leo Giugiuc's Exercise
  583. Leo Giugiuc's Inequality with Constraint
  584. Leo Giugiuc's Optimization with Constraint
  585. Leo Giugiuc's Trigonometric Lemma
  586. Less than, Equal to, Greater Than Symbols
  587. Lewis Carroll's Game of Logic [Java]
  588. Light Bulbs in a Circle [Java]
  589. Lights Out [Java]
  590. Limit of a Recursive Sequence
  591. Linear Algebra Tools for Proving Inequalities
  592. Linear Algebra Tools for Proving Inequalities: Cauchy-Binet Formula
  593. Linear Function with Coefficients in Arithmetic Progression [GeoGebra]
  594. Linear functions
  595. Linear Programming
  596. Linear System with Parameter
  597. Listing All Combinations [Java]
  598. Listing all permutations [Java]
  599. Listing All Permutations: Johnson-Trotter Method [Java]
  600. Logistic Model applet [Java]
  601. Long Runs of Composite Numbers
  602. Long Up Short Down
  603. Looking your best
  604. Loop or Halt [Java]
  605. Lucas' Theorem
  606. Lucas' Theorem II
  607. Lucky Numbers [Java]
  608. Marian Dinca's Refinement of Nesbitt's Inequality
  609. Marriage Theorem
  610. Marriage Problem, simulation [Java]
  611. Math Lotto
  612. Mathematical Induction
  613. Mathematical Proof
  614. Mathematicians Doze Off
  615. Matrices and Determinants as Optimization Tools: an Example
  616. Matrix Groups
  617. Maximum Volume of a Cut Off Box [Java]
  618. Means [Java]
  619. Mediant Fractions
  620. Method of Markers II [Java]
  621. Michael Rozenberg's Inequality in Three Variables with Constraints
  622. Michael Rozenberg's Inequality in Two Variables
  623. Minesweeper Theorem
  624. Mini Lights Out [Java]
  625. Minimax Principle [Java]
  626. Minimax Theorem [Java]
  627. Minimum of a Cyclic Sum with Logarithms
  628. Minumum Spanning Trees and Kruskal's Algorithm [Java]
  629. Minimum under Two Constraints
  630. Minus One But What a Difference
  631. Modular Arithmetic [Java]
  632. Monotone Convergence Theorem
  633. Monotone Number Sequences [Java]
  634. Monthly Problem 11199
  635. Much from Little
  636. Muirhead's Inequality
  637. Multiplication
  638. Multiplication of Points on a Circle [JavaScript, GeoGebra]
  639. Multiplication of Points on an Ellipse [JavaScript, GeoGebra]
  640. Napoleon's Relatives [Java]
  641. Negative on the left, positive on the right: how come?
  642. Nested Radicals
  643. Nested Subsets
  644. New Proof of Euclid's Theorem
  645. Newton's Method [Java]
  646. Newton's and Maclaurin's Inequalities [JavaScript]
  647. Nine Squares [Java]
  648. No Flipping without End [Java]
  649. NonSquare Matrix as a Tool for Proving an Inequality
  650. Number 8 Is Interesting
  651. Number Comparison - from the 1969 Canadian MO
  652. Number sequences and Pigeonhole [Java]
  653. Number System With Base 36
  654. Modern Vietnamese Classic
  655. On Dedekind Cuts
  656. On the Games of Hex and Y
  657. On the Intersection of kx and |sin(x)| [JavaScript]
  658. On the Intersection of 1+x and e^x and Trigonometric Consequences [JavaScript]
  659. One Dimensional Ants
  660. One Inequality - Two Domains
  661. One Trigonometric Formula and Its Consequences
  662. Optimization in Four Variables with Two Constraints
  663. Optimization with Many Variables
  664. Order. Well-ordered sets.
  665. Order and Chaos in Multiple Pendulums
  666. Ordinal Numbers
  667. Orthocenter in Complex Plane
  668. Outline Mathematics
  669. p-adic Expansions
  670. p-adic Numbers
  671. Pairing Monotone and Convex Functions
  672. Parabola and Abutting Equilateral Triangles
  673. Parabolic Sieve of Prime Numbers
  674. Parity counting [Java]
  675. Parrondo Paradox Via Simpson Paradox [Java]
  676. Partial But General Case Of Fermat's Last Theorem
  677. Party Acquaintances [Java]
  678. Patterns in Pascal's Triangle
  679. Peano axioms
  680. Pennies in Boxes [Java]
  681. Percent
  682. Permutations
  683. Perrin Sequence
  684. Pi in Pascal's Triangle
  685. Pi in Pascal's Triangle via Triangular Numbers
  686. pi^e vs e^pi [JavaScript]
  687. Pigeonhole Principle
  688. Pinocchio as Epimenides
  689. Place Value [Java]
  690. Powers and Fractions Inequality
  691. Presence of the Antipodes on a Circle
  692. Prime Sums in Pairs
  693. Probability of Divisibility
  694. Problem 1 from the IMO 2009
  695. Problem 11804 from the AMM
  696. Problem A1, 69th Putnam Competition
  697. Problem A2, 69th Putnam Competition
  698. Problem of Divisibility by 5n
  699. Product of Diagonals in Regular N-gon
  700. Product Of Four Consecutive Integers
  701. Product of Simple Fractions [Java]
  702. Product of permutations [Java]
  703. Proizvolov's Identity [Java]
  704. Proofs Without Words [Java]
  705. Property of an
  706. Property of Quadratic Polynomials
  707. Property of the Powers of 2 [Java]
  708. Puzzlist vs. Mathematician: A Practical Arithmetic Problem
  709. Pythagorean Triples [Java]
  710. Quadratic Function for Solving Inequalities
  711. An Inequality Where One Term Is More Equal Than Others
  712. Quasigroups and block designs from triangular grid [Java]
  713. Rabbits Reproduce; Integers Don't
  714. Radicals, Radicals, And More Radicals in an Inequality
  715. Radon's Inequality and Applications
  716. Ramsey's Number R(3, 3, 3)
  717. Ramsey Number R(4, 3)
  718. Ramsey Number R(5, 3)
  719. Ramsey Number R(4, 4)
  720. Ramsey's Theorem
  721. Random Walks [Java]
  722. Rascal Triangle
  723. Rational Solutions to xy = yx
  724. Rearrangement Inequality [JavaScript]
  725. Recombining Permutations [Java]
  726. Rectangle on a Chessboard [Java]
  727. Rectilinear Paradox
  728. Reflection Lemma [Java]
  729. Riemann Rearrangement Theorem
  730. Riemann Sphere and Möbius Transformation
  731. Riemann Sums - Function Integration [Java]
  732. Right Replacement [Java]
  733. Rolle's and Mean Value Theorems [Java]
  734. Rolle's Theorem in Action
  735. RomanoNorwegian Inequality
  736. Rooks on a Chessboard
  737. Roots and Tangents [GeoGebra]
  738. Round Robin Tournament
  739. Russell's Paradox
  740. Solve A Fourth Degree Equation
  741. Sophie Germain's Identity
  742. Square Roots and Triangle Inequality
  743. Squares in Pascal's Triangle
  744. Squeeze Theorem
  745. Sum of Absolute Values [JavaScript]
  746. Sum of All Natural Numbers
  747. Sums of Binomial Reciprocals in Pascal's Triangle
  748. Squeeze (Sandwich, Pinching ...) Theorem (Lemma): an Example
  749. Scalar Product Optimization [JavaScript]
  750. Scheduling and Critical Path Algorithm [Java]
  751. Schur's Inequality
  752. Scrub Tile Puzzle [Java]
  753. Self-descriptive Strings [Java]
  754. Self-documenting Iterations [JavaScript]
  755. Self-reference
  756. Seven integers under 127 and their Ratios
  757. Seven Terms Periodic Sequence
  758. Shapley-Shubik Power Index [Java]
  759. Shapley-Shubik Power Index Calculator [Java]
  760. Shifting Digits and a Point of View
  761. Ship-Dock Theorem
  762. Shuttle Puzzle Practice II [Java]
  763. Sicherman Dice
  764. Sierpinski Gasket and Tower of Hanoi
  765. Sierpinski's Gasket and Dihedral Symmetry [Java]
  766. Sieve of Squares [Java]
  767. Σ2-n = Σn·2-n
  768. Signs and Sums in a Sequence [Java]
  769. Signs in a Matrix [Java]
  770. Simple But Uncommon Equation from 1968 IMO
  771. Simple Cellular Automaton [Java]
  772. Simple Graph Practice [Java]
  773. Simple Inequality with a Variety of Solutions
  774. Simple Inequality with Many Faces And Variables
  775. Simple Somos Sequence - a Russian Olympiad problem
  776. Simple Somos Sequence Calculator [Java]
  777. Simple Yet Uncommon Inequalities with Absolute Value
  778. Simpson Paradox [Java]
  779. $\sin 1^{\circ}+\sin {2^\circ}+\sin 3^{\circ}+\cdots+\sin 180^{\circ}=\tan 89.5^{\circ}$
  780. Sine And Cosine Are Continuous Functions
  781. Sine of a Sum Formula, Proof Without Words
  782. Single formula to express any integer with digits 0-9 each used just once
  783. Sitaru-Schweitzer Inequality
  784. Six Balls, Two Weighings
  785. Size of a Class: Two Viewpoints
  786. Sladjan Stankovik's Inequality In Four Variables
  787. Sladjan Stankovik's Inequality With Constraint
  788. Sladjan Stankovik's Inequality With Constraint II
  789. Slider Puzzles [Java]
  790. Solitaire in Two Colors [Java]
  791. Solutions of F(x)=x and F(F(x))=x [JavaScript]
  792. Solve - an Olympiad Problem - in Your Mind
  793. Solve xx3 = 3
  794. Solve cos²x + cos²2x + cos²3x = 1
