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

    Cute Exercise by Dorin Marghidanu

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

|Contact| |Front page| |Contents|

Copyright © 1996-2018 Alexander Bogomolny

 63030190

Search by google: