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• AMC historical results
  This is the '''AMC historical results''' page. This page should include results for *Mean: 68.3
  17 KB (1,921 words) - 13:00, 28 April 2024
• Inequality
  ...with [[optimization]] methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in [[math ...f <math>a</math> is greater than <math>b</math>, that is, <math>a-b</math> is positive.
  12 KB (1,798 words) - 16:20, 14 March 2023
• United States of America Mathematical Talent Search
  The '''United States of America Mathematical Talent Search''' ('''USAMTS''') is a [[mathematics competition]] in which students are challenged to write ful The USAMTS is administered by the [[Art of Problem Solving Foundation]] with support and
  4 KB (613 words) - 13:08, 18 July 2023
• AMC 10
  ...rican Mathematics Contest 10''' ('''AMC 10'''), along with the [[AMC 12]], is one of the first exams in the series of exams used to challenge bright stud ...rican Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC.
  4 KB (574 words) - 15:28, 22 February 2024
• AMC 12
  The '''American Mathematics Contest 12''' ('''AMC 12''') is the first exam in the series of exams used to challenge bright students, gr ...rican Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC!
  4 KB (520 words) - 12:11, 13 March 2024
• 1955 AHSME Problems/Problem 38
  ...21</math>, and <math>17</math> are obtained. One of the original integers is: ...ystem of equation should be constructed. (It doesn't matter which variable is which.)
  1 KB (200 words) - 23:35, 28 August 2020
• American Invitational Mathematics Examination
  The '''American Invitational Mathematics Examination''' ('''AIME''') is the second exam in the series of exams used to challenge bright students on ...matical Association of America]] (MAA). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC!
  8 KB (1,057 words) - 12:02, 25 February 2024
• 1954 AHSME Problems/Problem 46
  dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
  3 KB (415 words) - 18:01, 24 May 2020
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...t points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (692 words) - 22:33, 15 February 2021
• United States of America Mathematical Olympiad
  The '''United States of America Mathematical Olympiad''' ('''USAMO''') is the third test in a series of exams used to challenge bright students on th ...rican Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC and of the recent expansion of USAMO participant
  6 KB (869 words) - 12:52, 20 February 2024
• Joining an ARML team
  ...e Spring Semester to determine the team each year. The 6 practices include 3 individual tests to help determine the team and some lectures on certain ma ...ent process of selecting team members has yet to be decided upon. The team is organized by and practices at the San Diego Math Circle (SDMC), and most of
  21 KB (3,500 words) - 18:41, 23 April 2024
• AoPS Online School
  ...hosts classes for outstanding middle and high school students. The school is also accredited by the Western Association of Schools and Colleges. Each of ...ine School/Intermediate Algebra | Intermediate Algebra]] (formerly Algebra 3) — [https://artofproblemsolving.com/school/course/catalog/intermediate-al
  8 KB (965 words) - 03:41, 17 September 2020
• Wilson's Theorem
  ...)! + 1</math> is divisible by <math>p</math> if and only if <math>p</math> is prime. It was stated by John Wilson. The French mathematician Lagrange prov ...h> is composite. Then <math>p</math> has a factor <math>d > 1</math> that is less than or equal to <math>p-1</math>. Then <math>d</math> divides <math>
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  ...ity''' is an [[inequality]] that states that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness. ...al inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.
  3 KB (560 words) - 22:51, 13 January 2024
• Arithmetic mean
  The '''arithmetic mean''' of a [[set]] of numbers (or variables) is the sum of all the numbers, divided by the number of numbers - the [[averag is the arithmetic mean of the <math>{n}</math> numbers <math>x_1,x_2,\ldots,x_
  699 bytes (110 words) - 12:44, 20 September 2015
• Completing the square
  The idea of '''completing the square''' is to add something to an equation to make that equation a [[perfect square]]. ...math> was added to this, then we would have a [[perfect square]], <math>(x-3)^2=x^2-6x+9</math>. To do this, add <math>7</math> to each side of the equ
  2 KB (422 words) - 16:20, 5 March 2023
• Heron's Formula
  '''Heron's Formula''' (sometimes called Hero's formula) is a [[mathematical formula | formula]] for finding the [[area]] of a [[triang ...serve as a reason for why the area <math>A</math> is never imaginary. This is equivalent of ending at step <math>4</math> in the proof and distributing.
