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. 6th European Mathematical Cup (2017), Junior Problem 4
  10. A 49th Degree Challenge
  11. A Case for Determinants
  12. A Case of Divergence
  13. A Clique of Acquaintances
  14. A Curious Property of Numbers 3 and 5
  15. A Cyclic But Not Symmetric Inequality in Four Variables
  16. A Cyclic Inequality from Math Phenomenon
  17. A Cyclic Inequality in Three Variables
  18. A Cyclic Inequality in Three Variables II
  19. A Cyclic Inequality in Three Variables III
  20. A Cyclic Inequality in Three Variables IV
  21. A Cyclic Inequality in Three Variables V
  22. A Cyclic Inequality in Three Variables VI
  23. A Cyclic Inequality in Three Variables VII
  24. A Cyclic Inequality in Three Variables VIII
  25. A Cyclic Inequality in Three Variables IX
  26. A Cyclic Inequality in Three Variables X
  27. A Cyclic Inequality in Three Variables XI
  28. A Cyclic Inequality in Three Variables XII
  29. A Cyclic Inequality in Three Variables XIII
  30. A Cyclic Inequality in Three Variables XIV
  31. A Cyclic Inequality in Three Variables XV
  32. A Cyclic Inequality in Three Variables XVI
  33. A Cyclic Inequality in Three Variables XVII
  34. A Cyclic Inequality in Three Variables XVIII
  35. A Cyclic Inequality in Three Variables XIX
  36. An Inequality with Constraint XX
  37. A Cyclic Inequality in Three Variables XX
  38. A Cyclic Inequality in Three Variables XXI
  39. A Cyclic Inequality in Three Variables XXII
  40. A Cyclic Inequality in Three Variables XXIII
  41. A Cyclic Inequality in Three Variables XXIV
  42. A Cyclic Inequality in Three Variables XXV
  43. A Cyclic Inequality in Three Variables And One More
  44. >A Cyclic Inequality In Three Variables by Sorin Radulescu
  45. A Cyclic Inequality in Three Variables by Uche E. Okeke
  46. A Cyclic Inequality from India In Three Variables And More
  47. A Cyclic Inequality in Three Variables with Constraint
  48. A Cyclic Inequality in Three or More Variables
  49. A Cyclic Inequality of Degree Four
  50. A Cyclic Inequality With Constraint in Two Triples of Variables
  51. A Cycling Inequality with Integrals
  52. A Cycling Inequality with Integrals II
  53. A Cyclic Inequality with Many Sums
  54. A Cyclic Inequality with Powers 2 through 7
  55. A Cyclic Inequality in Three Variables with a Variable Hierarchy
  56. A Cyclic Inequality on [-1,1]
  57. A Diophantine Equation - Problem from 2000 Belarus MO
  58. A Diophantine Equation with Square Roots - Problem from the 2008 British MO
  59. A DIY Proof of the Irrationality of sqrt(2)
  60. A Formula for sin(3x)
  61. A Functional Equation from 1968 IMO
  62. A Functional Equation from the 2012 IMO
  63. A Generalization of an Inequality from a Romanian Olympiad
  64. A Limit of a Sequence
  65. A Limit of Another Sequence
  66. A Limit of a Series of Determinants
  67. A Limit with Fractions, Roots, Powers and Series
  68. A Limit With Sums, Roots, And Binomial Coefficients
  69. A Long Cyclic Inequality of Degree 4
  70. A Low Bound for 1/2 * 3/4 * 5/6 * ... * (2n-1)/2n
  71. A Mathematical Rabbit out of an Algebraic Hat
  72. A Memorable Fact
  73. A Middle School Problem from a Moscow Olympiad
  74. A New Proof That the Rationals Are Countable
  75. A Not Quite Cyclic Inequality
  76. A Not Quite Cyclic Inequality from Tibet
  77. A Not Symmetric Inequality [JavaScript]
  78. A Not So Simple Inequality in Three Variables
  79. A One-Line Proof of the Infinitude of Primes
  80. A Partly Cyclic Inequality in Four Variables
  81. A Problem for the 2015 New Year's Eve
  82. A Problem From a Mongolian Olympiad for Grade 11
  83. A Problem from Croatian TST
  84. A Problem from the Danubius Contest 2016
  85. A Problem from the Danubius-XI Contest
  86. A Problem From the Romanian Mathematical Competitions (2016)
  87. A Problem in Complex Numbers Sets 3 Aside
  88. A Problem in Division of Polynomials
  89. A Problem with a Magical Solution from Secrets in Inequalities
  90. A Proof by Game for a Sum of a Convergent Series
  91. A Property of Number 4 li>A Property of Product of Special Matrices, Probably Folklore
  92. A Property of the fusc() Function
  93. A Refinement of Turkevich's Inequality
  94. A Representation of Rational Numbers
  95. A Quartic Equation
  96. A Series with k-Permutations
  97. A Simple Cyclic Inequality in Three Variables
  98. A Simple Inequality in Three Variables
  99. A simple integral
  100. A simple integral, III
  101. A Simplified Version of Leo Giugiuc's Inequality from the AMM
  102. A Sum by Dorin Marghidanu
  103. A Sum of a Double Series
  104. A System of Equations Begging for Generalization
  105. A System of Equations in Determinants
  106. A System of Two Equations Replete with Squares
  107. A Telescoping Series
  108. A Tricky Integral Inequality
  109. A Trigonometric Inequality from the RMM
  110. >A Trigonometric Observation in Right Triangle
  111. A Triple Integral Inequality
  112. A Warmup Inequality from Vasile Cirtoaje
  113. ab + bc + ca ≤ aa + bb + cc
  114. About numbers
  115. Absolute Value
  116. Absolute Value and 1998 integers
  117. Achieve the Equilibrium [Flash]
  118. Addition
  119. Addition and Subtraction Formulas for Sine and Cosine III
  120. Addition and Subtraction Formulas for Sine and Cosine IV
  121. Advancing a Millenium Problem [JavaScript]
  122. Affirmative Action Problem [Java]
  123. Algebraic-Geometric Inequality
  124. Algebraic Structure of Complex Numbers
  125. All Integers Are Equal to 1
  126. All Integers Are Even
  127. All Powers of 2 Are Equal to 1
  128. All Powers of x are Constant
  129. Almost every integer has the digit 3 in it
  130. Almost Integers - Gaussian Integers
  131. Amazing Mix of Triangular Numbers and Golden Ratio
  132. Ambassadors at a Round Table [Java]
  133. An Acyclic Inequality in Three Variables
  134. An Algebraic Lemma with Geometric Consequences [JavaScript]
