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• Fibonacci sequence
  ...s the sum of the two preceding it. The first few terms are <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...</math>. ...ivial example of a [[linear recursion]] with constant coefficients. There is also an explicit formula [[#Binet's formula|below]].
  6 KB (957 words) - 23:49, 7 March 2024
• Muirhead's Inequality
  The inequality is easier to understand given an example. Since the sequence <math>(5,1)</mat ...lympiad solution; one should use an application of AM-GM instead. Thus, it is suggested that Muirhead be used only to verify that an inequality ''can'' b
  8 KB (1,346 words) - 12:53, 8 October 2023
• Cardinality
  ..., 3\}, \{1, 2, 3\}\}</math> is 3, and the cardinality of the [[empty set]] is 0. ...In the above example, the cardinality of <math>\{3, 4\}</math> is <math>|\{3, 4\}| = 2</math>. Sometimes, the notations <math>n(A)</math> and <math>\# (
  2 KB (263 words) - 00:54, 17 November 2019
• Fundamental Theorem of Calculus
  This section is for people who know what [[integral]]s are but don't know the Fundamental T * Evaluate: <math>\int_2^5 x^3 dx</math> and <math>\int_{.2}^{.4} \cos(x) dx</math>. (The next few questi
  11 KB (2,082 words) - 15:23, 2 January 2022
• Polygon
  A '''polygon''' is a closed [[planar figure]] consisting of straight [[line segment]]s. There A polygon can be [[regular polygon| regular]] or irregular. A polygon is regular if all sides are the same length and all angles are [[congruent]].
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• 2011 AMC 10B Problems/Problem 22
  ...opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? ...27} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad\textbf{(E)}\ \frac{\sqrt{3}}{9}</math>
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• 2011 AMC 10B Problems/Problem 4
  ...the costs equally, LeRoy must give Bernardo half of the difference, which is <math>\boxed{\textbf{(C) } \;\frac{B-A}{2}}</math> .... Quickly, we realize the only way they could pay the same amount of money is if they both pay 45 dollars. This means LeRoy must give Bernardo <math>50 -
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• Vector
  ...x). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space. ...es, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  7 KB (1,265 words) - 13:22, 14 July 2021
• Logarithm
  ...come down to never having to deal with massive numbers. ex. :<cmath>((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}< left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers t
  4 KB (680 words) - 12:54, 16 October 2023
• Law of Cosines
  The '''Law of Cosines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle In the case that one of the angles has measure <math>90^\circ</math> (is a [[right angle]]), the corresponding statement reduces to the [[Pythagorea
  6 KB (1,003 words) - 00:02, 20 May 2024
• Schur's Inequality
  ...Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur. ...ath>a=b=c</math> or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>.
  2 KB (398 words) - 16:57, 29 December 2021
• Trigonometric identities
  ...<math>(\cos (x), \sin (x))</math> is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, <math>\s * <math>\sin 3x = 3\sin x-4\sin^3 x</math>
  8 KB (1,397 words) - 21:55, 20 January 2024
• Irrational number
  An '''irrational number''' is a [[real number]] that cannot be expressed as the [[ratio]] of two [[intege ...entury <math>B.C</math>. The Pythagoreans lived by the doctrine that ''all is number'', or that all things could be explained by relationships between nu
  3 KB (368 words) - 19:26, 6 June 2015
• Complex number
  ...ive]], so this equation has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{- ...= \sqrt{-1} </math> is the [[imaginary unit]]. The set of complex numbers is denoted by <math>\mathbb{C}</math>. The set of complex numbers contains th
  5 KB (860 words) - 15:36, 10 December 2023
• Angle bisector
  ...math> such that the angle between this line and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</math>: D=(3,4);
  3 KB (575 words) - 15:27, 19 March 2023
• Without loss of generality
  ...ten abbreviated to WLOG, is a frequently used expression in math. The term is used to indicate that the following proof emphasizes on a particular case, If you use WLOG in a proof and the statement is not necessarily true, points will get marked off. For example, you can't sa
  2 KB (280 words) - 15:30, 22 February 2024
• Law of Sines
  The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines ...math>, <math>c</math> opposite to <math>C</math>, and where <math>R</math> is the circumradius:
  4 KB (658 words) - 16:19, 28 April 2024
• Geometric sequence
  ...hat the ratio between any two consecutive terms is constant. This constant is called the '''common ratio''' of the sequence. ...mon ratio <math>-1/2</math>; however, <math>1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between c
  4 KB (644 words) - 12:55, 7 March 2022
• Arithmetic sequence
  ...he difference between any two consecutive terms is constant. This constant is called the '''common difference''' of the sequence. ...ence with common difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with common difference <math>-8</math>; however, <ma
  4 KB (736 words) - 02:00, 7 March 2024
• Fermat's Last Theorem
  ...ting that for positive [[integers]] <math>a,b,c,n</math> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math> ...vered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.''"