  4 KB (675 words) - 00:05, 22 January 2024
• Polynomial
  ...abstract algebra]] often an arbitrary [[field]]). Note that a [[constant]] is also a polynomial. * <math>x^3 + 3x^2y + 3xy^2 + y^3</math>, in the variables <math>x</math> and <math>y</math>
  6 KB (1,100 words) - 01:44, 17 January 2024
• Simon's Favorite Factoring Trick
  ...3333</cmath>where <math>23333</math> is the constant term, <math>xy</math> is the product of the variables, <math>66x</math> and <math>-88y</math> are th ...>a</math> are integer constants, and the coefficient of xy must be 1(If it is not 1, then divide the coefficient off of the equation.). According to Simo
  7 KB (1,107 words) - 07:35, 26 March 2024
• Factoring
  ...mathematical toolbox. To factor, or to break an expression into factors, is to write the expression (often an [[integer]] or [[polynomial]]) as a produ This leads to the difference of cubes factorization, <cmath>a^3-b^3=(a-b)(a^2+ab+b^2)</cmath>
  3 KB (532 words) - 22:00, 13 January 2024
• Mathematical problem solving
  ...ehind The [[Art of Problem Solving]] as well as many [[math competitions]] is the use of creative methods to solve problems. In a way, students are disco An interesting example of this kind of thinking is the calculation of the sum of the [[series]] <math>\frac11 + \frac14 + \fra
  2 KB (314 words) - 06:45, 1 May 2014
• Pigeonhole Principle
  ...principle'''. A common phrasing of the principle uses balls and boxes and is that if <math>n</math> balls are to be placed in <math>k</math> boxes and < An intuitive proof of the pigeonhole principle is as follows: suppose for contradiction that there exists a way to place <mat
  11 KB (1,985 words) - 21:03, 5 August 2023
• 1965 AHSME Problems/Problem 40
  ...+ 11x^2 + 3x + 31</math> is the square of an integer. Then <math>n</math> is: \textbf{(B) }\ 3 \qquad
  3 KB (571 words) - 00:42, 22 October 2021
• Geometric mean
  ...while the geometric mean of the numbers <math>b</math> and <math>c</math> is the number <math>g</math> such that <math>g\cdot g = b\cdot c</math>. ...nd 2 is <math>\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}</math>.
  2 KB (282 words) - 22:04, 11 July 2008
• Principle of Inclusion-Exclusion
  ...is counted once and only once. In particular, memorizing a formula for PIE is a bad idea for problem solving. Here, we will illustrate how PIE is applied with various numbers of sets.
  9 KB (1,703 words) - 07:25, 24 March 2024
• Combination
  ...om a set of <math>n</math> where the order in which the objects are chosen is irrelevant. We are generally concerned with finding the number of combinat This video is a great introduction to permutations, combinations, and constructive counti
  4 KB (615 words) - 11:43, 21 May 2021
• 2020 CIME II Problems/Problem 10
  ...htarrow (a-1)(b-1)=2</math> from whence we have <math>(a,b,c)\in\{(2,3,1),(3,2,1)\}</math>. ...c|a+b</math>; hence <math>a+b</math> is a multiple of <math>c</math> which is no more than <math>2c+6</math>. It follows that <math>a+b\in\{c,2c,3c,4c,5c
  2 KB (332 words) - 09:37, 30 December 2021
• Cauchy-Schwarz Inequality
  ...Bunyakovsky–Schwarz Inequality''' or informally as '''Cauchy-Schwarz''', is an [[inequality]] with many ubiquitous formulations in abstract algebra, ca ...tion for inequality problems in intermediate and olympiad competitions. It is particularly crucial in proof-based contests.
  13 KB (2,048 words) - 15:28, 22 February 2024
• Factorial
  The '''factorial''' is an important function in [[combinatorics]] and [[analysis]], used to determ ...h>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>.
  10 KB (809 words) - 16:40, 17 March 2024
• Discriminant
  ...negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. We know that the discriminant of a polynomial is the product of the squares of the differences of the polynomial roots <math
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  It is named after Menelaus of Alexandria. ...gle ABC</math>, where <math>P</math> is on <math>BC</math>, <math>Q</math> is on the extension of <math>AC</math>, and <math>R</math> on the intersection
  5 KB (804 words) - 03:01, 12 June 2023
• ARML historical results
  This is a list of historical results from the [[American Regions Mathematics League ...ards. One indvididual [need name] from Taiwan would have placed in the top 3 students overall on the individual round tiebreaker but was not considered
  19 KB (2,632 words) - 14:31, 12 June 2022
• Fermat's Little Theorem
  ...if they have a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. ...n [[integer]], <math>{p}</math> is a [[prime number]] and <math>{a}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equi
  16 KB (2,658 words) - 16:02, 8 May 2024
• Parabola
  A '''parabola''' is a type of [[conic section]]. A parabola is a [[locus]] of points that are equidistant from a point (the [[focus]]) and ...: <math>y = a{x}^2+b{x}+c</math> where a, b, and c are [[constant]]s. This is useful for manipulating the polynomial.
