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• Newton's Inequality
  ...math> in the polynomial <math> \prod_{i=1}^{n}(t+x_i) </math> (see [[Viete's sums]]). We define the ''symmetric average'' <math>d_k </math> to be <math ...have a root between <math>x_i </math> and <math>x_{i+1} </math> by [[Rolle's theorem]] if <math>x_i \neq x_{i+1} </math>, and if <math> x_i = x_{i+1} =
  5 KB (830 words) - 04:05, 28 January 2023
• Maclaurin's Inequality
  ...ymmetric polynomial]]s. For notation and background, we refer to [[Newton's Inequality]]. By the lemma from [[Newton's Inequality]], it suffices to show that for any <math>n </math>,
  992 bytes (146 words) - 16:48, 29 December 2021
• Nesbitt's Inequality
  '''Nesbitt's [[Inequality]]''' is a theorem which, although rarely cited, has many instr If <math> a_1, \ldots a_n </math> are positive and <math> \sum_{i=1}^{n}a_i = s </math>, then
  7 KB (1,224 words) - 16:21, 24 October 2022
• Pascal's Theorem
  '''Pascal's Theorem''' is a result in [[projective geometry]]. It states that if a [[h Since it is a result in the projective plane, it has a dual, [[Brianchon's Theorem]], which states that the diagonals of a hexagon circumscribed about
  4 KB (712 words) - 21:57, 25 April 2020
• Russell's Paradox
  The '''Russell's Paradox''', credited to Bertrand Russell, was one of those which forced the
  580 bytes (100 words) - 17:57, 12 May 2023
• Green's Theorem
  '''Green's Theorem''' is a result in [[real analysis]]. It is continuous [[partial derivative]]s mapping an open set containing
  2 KB (381 words) - 12:12, 30 May 2019
• Cramer's Rule
  '''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices Cramer's Rule employs the [http://en.wikipedia.org/wiki/Determinant matrix determina
  2 KB (352 words) - 18:22, 11 October 2023
• Ceva's Theorem/Problems
  #REDIRECT[[Ceva's theorem/Problems]]
  36 bytes (4 words) - 16:16, 9 May 2021
• Ceva's theorem/Problems
  530 bytes (82 words) - 16:16, 9 May 2021
• L'Hôpital's Rule
  '''L'Hopital's Rule''' is a theorem dealing with [[limit]]s that is very important to [[calculus]]. ...cdot \epsilon(h)}</math>, which would hence prove our lemma for L'Hospital's rule.
  2 KB (475 words) - 15:04, 24 March 2022
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to We first prove some intermediate results, viz., Bourbaki's Theorem (also known as the Bourbaki-Witt theorem).
  9 KB (1,669 words) - 19:02, 1 August 2018
• Pascal's Identity
  ...coefficients). It can often be used to simplify complicated [[expression]]s involving binomial coefficients. ...is also known as Pascal's Rule, Pascal's Formula, and occasionally Pascal's Theorem.
  3 KB (496 words) - 15:43, 11 April 2024
• Binet's Formula
  '''Binet's formula''' is an explicit formula used to find the <math>n</math>th term of It's coefficients look like this:
  6 KB (953 words) - 21:37, 30 May 2024
• Carleman's Inequality
  '''Carleman's Inequality''' states that for [[nonnegative]] [[real numbers]] <math>\{a_n\
  2 KB (278 words) - 16:39, 29 December 2021
• Vieta's formulas
  ...its roots can be easily expressed as a ratio between two of the polynomial's coefficients. It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many math contests/t
  3 KB (515 words) - 19:04, 19 May 2024
• Euclid's Elements
  ...ct should have been either common notions or postulates, as some of Euclid's methods of proof were faulty. Euclid's work is split into thirteen volumes. It covers not only geometry, but numbe
  12 KB (2,094 words) - 15:42, 1 December 2015
• Euclid's proof on the infitude of primes
  #REDIRECT [[Euclid's proof of the infinitude of primes]]
  56 bytes (8 words) - 15:35, 11 February 2015
• Lagrange's Mean Value Theorem
  '''Lagrange's mean value theorem''' (often called "the mean value theorem," and abbreviat We reduce the problem to [[Rolle's theorem]] by using an auxiliary function.
  1 KB (210 words) - 12:53, 20 February 2024
• Zassenhaus's Lemma
  ...s proof of the lemma in 1934 to provide a more elegant proof of [[Schreier's Theorem]]. He was a doctorate student under Emil Artin at the time. In th ...up of <math>K' \cdot (H \cap K)</math>; furthermore, the [[quotient group]]s
  2 KB (414 words) - 12:13, 9 April 2019
• Schreier's Theorem
  ...[[Zassenhaus's Lemma | lemma]], which gives an improved proof of Schreier's Theorem. ...gma_1</math> and <math>\Sigma_2</math>, respectively. Again by Zassenhaus's Lemma, the quotients <math>H'_{im+j}/H'_{im+j+1}</math> and <math>K'_{jn+i}
  2 KB (337 words) - 12:13, 9 April 2019