  795. Solving an Equation with Inverse Functions
  796. Soriteses
  797. Sorting Algorithms
  798. Sperner's Theorem
  799. Splitting piles
  800. Square Root of 2 as a Continued Fraction
  801. Square root of 2 is irrational
  802. Star of David Theorems in Pascal Triangle
  803. Steinhaus' Problem [Java]
  804. Step into the Elliptic Realm
  805. Stern-Brocot Tree
  806. Straight Tromino on a Chessboard [Java]
  807. Structure of Hyperreal Numbers
  808. Subsets and Intersections
  809. Subtraction Game [Java]
  810. Sums of the Elements of Three Element Subsets [JavaScript]
  811. Sum of Three Squares [JavaScript]
  812. Sums and Products, a Generalization
  813. Sums, Products, and 1-1 Functions
  814. Sums of Geometric Series - Proofs Without Words [Java]
  815. Sums of Power Series
  816. Sums Of Successive Squares
  817. Sum without Adding
  818. Sums and Products [Java]
  819. Sums of powers of digits [Java]
  820. Superposition Principle
  821. Surreal Numbers
  822. Swapping Rows and Columns [Java]
  823. Sylvester's Problem
  824. Taking Games Seriously [Java]
  825. Tangential Chaos
  826. Telescoping Cubes
  827. Telescoping Squares
  828. Telescoping Sums and Series [JavaScript]
  829. Telescoping Tangents
  830. The ass and the mule [Java]
  831. The Basic Rules of Counting
  832. The Local-Global Principle
  833. The Regula Falsi - Iterated
  834. The Roads We Take
  835. The Size is in the Eyes of the Beholder
  836. Three Dimensional Cross
  837. Three Junior Problems from Vietnam
  838. Tool for Constructing Orthogonal Latin Squares [Java]
  839. Tower of Hanoi, the Hard Way [Java]
  840. Train on Bridge [JavaScript]
  841. Tran Hoang Nam's Cyclic Inequality in Three Variables
  842. Transpositions
  843. Traveling Salesman Problem [Java]
  844. Treatise on Arithmetical Triangle
  845. Triangular numbers that are also square
  846. Trilateral Diagrams [Java]
  847. Trinary Tree(s) underlying Primitive Pythagorean Triples
  848. Tromino Puzzle: Deficient Squares [Java]
  849. Twenty five boys and twenty five girls around a table.
  850. Twin Inequalities in Four Variables: Twin 1
  851. Twin Inequalities in Four Variables: Twin 2
  852. Two Circles and a Limit [Java]
  853. Two Colors in Two Rows [Java]
  854. Two Cyclic Inequalities [JavaScript]
  855. Two Equations in Three Variables [JavaScript]
  856. Two Men of Tibet [Java]
  857. Two-Parameter Solutions to Three Almost Fermat Equations
  858. Two Problems from a Serbian 7 Grade Math Competition
  859. Two Products: Constraint and Inequality
  860. Two Sequences Defined by Recurrence
  861. Two-Sided Inequality - One Provenance
  862. Two Solutions: One Correct, One Illuminating. An Example
  863. Uncountability of the Reals - via a Game
  864. Undefined vs Indeterminate in Mathematics
  865. (Un)fair Division by Consensus
  866. Universal Coloring [Java]
  867. Universal Product Code [Java]
  868. Unproperty of the Powers of 2
  869. Unsolved Problem from Crux Solved
  870. USA 1975 Problem 3
  871. Vandermonde's Convolution Formula
  872. Vandermonde Determinant
  873. Venn Diagrams
  874. Viète's Formulas
  875. Voronoi Diagram [JavaSCript]
  876. Ways To Count
  877. Weierstrass Product Inequality
  878. Weierstrass Substitution
  879. What, how, and the Web [Java]
  880. What 1, 6, and 14 Have to Do with 5?
  881. What Is Algorithm?
  882. What Is Determinant?
  883. What Is Finite?
  884. What Is Infinite Descent?
  885. What Is Infinity?
  886. What Is Limit?
  887. What is so natural about "natural logarithms"?
  888. What Is Space Completion?
  889. WHEN THE COUNTING GETS TOUGH, THE TOUGH COUNT ON MATHEMATICS
  890. When Two Segments Intersect? [JavaScript]
  891. Where To Start? [Java]
  892. White and Black Balls in Urn. 1 in, 2 out. What Color Remains?
  893. Why 17?
  894. Why sin(x)/x tends to 1
  895. Why The Number of Primes Could Not Be Finite?
  896. Wilson's Theorem
  897. Wonderful Inequality on Unit Circle
  898. Word Problems [Java]
  899. Wu's Inequality
  900. $x_n=\sin 1+\sin 3+\sin 5+\cdots+\sin (2n-1)$
  901. Yet Another Puzzle with Hats
  902. Zeros and Nines [Java]

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