  135. An Application of Quadratic Formula
  136. An Arctan Identity and its Consequence [JavaScript, GeoGebra]
  137. An Easy Cyclic Inequality And a Remark
  138. An Easy Inequality with Three Integrals
  139. An Elementary Inequality by Non-elementary Means
  140. An Elementary Proof for Euler's Series
  141. An Equation in Determinants
  142. An Equation in Factorials
  143. An Equation in Matrices [JavaScript]
  144. An Equation in Radicals
  145. An Equation in Reciprocals
  146. An Equation in RxR from Romania
  147. An Equation in Squares [JavaScript]
  148. An Equation Involving Four Collinear Points
  149. An Exercise without Calculating Aids
  150. An Identity between the Roots of a Cubic Polynomial - Problem from the 2007 Irish MO
  151. An Identity From the 2010 China Mathematical Competition
  152. An Identity Involving the Integer Part - Problem from the 1981 Canadian MO
  153. An Increasing Sequence from 2014 IMO
  154. An Inequality and Its Modifications
  155. An Inequality between AM, QM and GM
  156. An Inequality by Uncommon Induction
  157. An Inequality for Grade 8
  158. An Inequality from the 1967 IMO Shortlist
  159. An inequality from 1971 Canadian Mathematical Olympiad
  160. An Inequality from Marocco [JavaScript]
  161. An Inequality from Tibet
  162. An inequality in 2+2 variables from SSMA magazine
  163. Inequality in Convex Quadrilateral
  164. An Inequality in Determinants
  165. An Inequality in Four Weighted Variables
  166. An Inequality in Fractions with Absolute Values
  167. An Inequality in Integers [JavaScript]
  168. An Inequality in Integers II
  169. An Inequality in Parallelogram of Unit Area
  170. An Inequality in Reciprocals
  171. An Inequality in Reciprocals II An Inequality in Reciprocals III
  172. An Inequality in Three (Or Is It Two) Variables
  173. An Inequality in Triangle and In General
  174. An Inequality In Triangle and Without
  175. An Inequality In Triangle and Without II
  176. An Inequality in Two Or More Variables
  177. An Inequality in Two Or More Variables II
  178. An Inequality Involving Arithmetic And Geometric Means
  179. An Inequality Solved by Changing Appearances
  180. An Inequality with Angles and Integers
  181. An Inequality with Arbitrary Roots
  182. An Inequality with Areas, Norms, and Complex Numbers
  183. An Inequality with Complex Numbers of Unit Length
  184. An Inequality with Complex Numbers of Unit Length II
  185. An Inequality with Cyclic Sums And Products
  186. An Inequality with Cyclic Sums on Both Sides
  187. An Inequality with Cyclic Sums on Both Sides II
  188. An Inequality with Cyclic Sums on Both Sides III
  189. An Inequality with Exponents
  190. An Inequality with Fractions
  191. An Inequality with Integrals
  192. An Inequality with Integrals and Radicals
  193. An Inequality with Just Two Variable And an Integer
  194. An Inequality with Sines But Not in a Triangle
  195. An Inequality with One Tangent and Six Sines
  196. An Inequality with Powers And Logarithm
  197. An Inequality With Six Variables and Constraints
  198. An Inequality Where One Term Is More Equal Than Others
  199. An Inequality with a Parameter and a Constraint
  200. An Inequality with Absolute Values
  201. An Inequality with an Infinite Series
  202. An Inequality with Constraint
  203. An Inequality with Constraints II
  204. An Inequality with Constraint III
  205. An Inequality with Constraint IV
  206. An Inequality with Constraint V
  207. An Inequality with Constraint VI
  208. An Inequality with Constraint VII
  209. An Inequality with Constraint VIII [JavaScript]
  210. An Inequality with Constraint IX [JavaScript]
  211. An Inequality with Constraint X [JavaScript]
  212. An Inequality with Constraint XI [JavaScript]
  213. An Inequality with Constraint XII
  214. An Inequality with Constraint XIII
  215. An Inequality with Constraint XIV
  216. Inequalities with Constraint XV and XVI
  217. An Inequality with Constraint XVII
  218. An Inequality with Constraint XVIII
  219. An Inequality with Constraint XIX
  220. An Inequality with Constraint in Four Variables
  221. An Inequality with Constraint in Four Variables II
  222. An Inequality with Constraint in Four Variables III
  223. An Inequality with Constraint in Four Variables IV
  224. An Inequality with Constraint in Four Variables V
  225. An Inequality with Constraint in Four Variables VI
  226. An Inequality with Cot, Cos, and Sin
  227. An Inequality with Cycling Sums
  228. An Inequality with Determinants [JavaScript]
  229. An Inequality with Determinants II
  230. An Inequality with Determinants III
  231. An Inequality with Determinants IV
  232. An Inequality with Determinants V
  233. An Inequality with Determinants VI
  234. An Inequality with Determinants VII
  235. An Inequality with Exponents [JavaScript]
  236. An Inequality with a Generic Proof [JavaScript]
  237. An Inequality with Nested Radicals
  238. An Inequality via Young's Theorem
  239. An Inequality with Just Two Variable
  240. An Inequality with Just Two Variable II
  241. An Inequality with Just Two Variable III
  242. An Inequality with Just Two Variables IV
  243. An Inequality with Just Two Variables V
  244. An Inequality with Just Two Variables VI
  245. An Inequality with Just Two Variable VII
  246. An Inequality with Just Two Variable VIII
  247. An Inequality with Permutations, I [JavaScript]
  248. An Inequality with Permutations, II [JavaScript]
  249. An Inequality with Two Cyclic Sums
  250. An Inequality with Two Pairs of Triplets An Inequality with Two Triples of Variables
  251. An Inequality with Two Triples of Variables II
  252. Inequality with Two Variables? Think Again
  253. An Infinite Product from Recursion
  254. An Integral Estimate
  255. An Integral Identity with Two Functions
  256. An Integral from the RMM
  257. An Integral Inequality from the RMM
  258. An Old Inequality