Page text matches

• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (709 words) - 15:00, 1 June 2024
• United States of America Mathematical Olympiad
  ...istered to approximately 500 of the best and brightest students from the U.S. and Canada. Qualification is based on [[AMC 10]], [[AMC 12]], and [[AIME]] ...Olympiad?" in the ''American Mathematical Monthly'' 78 (1971), the [[MAA]]'s National Contest Committee revived an Olympiad Subcommittee, which voted to
  6 KB (869 words) - 12:52, 20 February 2024
• Informatics competitions
  '''Informatics competitions''' test a student's ability to understand, organize, and work with information on computers. * [[St Mary’s University High School Programming Competition]] [http://cs.stmarys.ca/hspc
  7 KB (932 words) - 12:13, 15 January 2024
• Joining an ARML team
  ...nds. Please email Xinke Guo-Xue at xinkeguoxue@gmail.com, or message Xinke's AoPS account "hurdler", if you are interested in trying out for the Alabama ...at the San Diego Math Circle (SDMC), and most of the students on last year's team were regular attendees at SDMC. Also, since the 2007 team contained no
  22 KB (3,533 words) - 20:58, 2 June 2024
• Community Service Scholarships
  * Kohl's Kids Who Care Scholarship Program [http://www.kohlscorporation.com/communit
  3 KB (350 words) - 01:18, 19 June 2016
• Mu Alpha Theta
  #10 The set S is {#, !, @, *, $, %}. How many different proper subsets are possible?
  4 KB (632 words) - 17:09, 11 October 2020
• Power Mean Inequality
  The Power Mean Inequality follows from [[Jensen's Inequality]]. As <math>\ln(x)</math> is concave, by [[Jensen's Inequality]], the last inequality is true, proving <math>M(t)\ge M(0)</math
  3 KB (606 words) - 23:59, 1 July 2022
• Wilson's Theorem
  In [[number theory]], '''Wilson's Theorem''' states that if [[integer ]]<math>p > 1</math> , then <math>(p-1) ...e. Consider the [[field]] of integers modulo <math>p</math>. By [[Fermat's Little Theorem]], every nonzero element of this field is a root of the [[po
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  For all [[real number]]s <math>x</math>, <math>x^2 \ge 0</math>. ...h> and <math>s</math> are relatively prime positive integers. Find <math>r+s</math>. (Solution [[User:Ddk001#Solution_1.28Probably_official_MAA.2C_lots_
  3 KB (560 words) - 22:51, 13 January 2024
• Heron's Formula
  '''Heron's Formula''' (sometimes called Hero's formula) is a [[mathematical formula | formula]] for finding the [[area]] o <math>A=\sqrt{s(s-a)(s-b)(s-c)}</math>
  4 KB (675 words) - 00:05, 22 January 2024
• Polynomial
  ...onnegative]], [[integer|integral]] powers and multiplied by [[coefficient]]s from a predetermined [[set]] (usually the set of integers; [[rational]], [[ ...ly one way (not counting re-arrangements of the terms of the product). It's very easy to find the roots of a polynomial in this form because the roots
  6 KB (1,100 words) - 01:44, 17 January 2024
• Simon's Favorite Factoring Trick
  Simon's Favorite Factoring Trick (SFFT) (made by AoPS user [https://artofproblemsol ...t 1, then divide the coefficient off of the equation.). According to Simon's Favorite Factoring Trick, this equation can be transformed into: <cmath>(x+
  7 KB (1,130 words) - 18:48, 3 June 2024
• Factoring
  Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for difference of powers: == Vieta's/Newton Factorizations ==
  3 KB (532 words) - 22:00, 13 January 2024
• Proof by contradiction
  ===[[Euclid's proof of the infinitude of primes]]===
  2 KB (374 words) - 14:01, 21 August 2022
• Mathematical problem solving
  ...king. Mathematical [[problem solving]] involves using all the tools at one's disposal to attack a problem in a new way.
  2 KB (314 words) - 06:45, 1 May 2014
• Pigeonhole Principle
  ...ts, 2n}</math>. Show that if we choose <math>n+1</math> numbers from <math>S</math>, then there exist two numbers such that one is a multiple of the oth ...ath> integers. Prove that there exists distinct <math>a, b</math> in <math>S</math> such that <math>a - b</math> is a multiple of <math>n</math>.''
  11 KB (1,985 words) - 21:03, 5 August 2023
• Quadratic equation
  ...>, where <math>a</math>, <math>b</math> and <math>c</math> are [[constant]]s (that is, they do not depend on <math>x</math>) and <math>x</math> is the u ...of factoring is to turn a general quadratic into a product of [[binomial]]s. This is easier to illustrate than to describe.
  2 KB (264 words) - 12:04, 15 July 2021
• Relatively prime
  Two [[positive]] [[integer]]s <math>m</math> and <math>n</math> are said to be '''relatively prime''' or [[Euler's totient function]] determines the number of positive integers less than any
  2 KB (245 words) - 15:51, 25 February 2020
• Vieta's Formulas
  #REDIRECT[[Vieta's formulas]]
  29 bytes (3 words) - 14:40, 5 November 2021
• Minnesota MathCounts
  * 2015 - Frank Han (11th written, S)
  995 bytes (131 words) - 18:02, 12 March 2023

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