  3 KB (551 words) - 16:22, 13 September 2023
• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...me to <math>n</math>. If <math>{a}</math> is an integer and <math>m</math> is a positive integer [[relatively prime]] to <math>a</math>, then <math>{a}^{
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  A '''geometric inequality''' is an [[inequality]] involving various measures ([[angle]]s, [[length]]s, [[ar ...e]] triangle is greater than the length of the third side. This inequality is particularly useful and shows up frequently on Intermediate level geometry
  7 KB (1,296 words) - 14:22, 22 October 2023
• Brahmagupta's Formula
  '''Brahmagupta's Formula''' is a [[formula]] for determining the [[area]] of a [[cyclic quadrilateral]] gi ...formula which Brahmagupta derived for the area of a general quadrilateral is
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's theorem
  ...tween the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] of [[Ptolemy's Inequality]]. Pto ...\angle ABC+m\angle ADC=180^\circ .</math> However, <math>\angle ADP</math> is also supplementary to <math>\angle ADC,</math> so <math>\angle ADP=\angle A
  7 KB (1,198 words) - 20:39, 9 March 2024
• Elementary symmetric sum
  An '''elementary symmetric sum''' is a type of [[summation]]. ...leq n</math>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
  2 KB (275 words) - 12:51, 26 July 2023
• Number theory/Advanced
  ...ory from the perspective of [[abstract algebra]]. In particular, heavy use is made of [[ring theory]] and [[Galois theory]]. Algebraic methods are partic ...erties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].
  5 KB (849 words) - 16:14, 18 May 2021
• 1959 IMO Problems/Problem 2
  For what real values of <math>x</math> is Since the term inside the square root is a perfect square, and by factoring 2 out, we get
  3 KB (466 words) - 12:04, 12 April 2024
• Harmonic mean
  ...math>n</math> [[positive]] [[real number]]s <math> x_1, x_2... x_n </math> is defined to be: <math> \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_ ...ate <math>\frac 3{\frac 13 + \frac 16 - \frac 12} = \frac 30</math>, which is obviously problematic.
  1 KB (196 words) - 00:49, 6 January 2021
• LaTeX
  ...d [[math|mathematical]] and scientific writing. <math>\text{\LaTeX}</math> is very handy for producing equations such as <cmath>1+2+3+4+5+\sin \pi = \frac{5\cdot 6}{2}+0=15.</cmath>
  1 KB (164 words) - 19:09, 14 February 2024
• North Carolina MathCounts
  In the North Carolina MathCounts State Competition, the Countdown Round is unofficial in that it doesn't affect individual results. * 1987 - Ashley Reiter (3), Stephen London (41), Tim Ross (37), Ghene Faulcon, Coach: Caroline Wolfe
  4 KB (580 words) - 15:33, 2 April 2024
• Divisibility
  In [[number theory]], '''divisibility''' is the ability of a number to evenly divide another number. The study of divis ...th>a</math> is a '''multiple''' of <math>b</math>, and that <math>a</math> is '''divisible''' or '''evenly divisible''' by <math>b</math>.
  2 KB (277 words) - 16:21, 29 April 2023
• Real number
  ...s that are not real are <math>\ 3i</math>, <math>\ 3+2.5i</math>, <math>\ 3+2i+2j+k</math>, i.e. [[complex number]]s, and [[quaternion]]s. The set of real numbers, denoted by <math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used sub
  3 KB (496 words) - 23:22, 5 January 2022
• Divisibility rules
  ..., in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. === Divisibility Rule for 3 and 9===
  8 KB (1,315 words) - 18:18, 2 March 2024
• Induction
  ...rks for <math>n=1+1=2</math>, which in turn means it works for <math>n=2+1=3</math>, and so on. ...e. If a problem asks you to prove something for all integers greater than 3, you can use <math>n=4</math> as your base case instead. You might have to
  5 KB (768 words) - 20:45, 1 September 2022
• Triangle
  A '''triangle''' is a type of [[polygon]]. {{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}
  4 KB (628 words) - 17:17, 17 May 2018
• Greatest common divisor
  ...common factor''')) of two or more [[integer]]s is the largest integer that is a [[divisor]] of all the given numbers. The GCD is sometimes called the '''greatest common factor''' ('''GCF''').
  2 KB (288 words) - 22:40, 26 January 2021

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