  259. An Unsolvable Riccati Equation
  260. An Unusual Quadratic Equation
  261. Another Face and Proof of a Trigonometric Identity
  262. Another Inequality with Logarithms, But Not Really
  263. Another Integral Inequality from the RMM
  264. Another Problem from the 2016 Danubius Contest
  265. Another Property of Number 4
  266. Another simple integral
  267. An Unusual Problem by Leo Giugiuc
  268. Any Integer with Three 2s
  269. Application of Cauchy-Schwarz Inequality
  270. Applications of Schweitzer's Inequality
  271. Approximation Of Roots
  272. Area of a Circle by Rabbi Abraham bar Hiyya Hanasi [Java]
  273. Areas on the Graphs of Power Functions
  274. Arithmetic-Geometric Means Inequality
  275. Arithmetic of Pie Sharing
  276. Ascending Bases and Exponents in Pascal's Triangle
  277. Assigning Numbers to Points in the Plane
  278. Associativity from 1977 IMO
  279. Averages, Arithmetic and Harmonic Means [Java]
  280. Averages in a sequence
  281. Ballot Lemma [Java]
  282. Banzhaf Power Index Calculator [Java]
  283. Barcode Encoding
  284. Barcode Magic [Java]
  285. Base Converter [JavaScript]
  286. Bergström's inequality
  287. Birth of an Inequality
  288. Bayes' Theorem
  289. Beatty Sequences
  290. Beauty and the Beast - in Trigonometry
  291. Bertrand's Postulate
  292. Bicubal Domino [JavaScript]
  293. Bilateral Diagrams [Java]
  294. Binet's Formula via Generating Functions
  295. Binet's Formula with Cosines
  296. Binomial Distribution [Java]
  297. Binomial Theorem
  298. Bisector of an imaginary angle may be real
  299. Brahmagupta-Fibonacci Identity [JavaScript]
  300. Byzantine Basketball
  301. Calculation of the Digits of pi by the Spigot Algorithm [Java]
  302. Calculus Proof of the Pythagorean Theorem
  303. Candido's Identity
  304. Cantor-Bernstein-Schroeder theorem
  305. Cantor's First Uncountability Proof
  306. Cardinal Numbers
  307. Cartesian Coordinate System [Java]
  308. Cassini's Identity
  309. Catalan Numbers in Pascal's Triangle
  310. Cavalieri's Principle [Java]
  311. Central and Inscribed Angles in Complex Numbers
  312. Chebyshev Inequality
  313. Chebyshev Holds a Key
  314. Chess tournament with 1.5 Points Winners
  315. Chinese Remainder Theorem
  316. Clifford Algebras
  317. Clubs in a Vector Space
  318. Collatz Conjecture [Java]
  319. Coloring a Graph [Java]
  320. Combinatorial Proofs
  321. Combo Playground: Calculating the Number of Paths on a Square Grid
  322. Committee Chairs
  323. Comparing complex numbers
  324. Completing a Square and Quadratic Formula
  325. Completing the Square Where It Matters
  326. Complex Numbers: A Dynamic Tool [Java]
  327. Complex number to a complex power may be real
  328. Complicated Constraint - Simple Inequality
  329. Composition of Functions, an Exercise Constrained Inequality with Reciprocals
  330. Constrained Intermarriages [JavaScript]
  331. Constructible Numbers
  332. Construction of Pascal's Triangle [Java]
  333. Constructive Existence of the de Bruijn Cycles
  334. Continued fractions
  335. cos/sin inequality
  336. cos(π/7)-cos(2π/7)+cos(3π/7) = 1/2
  337. Cospherical Points
  338. Countability of Rational Numbers
  339. Countability Principle
  340. Counting Chips On a Circle [Java]
  341. Counting Fat Sets
  342. Counting Squares in a Square [Java]
  343. Counting Triangles [Java]
  344. Counting Triangles II [Java]
  345. Covering a Chessboard with a Hole with L-Trominoes [Java]
  346. Covering A Chessboard With Domino
  347. Cubes Constrained
  348. Cubes in Pascal's Triangle
  349. Cubic Equation for the Sides of a Triangle
  350. Cubic Formula [JavaScript]
  351. Cubics and Bézout's Theorem
  352. Cubic Optimization with Linear Constraints
  353. Cubic Optimization with Partly Linear Constraints
  354. Curious Identities In Pythagorean Triangles
  355. Curious Identities Involving Integer Products

    Cute Exercise by Dorin Marghidanu

  356. Cyclic Inequality in Three Variables by Marian Cucoanes
  357. Cyclic Inequality with Arctangents
  358. Cyclic Inequality with Logarithms
  359. Cyclic Inequality with Square Roots
  360. Cyclic Inequality with Square Roots And Absolute Values
  361. Dan Sitaru's Amazing, Never Ending Inequality
  362. Dan Sitaru's Cyclic Inequality in Four Variables
  363. Dan Sitaru's Cyclic Inequality In One Variable
  364. Dan Sitaru's Cyclic Inequality In Many Variables
  365. Dan Sitaru's Cyclic Inequality In Three Variables
  366. Dan Sitaru's Cyclic Inequality in Three Variables II
  367. Dan Sitaru's Cyclic Inequality in Three Variables III
  368. Dan Sitaru's Cyclic Inequality in Three Variables IV
  369. Dan Sitaru's Cyclic Inequality in Three Variables V
  370. Dan Sitaru's Cyclic Inequality in Three Variables VI
  371. Dan Sitaru's Cyclic Inequality in Three Variables VII
  372. Dan Sitaru's Cyclic Inequality in Three Variables VIII
  373. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints
  374. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints IV
  375. Dan Sitaru's Exercise with Pi and Ln
  376. Dan Sitaru's Inequality: From Three Variables to Many in Two Ways
  377. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints II
  378. Dan Sitaru's Cyclic Inequality In Three Variables with Constraints III
  379. Dan Sitaru's Inequality by Induction
  380. Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots
  381. Dan Sitaru's Cyclic Inequality with a Constraint and Cube Roots II
  382. Dan Sitaru's Inequality with a Double Integral
  383. Dan Sitaru's Inequality with Roots and Powers
  384. Dan Sitaru's Inequality with Tangents
  385. Dan Sitaru's Inequality with Three Related Integrals and Derivatives
  386. Dan Sitaru's Integral Inequality with Powers of a Function
  387. Dan Sitaru's Sum of a Series
  388. Daniel Dan's Optimization in Three Variables
  389. Darij Grinberg's Inequality In Three Variables
  390. Deceptive Appearances