  3 KB (453 words) - 11:13, 9 June 2023
• 2005 USAMO Problems/Problem 4
  ...piece of length <math>k_i</math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
  7 KB (1,276 words) - 20:51, 6 January 2024
• 2000 AMC 12 Problems/Problem 1
  ...ger]]s such that the product <math>I \cdot M \cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? ...process on <math>2001</math> to get <math>667 * 3 * 1</math> as our <math>3</math> factors.
  2 KB (276 words) - 05:25, 9 December 2023
• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or sometimes [[natural number]] or [[ ...a Diophantine equation has infinitely many solutions, [[parametric form]] is used to express the relation between the variables of the equation.
  9 KB (1,434 words) - 13:10, 20 February 2024
• Fraction
  A '''fraction''' is the [[ratio]] of two [[number]]s. Most commonly, we consider [[rational nu ...numerator is the same as the denominator such as <math>\frac{3}{3}</math> is always equal to <math>1</math>.
  3 KB (432 words) - 19:34, 11 June 2020
• Functional equation
  A '''functional equation''', roughly speaking, is an equation in which some of the unknowns to be solved for are [[function]] ...he '''inverse function'''.) Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math>.
  2 KB (361 words) - 14:40, 24 August 2021
• Exponential form
  ...(yes, again!) rewrite <math>z</math> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. D=(1/2,sqrt(3)/2);
  1 KB (238 words) - 22:51, 20 February 2022
• Floor function
  ...ion is the same as "dropping everything after the decimal point," but this is ''not'' true for negative values. *<math>\lfloor 3.14 \rfloor = 3</math>
  3 KB (508 words) - 21:05, 26 February 2024
• Pascal's triangle
  '''Pascal's triangle''' is a triangle which contains the values from the [[binomial expansion]]; its v ...n</math>, the sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.
  5 KB (838 words) - 17:20, 3 January 2023
• Exponentiation
  .../math>, where <math>b</math> is the exponent (or power) and <math>a</math> is the [[base]]. ...ed if a equation has [[parentheses]] or the first one performed when there is no parentheses.
  5 KB (803 words) - 16:25, 10 August 2020
• Trigonometry
  ...gths and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way toward In contest math, trigonometry is an integral subfield of both [[geometry]] and [[algebra]]. Many essential r
  8 KB (1,217 words) - 20:15, 7 September 2023
• Mathematical Olympiad Summer Program
  ...especially the [[International Mathematical Olympiad]]. While the program is free to participants, invitations are limited to the top finishers on the [ ...d train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results
  6 KB (936 words) - 10:37, 27 November 2023
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality''' (EM-AM-GM-HM), is an [[inequality]] of the [[root-mean power]], [[arithmetic mean]], [[geomet ...where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power.
  5 KB (912 words) - 20:06, 14 March 2023
• Harmonic series
  Generally, a '''harmonic series''' is a [[series]] whose terms involve the [[reciprocal]]s of the [[positive inte The the most basic harmonic series is the infinite sum
  2 KB (334 words) - 20:52, 13 March 2022
• Goldbach Conjecture
  ...proven [[conjecture]] stating that every [[even integer]] greater than two is the sum of two [[prime number]]s. The conjecture has been tested up to 400 Goldbach's conjecture is one of the oldest unsolved problems in [[number theory]] and in all of math
  7 KB (1,201 words) - 16:59, 19 February 2024
• Twin Prime Conjecture
  The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) that states that there are [[inf One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of [[Dirichlet's Theorem]]. If one can show
  2 KB (308 words) - 02:27, 1 May 2024
• Quadratic residues
  ...</math> if there is some integer <math>n</math> so that <math>n^2-a</math> is [[divisibility | divisible]] by <math>m</math>. ...modulo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</math>
  5 KB (778 words) - 13:10, 29 November 2017
• Power of a Point Theorem
  The '''Power of a Point Theorem''' is a relationship that holds between the lengths of the [[line segment]]s form # One of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2
  5 KB (827 words) - 17:30, 21 February 2024
• Collatz Problem
  ...th>\{n,f(n),f(f(n)),f(f(f(n))),\ldots\}</math> contains 1. This conjecture is still open. Some people have described it as the easiest unsolved problem i ...6m+4\over 2}=3m+2</cmath> we can then observe that; only if <math>m</math> is even will another division by 2 be possible.
  1 KB (231 words) - 19:45, 24 February 2020
• Algebraic number
  ...[27]{19}}{\sqrt[3]{4}+\sqrt[7]{97}}</math>. A number that is not algebraic is called a [[transcendental number]], such as <math>e</math> or <math>\pi</ma ...mbers is large, there are only [[countable|countably]] many of them. That is, the algebraic numbers have the same [[cardinality]] as the [[natural numbe
  1,006 bytes (151 words) - 21:56, 22 April 2022
• International Mathematical Olympiad
  The '''International Mathematical Olympiad''' is the pinnacle of all high school [[mathematics competition]]s and the oldest ...eakdown=<u>Problem 1/4</u>: 6.5<br><u>Problem 2/5</u>: 7.5-8<br><u>Problem 3/6</u>: 9.5<br><u>Problem SL1-2</u>: 5.5-7<br><u>Problem SL3-4</u>: 7-8<br><
  3 KB (490 words) - 03:32, 23 July 2023
• Prime Number Theorem
  The '''Prime Number Theorem''' (PNT) is one of the most celebrated results in [[analytic number theory]]. Indeed, it is
  10 KB (1,729 words) - 19:52, 21 October 2023
• Infinite
  ...n]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element that y
  1 KB (186 words) - 23:19, 16 August 2013
• Isosceles trapezoid
  ...isosceles trapezoid''' is a geometric figure that lies in a [[plane]]. It is a specific type of [[trapezoid]] in which the legs have the same length. I * the segment joining the midpoints of the bases is perpendicular to the bases
  577 bytes (81 words) - 10:33, 18 April 2019
• Mock AMC
  A '''Mock AMC''' is a contest intended to mimic an actual [[AMC]] (American Mathematics Competi ...popular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year,
  51 KB (6,175 words) - 20:58, 6 December 2023
• Mock AIME
  A '''Mock AIME''' is a contest that is intended to mimic the [[AIME]] competition. (In more recent years, recurrin ...Y2QwOTc3NWZiYjY0LnBkZg==&rn=TWlsZG9yZiBNb2NrIEFJTUUucGRm Mildorf Mock AIME 3]
  8 KB (906 words) - 17:30, 26 April 2024
• Permutation
  A '''permutation''' of a [[set]] of <math>r</math> objects is any rearrangement (linear ordering) of the <math>r</math> objects. There a ...of [[infinite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.
  3 KB (422 words) - 11:01, 25 December 2020
• Riemann zeta function
  The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture
  9 KB (1,547 words) - 03:04, 13 January 2021
• Mock USAMO
  ...hen a mock USAMO is run on [[AoPS]]/[[MathLinks]], a very wide time window is often allowed to take the mock USAMO. ** [http://www.artofproblemsolving.com/blog/2712 Mock USAMO 3 2006]
  2 KB (205 words) - 19:56, 4 March 2020
• Modular arithmetic/Introduction
  ...of arithmetic that involves only [[integers]]. This goal of this article is to explain the basics of modular arithmetic while presenting a progression <math>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots </math>
  15 KB (2,396 words) - 20:24, 21 February 2024
• Perfect square
  An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect ...of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>
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