  391. Decimal Sums of Successive Integers
  392. Derivatives without Limits
  393. Descartes' Rule of Signs
  394. Determinant and Divisibility
  395. Determinant on a Circle
  396. Determinants in Pascal's Triangle
  397. Diagonal Argument
  398. Diophantine Equations
  399. Distance between the Orthocenter and Circumcenter
  400. Distance Between Strings [Java]
  401. Distance From a Point to a Straight Line [Java]
  402. Distance Inequality
  403. Distances to Three Points on a Circle
  404. Distinct Multisets with the Same Pairwise Sum
  405. Divergence of {cos(n)} [JavaScript]
  406. Divergence of the Harmonic Series (Details)
  407. Divide And Conquer in Cyclic Sums
  408. Dividing Loot Fairly
  409. Divisibility and Modulo Arithmetic at IMO 2007
  410. Divisibility by Powers of 2
  411. Dollars in Pockets
  412. Doodling on April 1st
  413. Dorin Marghidanu's Believe It Or Not
  414. Dorin Marghidanu's Calculus Lemma
  415. Dorin Marghidanu's Cyclic Inequality in Three Variables
  416. Dorin Marghidanu's Cyclic Inequality in Three Variables II
  417. Dorin Marghidanu's Cyclic Inequality in Three Variables III
  418. Dorin Marghidanu's Cyclic Inequality with Constraint
  419. Dorin Marghidanu's Inequality in Complex Plane
  420. Dorin Marghidanu's Inequality in Integer Variables
  421. Dorin Marghidanu's Inequality in Many Variables
  422. Dorin Marghidanu's Inequality in Many Variables Plus Two More
  423. Dorin Marghidanu's Inequality with Constraint
  424. Dorin Marghidanu's Inequality with Generalization
  425. Dorin Marghidanu's Inequality with Radicals
  426. Dorin Marghidanu's Light Elegance in Four Variables
  427. Dorin Marghidanu's Permuted Inequality
  428. Dorin Marghidanu's Spanish Problem Dorin Marghidanu's Sums and Products
  429. Dorin Marghidanu's Tribute to Descartes
  430. Dorin Marghidanu's Two-Sided Inequality
  431. Dot Patterns and Sierpinski Gasket [Java]
  432. Dots and fractions [Java]
  433. Double and Half Angle Formulas
  434. Double Sums by Dorin Marghidanu
  435. Ducci Sequences [Java]
  436. e in the Pascal Triangle
  437. EAN-13 Encoding [Java]
  438. Eigenvectors by Inspection
  439. Elves and Gnomes [JavaScript]
  440. Entry 1.414... in Ramanujan's Notebooks
  441. Equation in Permutations
  442. Equation in Radicals as a System of Equations
  443. Equations of the Staight Line [Java]
  444. Equivalence Relations
  445. Escape of the Clones
  446. Euclid's Algorithm
  447. Euclid's Algorithm: An Interactive Illustration [Java]
  448. Euler Cycles in Digraph
  449. Euler's formula for e
  450. Euler's Function and Congruence Theorem
  451. Evolution of Algebraic Symbolism
  452. Exercise with Square Spiral [GeoGebra]
  453. Existence of the de Bruijn Cycles via Graphs
  454. Exponential Inequalities for Means
  455. Extension of Euclid's Algorithm
  456. Extrema of x^3+y^3+z^3-xyz on the unit sphere
  457. Factorial Products
  458. Factorials and division
  459. Fair Division: Method of Sealed Bids II [Java]
  460. Falsity implies anything
  461. Farey Series
  462. Fermat's Like Equation
  463. Fermat's Little Theorem
  464. Fermat's Only Published Proof
  465. Fibonacci Idendtities with Matrices
  466. Fibonacci's Quickies
  467. Fibonacci Tilings
  468. Filling a Grid with Good Neighbors [Java]
  469. Filling an Array with 0s and 1s - and Counting [Java]
  470. Filling Pool with Fluids
  471. Find Minimum without Calculus
  472. Finding Hamilton Paths and Circuits [Java]
  473. Finite Sums of Terms 2^(n-i) i²
  474. First 1008 Digits
  475. First Geometric Interpretation of Complex Numbers [Java]
  476. Five Methods of Apportionment [Java]
  477. Five Lattice Points [Java]
  478. Five Squares in Complex Numbers [GeoGebra]
  479. Fleury's Algorithm and Euler's Paths and Cycles [Java]
  480. Flipping Items Simultaneously [Java]
  481. Flipping Items Simultaneously II [Java]
  482. Flipping Items Simultaneously III [Java]
  483. Flipping pancakes [Java]
  484. Floor Function
  485. Formal Systems - an Example
  486. Formula for Primes
  487. Four Hinged Squares: using complex numbers
  488. Four Integrals in One Inequality
  489. Four Numbers, Six Differences, GCD of the Products
  490. Four Pegs That Form a Square
  491. Four Voting Methods [Java]
  492. From The Sum of Angles To The Sum of Sines
  493. Function
  494. Function in the Plane That Vanishes
  495. Functional Inequality from Vietnam
  496. Fundamental Theorem of Algebra
  497. Futurama Theorem and Puzzle [Java]
  498. Gasoline Stations on a Circular Trek
  499. Gauss and Euler Integrals
  500. Gauss' Lemma
  501. Gauss' theorem on constructibility of regular polygons
  502. GCD of Fibonacci Numbers
  503. GCD(M, N) × LCM(M, N) = M × N [Java]
  504. Generalization That Solves the Problem
  505. Generating Functions
  506. Geometric Construction of Roots of Quadratic Equation [Java]
  507. Getting a Scout out of Desert [Java]
  508. Getting Your Rightful Share Back [JavaScript]
  509. Gheorghe Stoica's Problem from Gazeta Matematica Gireaux's Theorem
  510. Gödel's Incompleteness Theorem
  511. Golden Ratio is Irrational
  512. Graph and Roots of a Third Degree Polynomial [Java]
  513. Graph and Roots of Quadratic Polynomial [Java]
  514. Graph of a Polynomial [Java]
  515. Graph of a Polynomial Defined by Its Roots [Java]
  516. Graph with Nodes of Even Degree
  517. Graph without 3-Cycles
  518. Graphing Equations Is Useful
  519. Graphing Equations Is Useful, II [Java]
  520. Graphing Equations Is Useful, III
  521. Graphing Equations Is Useful, IV
  522. Graphing Equations Is Useful, V
  523. Greatest Difference in Arithmetic Progression
  524. Group multiplication of permutations [Java]
  525. 1/2 = 1
  526. Hadamard's Determinant Inequalities and Applications I
  527. Hadamard's Determinant Inequalities and Applications II
  528. Harmonic Serires And Its Parts
  529. 2006 Hellenic Mathematical Olympiad, Problem 4 [JavaScript]
  530. Hlawka's Inequality
  531. Horner's Method [Java]
  532. How Jost Burgi Computed the Sines of All Integer Angles in 1586
  533. How to Ask an Embarrassing Question
  534. How to define a permutation? [Java]
  535. How to write an equation of the union of two sets
  536. However You Solve It ... A Wonderful Equation
  537. Hung Viet's Inequality
  538. Hung Viet's Inequality II
  539. Hung Viet's Inequality III
  540. Hung Viet's Inequality IV
  541. Hyperintegers and Hyperreal Numbers
  542. Identities with Triangular Numbers
  543. Imad Zak's Cyclic Inequality in Three Variables
  544. Imad Zak's Cyclic Inequality in Three Variables II
  545. Implementation of Base Conversion
  546. Importance of the Absolute Value
  547. Improving an Inequality [JavaScript]
  548. Improving on an Escalator [JavaScript]
  549. In Calculus too 1 = 0
  550. Inclusion-Exclusion Principle
  551. Inclusion-Exclusion Principle: an Example
  552. Inequalities and Quadratic Forms
  553. An Inequality for Mixed Means
  554. An Inequality from RMM with a Generic 5
  555. An Inequality from RMM with Powers of 2
  556. Inequality 101 from the Cyclic Inequalities Marathon
  557. Inequality 1/2·3/4·5/6· ... ·99/100 < 1/10
  558. Inequality (1 + 1-3)(1 + 2-3)(1 + 2-3)...(1 + n-3) < 3
  559. Inequality 1 + 2-2 + 3-2 + 4-2 + ... + n-2 < 2.
  560. Inequality by Calculus
  561. Inequality for Integer Sequences
  562. Inequality for Integer Sequences II
  563. Inequality for Mixed Means
  564. An Inequality from Gazeta Matematica, March 2016
  565. An Inequality from Gazeta Matematica, March 2016 II
  566. An Inequality from Gazeta Matematica, March 2016 III
  567. An Inequality from Gazeta Matematica, March 2016 IV
  568. Inequalities with Double And Triple Integrals
  569. Inequality from Math Magazine 2009
  570. Inequality from Math Phenomenon
  571. Inequality in Acute Triangle
  572. Inequality of a Series of Square Roots
  573. Inequality with Constraint from Dan Sitaru's Math Phenomenon
  574. Inequality with Csc And Sin
  575. An Inequality with Exponents [JavaScript]
  576. An Inequality with Finite Sums
  577. An Inequality with Integrals and Rearrangement
  578. Inequality with Logarithms
  579. Inequality with Powers And Radicals
  580. Inequality with Three Linear Constraints
  581. Inequality with Three Numbers, Not All Zero
  582. Inequality with Two Minima
  583. Infinite Discent for x4 + y4 = z4
  584. Infinitesimals. Non-standard Analysis
  585. Infinitude of Primes
  586. Infinity As a Limit
  587. Inflection Points of Fourth Degree Polynomials [Java]
  588. An innovative recurrence
  589. Integer Iterations on a Circle [Java]
  590. Candy Game (Change Discharged) [Java]
  591. Integer Iterations on a Circle III [Java]
  592. Integer roots of integers are irrational
  593. Integer Triangle That Sums to Cube
  594. Integers and Rectangles [Java]
  595. Integers and Rectangles: a Proof by Induction [Java]
  596. Integers and Rectangles: Two Simple Proofs [Java]
  597. Integers With Digits 1 and 2 and Residues Modulo Powers of 2
  598. Integral in Normal Distribution
  599. Integral Is Area
  600. Integrals of a Function And Its Inverse
  601. Interest Calculations [Java]
  602. Integral of a Piece-wise Function
  603. Irrationality of k-th roots
  604. Irrationality of sqrt(2) via a Difference Equation
  605. Is GLOBALHELLFRY a Prime? [JavaScript]
  606. Is the Triangle Inequality Necessary?
  607. Is There a Limit?
  608. ISBN Encoding [Java]
  609. Irrational number to an irrational power may be rational
  610. Is 0.999... = 1?
  611. It's All About Powers of 10
  612. Iterations on Monotone Functions
  613. Jensen's Inequality [JavaScript]
  614. Ji Chen's Inequality
  615. Joachimsthal's Notations
  616. Jumping Isn't Everything [Java]
  617. Kaprekar's Iterations and Numbers [Java]
  618. Kulikowski's Theorem
  619. Kunihiko Chikaya's Inequality
  620. Kunihiko Chikaya's Inequality with a Constraint
  621. Kunihiko Chikaya's Inequality with Parameter
  622. Lagrange's identity, Proof
  623. Lagrange interpolation
  624. Lamé's Theorem - First Application of Fibonacci Numbers
  625. Langford and Skolem Sequences [Java]
  626. Latin Squares [Java]
  627. Le Khanh Sy's Problem
  628. Leech Trees and Taylor's Criterion
  629. Legendre's Formula
  630. Legendre's Theorem - The Prime Factorization of Factorials
  631. Leibniz and Pascal Triangles [Java]
  632. Leo Giugiuc's Cyclic Inequality in Three Variables
  633. Leo Giugiuc's Cyclic Quickie in Four Variables
  634. Leo Giugiuc's Exercise
  635. Leo Giugiuc's Inequality with Constraint
  636. Leo Giugiuc's Optimization with Constraint
  637. Leo Giugiuc's Trigonometric Lemma
  638. Less than, Equal to, Greater Than Symbols
  639. Lewis Carroll's Game of Logic [Java]
  640. Light Bulbs in a Circle [Java]
  641. Lights Out [Java]
  642. Limit of a Recursive Sequence
  643. Linear Algebra Tools for Proving Inequalities
  644. Linear Algebra Tools for Proving Inequalities: Cauchy-Binet Formula
  645. Linear Function with Coefficients in Arithmetic Progression [GeoGebra]
  646. Linear functions
  647. Linear Programming
  648. Linear System with Parameter
  649. Listing All Combinations [Java]
  650. Listing all permutations [Java]
  651. Listing All Permutations: Johnson-Trotter Method [Java]
  652. Logistic Model applet [Java]
  653. Long Huynh Huu's Inequality and Solution
  654. Long Runs of Composite Numbers
  655. Long Up Short Down
  656. Looking your best
  657. Loop or Halt [Java]
  658. Lucas' Theorem
  659. Lucas' Theorem II
  660. Lucky Numbers [Java]
  661. Marian Dinca's Refinement of Nesbitt's Inequality
  662. Marriage Theorem
  663. Marriage Problem, simulation [Java]
  664. Math Lotto
  665. Mathematical Induction
  666. Mathematical Proof
  667. Mathematicians Doze Off
  668. Matrices and Determinants as Optimization Tools: an Example
  669. Matrix Groups
  670. Maximum Volume of a Cut Off Box [Java]
  671. Means [Java]
  672. Mediant Fractions
  673. Method of Markers II [Java]
  674. Michael Rozenberg's Inequality in Three Variables with Constraints
  675. Michael Rozenberg's Inequality in Two Variables
  676. Minesweeper Theorem
  677. Mini Lights Out [Java]
  678. Minimax Principle [Java]
  679. Minimax Theorem [Java]
  680. Minimum of a Cyclic Sum with Logarithms
  681. Minumum Spanning Trees and Kruskal's Algorithm [Java]
  682. Minimum under Two Constraints
  683. Minus One But What a Difference
  684. Modular Arithmetic [Java]
  685. Monotone Convergence Theorem
  686. Monotone Number Sequences [Java]
  687. Monthly Problem 11199
  688. Much from Little
  689. Muirhead's Inequality
  690. Multiplication
  691. Multiplication of Points on a Circle [JavaScript, GeoGebra]
  692. Multiplication of Points on an Ellipse [JavaScript, GeoGebra]
  693. Napoleon's Relatives [Java]
  694. Negative on the left, positive on the right: how come?
  695. Nested Radicals
  696. Nested Subsets
  697. New Proof of Euclid's Theorem
  698. Newton's Method [Java]
  699. Newton's and Maclaurin's Inequalities [JavaScript]
  700. Nine Squares [Java]
  701. No Flipping without End [Java]
  702. NonSquare Matrix as a Tool for Proving an Inequality
  703. Number 8 Is Interesting
  704. Number Comparison - from the 1969 Canadian MO
  705. Number sequences and Pigeonhole [Java]
  706. Number System With Base 36
  707. Modern Vietnamese Classic
  708. On Dedekind Cuts
  709. On the Games of Hex and Y
  710. On the Intersection of kx and |sin(x)| [JavaScript]
  711. On the Intersection of 1+x and e^x and Trigonometric Consequences [JavaScript]
  712. One Dimensional Ants
  713. One Inequality - Two Domains
  714. One Trigonometric Formula and Its Consequences
  715. Optimization in Four Variables with Two Constraints
  716. Optimization with Many Variables
  717. Order. Well-ordered sets.
  718. Order and Chaos in Multiple Pendulums
  719. Ordinal Numbers
  720. Orthocenter in Complex Plane
  721. Outline Mathematics
  722. Problem 1 From the 2016 Pan-African Math Olympiad
  723. Problem 11867 from the American Mathematical Monthly
  724. Problem 2, the 36th IMO (1995)
  725. Problem 2 from the 2017 IMO
  726. Problem 3824 from Crux Mathematicorum
  727. Problem 3980 from Crux Mathematicorum
  728. Problem 4002 from Crux Mathematicorum
  729. Problem 4010 from Crux Mathematicorum
  730. Problem 4020 from Crux Mathematicorum
  731. Problem 4033 from Crux Mathematicorum
  732. Problem 4117 from Crux Mathematicorum
  733. Problem 4142 From Crux Mathematicorum
  734. Problem 4165 from Crux Mathematicorum
  735. Problem 4186 from Crux Mathematicorum
  736. Problem 6 from Dan Sitaru's Algebraic Phenomenon
  737. Problem 790 from Pentagon: an Inequality in Two Variables
  738. Problem From the 2016 IMO Shortlist
  739. Problem M317 from Crux Mathematicorum
  740. Proofs of Schweitzer's inequality
  741. Prove 2[sqrt(n) - sqrt(n-1)] > 1/sqrt(n)
  742. Sam Walters' Four Questions
  743. Same Integral, Three Intervals
  744. Samuel G. Walters' Congruence
  745. Simple Nameless Inequality
  746. Small Change Makes Big Difference
  747. Smallest multiple of 9 with no odd digits [Java]
  748. Snake Permutations And Their Number
  749. Solving a Peculiar Third Degree Equation
  750. The Basics of Infinite Descent
  751. The Joy of Homogeneity
  752. The Lucky Find [Java]
  753. The Mean Property of the Mean
  754. The Number of Judges and the Number of Votes [JavaScript]
  755. The power of substitution: solving an equation in a right triangle
  756. The power of substitution II: proving an inequality with three variables
  757. The power of substitution III: proving an inequality with two variables
  758. The Very First Functional Equation at IMO
  759. The Very Last, for 2014, proof of the Irrationality of sqrt(2)
  760. Three Colors on a Chessboard
  761. Three digit twister [Java]
  762. Who Has Beard? [Java]
  763. Outline Mathematics: Diophantine Equation I [JavaScript]
  764. p-adic Expansions
  765. p-adic Numbers
  766. Pairing Monotone and Convex Functions
  767. Parabola and Abutting Equilateral Triangles
  768. Parabolic Sieve of Prime Numbers
  769. Parity counting [Java]
  770. Parrondo Paradox Via Simpson Paradox [Java]
  771. Partial But General Case Of Fermat's Last Theorem
  772. Party Acquaintances [Java]
  773. Patterns in Pascal's Triangle
  774. Peano axioms
  775. Pennies in Boxes [Java]
  776. Percent
  777. Permutations
  778. Perrin Sequence
  779. Pi in Pascal's Triangle
  780. Pi in Pascal's Triangle via Triangular Numbers
  781. pi^e vs e^pi [JavaScript]
  782. Pigeonhole Principle
  783. Pinocchio as Epimenides
  784. Place Value [Java]
  785. Powers and Fractions Inequality
  786. Presence of the Antipodes on a Circle
  787. Prime Sums in Pairs
  788. Probability of Divisibility
  789. Problem 1 from the IMO 2009
  790. Problem 11804 from the AMM
  791. Problem A1, 69th Putnam Competition
  792. Problem A2, 69th Putnam Competition
  793. Problem of Divisibility by 5n
  794. Product of Diagonals in Regular N-gon
  795. Product Of Four Consecutive Integers
  796. Product of Simple Fractions [Java]
  797. Product of permutations [Java]
  798. Proizvolov's Identity [Java]
  799. Proofs Without Words [Java]
  800. Property of an
  801. Property of Quadratic Polynomials
  802. Property of the Powers of 2 [Java]
  803. Puzzlist vs. Mathematician: A Practical Arithmetic Problem
  804. Pythagorean Triples [Java]
  805. Quadratic Function for Solving Inequalities
  806. Quadratic Inequalities on [-1,1]
  807. An Inequality Where One Term Is More Equal Than Others
  808. Quasigroups and block designs from triangular grid [Java]
  809. Rabbits Reproduce; Integers Don't
  810. Radicals, Radicals, And More Radicals in an Inequality
  811. Radon's Inequality and Applications
  812. Ramsey's Number R(3, 3, 3)
  813. Ramsey Number R(4, 3)
  814. Ramsey Number R(5, 3)
  815. Ramsey Number R(4, 4)
  816. Ramsey's Theorem
  817. Random Walks [Java]
  818. Rascal Triangle
  819. Rational Solutions to xy = yx
  820. Rearrangement Inequality [JavaScript]
  821. Recombining Permutations [Java]
  822. Rectangle on a Chessboard [Java]
  823. Rectilinear Paradox
  824. Reflection Lemma [Java]
  825. Riemann Rearrangement Theorem
  826. Riemann Sphere and Möbius Transformation
  827. Riemann Sums - Function Integration [Java]
  828. Right Replacement [Java]
  829. Rolle's and Mean Value Theorems [Java]
  830. Rolle's Theorem in Action
  831. RomanoNorwegian Inequality
  832. Rooks on a Chessboard
  833. Roots and Tangents [GeoGebra]
  834. Round Robin Tournament
  835. Russell's Paradox
  836. Solve A Fourth Degree Equation
  837. Sophie Germain's Identity
  838. Square Root of 2 is Irrational, Proof 29
  839. Square Roots and Triangle Inequality
  840. Squares in Pascal's Triangle
  841. Squeeze Theorem
  842. Sum of Absolute Values [JavaScript]
  843. Sum of All Natural Numbers
  844. Sums of Binomial Reciprocals in Pascal's Triangle
  845. Squeeze (Sandwich, Pinching ...) Theorem (Lemma): an Example
  846. Scalar Product Optimization [JavaScript]
  847. Scheduling and Critical Path Algorithm [Java]
  848. Schur's Inequality
  849. Scrub Tile Puzzle [Java]
  850. Self-descriptive Strings [Java]
  851. Self-documenting Iterations [JavaScript]
  852. Self-reference
  853. Seven integers under 127 and their Ratios
  854. Seven Terms Periodic Sequence
  855. Shapley-Shubik Power Index [Java]
  856. Shapley-Shubik Power Index Calculator [Java]
  857. Shifting Digits and a Point of View
  858. Ship-Dock Theorem
  859. Shuttle Puzzle Practice II [Java]
  860. Sicherman Dice
  861. Sierpinski Gasket and Tower of Hanoi
  862. Sierpinski's Gasket and Dihedral Symmetry [Java]
  863. Sieve of Squares [Java]
  864. Σ2-n = Σn·2-n
  865. Signs and Sums in a Sequence [Java]
  866. Signs in a Matrix [Java]
  867. Simple But Uncommon Equation from 1968 IMO
  868. Simple Cellular Automaton [Java]
  869. Simple Graph Practice [Java]
  870. Simple Inequality with a Variety of Solutions
  871. Simple Inequality with Many Faces And Variables
  872. Simple Somos Sequence - a Russian Olympiad problem
  873. Simple Somos Sequence Calculator [Java]
  874. Simple Yet Uncommon Inequalities with Absolute Value
  875. Simpson Paradox [Java]
  876. $\sin 1^{\circ}+\sin {2^\circ}+\sin 3^{\circ}+\cdots+\sin 180^{\circ}=\tan 89.5^{\circ}$
  877. Sine And Cosine Are Continuous Functions
  878. Sine of a Sum Formula, Proof Without Words
  879. Single formula to express any integer with digits 0-9 each used just once
  880. Sitaru-Schweitzer Inequality
  881. Six Balls, Two Weighings
  882. Size of a Class: Two Viewpoints
  883. Sladjan Stankovik's Inequality In Four Variables
  884. Sladjan Stankovik's Inequality With Constraint
  885. Sladjan Stankovik's Inequality With Constraint II
  886. Slider Puzzles [Java]
  887. Solitaire in Two Colors [Java]
  888. Solutions of F(x)=x and F(F(x))=x [JavaScript]
  889. Solve - an Olympiad Problem - in Your Mind
  890. Solve xx3 = 3
  891. Solve cos²x + cos²2x + cos²3x = 1
  892. Solving an Equation with Inverse Functions
  893. Soriteses
  894. Sorting Algorithms
  895. Sperner's Theorem
  896. Splitting piles
  897. Square Root of 2 as a Continued Fraction
  898. Square root of 2 is irrational
  899. Star of David Theorems in Pascal Triangle
  900. Stars in Array
  901. Steinhaus' Problem [Java]
  902. Step into the Elliptic Realm
  903. Stern-Brocot Tree
  904. Straight Tromino on a Chessboard [Java]
  905. Structure of Hyperreal Numbers
  906. Subsets and Intersections
  907. Subtraction Game [Java]
  908. Sums of the Elements of Three Element Subsets [JavaScript]
  909. Sum of Three Squares [JavaScript]
  910. Sums and Products, a Generalization
  911. Sums, Products, and 1-1 Functions
  912. Sums of Geometric Series - Proofs Without Words [Java]
  913. Sums of Power Series
  914. Sums Of Successive Squares
  915. Sum without Adding
  916. Sums and Products [Java]
  917. Sums of powers of digits [Java]
  918. Superposition Principle
  919. Surreal Numbers
  920. Swapping Rows and Columns [Java]
  921. Sylvester's Problem
  922. Taking Games Seriously [Java]
  923. Tangential Chaos
  924. Telescoping Cubes
  925. Telescoping Squares
  926. Telescoping Sums and Series [JavaScript]
  927. Telescoping Tangents
  928. The ass and the mule [Java]
  929. The Averages of Sides and Angles in a Triangle
  930. The Basic Rules of Counting
  931. The Local-Global Principle
  932. The Regula Falsi - Iterated
  933. The Roads We Take
  934. The Size is in the Eyes of the Beholder
  935. Three Complex Numbers Satisfy Fermat's Identity For Prime Powers
  936. Three Dimensional Cross
  937. Three Junior Problems from Vietnam
  938. Three Variables, Three Constraints, Two Inequalities (Only One to Prove) - by Leo Giugiuc
  939. Tool for Constructing Orthogonal Latin Squares [Java]
  940. Tower of Hanoi, the Hard Way [Java]
  941. Train on Bridge [JavaScript]
  942. Tran Hoang Nam's Cyclic Inequality in Three Variables
  943. Transpositions
  944. Traveling Salesman Problem [Java]
  945. Treatise on Arithmetical Triangle
  946. Triangular numbers that are also square
  947. >Trigonometric Inequality with Integrals
  948. A Trigonometric Inequality with Ordered Triple of Variables
  949. Trilateral Diagrams [Java]
  950. Trinary Tree(s) underlying Primitive Pythagorean Triples
  951. Tromino Puzzle: Deficient Squares [Java]
  952. Twenty five boys and twenty five girls around a table.
  953. Twin Inequalities in Four Variables: Twin 1
  954. Twin Inequalities in Four Variables: Twin 2
  955. Two Circles and a Limit [Java]
  956. Two Colors in Two Rows [Java]
  957. Two Cyclic Inequalities [JavaScript]
  958. Two Equations in Three Variables [JavaScript]
  959. Two Men of Tibet [Java]
  960. Two-Parameter Solutions to Three Almost Fermat Equations
  961. Two-Sided Inequality by Dorin Marghidanu
  962. Two Problems by Kunihiko Chikaya
  963. Two Problems from a Serbian 7 Grade Math Competition
  964. Two Products: Constraint and Inequality
  965. Two Sequences Defined by Recurrence
  966. Two-Sided Inequality - One Provenance
  967. Two Solutions: One Correct, One Illuminating. An Example
  968. Uncountability of the Reals - via a Game
  969. Undefined vs Indeterminate in Mathematics
  970. (Un)fair Division by Consensus
  971. Universal Coloring [Java]
  972. Universal Product Code [Java]
  973. Unproperty of the Powers of 2
  974. Unsolved Problem from Crux Solved
  975. USA 1975 Problem 3
  976. Vandermonde's Convolution Formula
  977. Vandermonde Determinant
  978. Venn Diagrams
  979. Viète's Formulas
  980. Voronoi Diagram [JavaSCript]
  981. Ways To Count
  982. Weierstrass Product Inequality
  983. Weierstrass Substitution
  984. What, how, and the Web [Java]
  985. What 1, 6, and 14 Have to Do with 5?
  986. What Is Algorithm?
  987. What Is Determinant?
  988. What Is Finite?
  989. What Is Infinite Descent?
  990. What Is Infinity?
  991. What Is Limit?
  992. What is so natural about "natural logarithms"?
  993. What Is Space Completion?
  994. WHEN THE COUNTING GETS TOUGH, THE TOUGH COUNT ON MATHEMATICS
  995. When Two Segments Intersect? [JavaScript]
  996. Where To Start? [Java]
  997. White and Black Balls in Urn. 1 in, 2 out. What Color Remains?
  998. Why 17?
  999. Why Can't They Be All Different?
  1000. Why sin(x)/x tends to 1
  1001. Why The Number of Primes Could Not Be Finite?
  1002. Wilson's Theorem
  1003. Without Definitions: Why $\displaystyle\frac{1}{3}+\frac{2}{5}=\frac{11}{15}?$
  1004. Wonderful Inequality on Unit Circle
  1005. Word Problems [Java]
  1006. Wu's Inequality
  1007. $x_n=\sin 1+\sin 3+\sin 5+\cdots+\sin (2n-1)$
  1008. Yet Another Puzzle with Hats
  1009. Zeros and Nines [Java]

|Contact| |Front page| |Contents|

Copyright © 1996-2018 Alexander Bogomolny

 63047944

Search by google: