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• 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
• 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
• 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 (1,107 words) - 07:35, 26 March 2024
• Vieta's Formulas
  #REDIRECT[[Vieta's formulas]]
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• Mill's Constant
  Mill's Constant is defined as the smallest real number <math>\theta</math> such th ...smallest element in that set. If the [[Riemann Hypothesis]] is true, Mill's constant is approximately <math>1.3063778838630806904686144926...</math> an
  794 bytes (105 words) - 01:59, 15 January 2022
• Ceva's Theorem
  #REDIRECT[[Ceva's theorem]]
  27 bytes (3 words) - 16:06, 9 May 2021
• Fermat's Little Theorem
  '''Fermat's Little Theorem''' is highly useful in [[number theory]] for simplifying the A frequently used corollary of Fermat's Little Theorem is <math>a^p \equiv a \pmod {p}</math>. As you can see, it i
  16 KB (2,658 words) - 16:02, 8 May 2024
• Chebyshev's Inequality
  '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if ...nce of the [[Rearrangement inequality]], which gives us that the sum <math>S=a_1b_{i_1}+a_2b_{i_2}+...+a_nb_{i_n} </math> is maximal when <math>i_k=k</m
  1 KB (214 words) - 20:32, 13 March 2022
• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]] Let <math>\phi(n)</math> be [[Euler's totient function]]. If <math>n</math> is a positive integer, <math>\phi{(n)
  3 KB (542 words) - 17:45, 21 March 2023
• Brahmagupta's Formula
  .../math>, <math>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. <cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath>
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's Theorem
  #REDIRECT[[Ptolemy's theorem]]
  30 bytes (3 words) - 17:37, 9 May 2021
• Ptolemy's theorem
  ...[equality condition | equality case]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving i ...ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• Vieta's Formula
  #REDIRECT[[Vieta's formulas]]
  29 bytes (3 words) - 00:31, 4 June 2022
• Euler's totient function
  '''Euler's totient function''' <math>\phi(n)</math> applied to a [[positive integer]] ...p_m^{e_m} </math> where the <math>p_i </math> are distinct [[prime number]]s. Now, we can use a [[PIE]] argument to count the number of numbers less th
  5 KB (898 words) - 19:12, 28 January 2024
• Stewart's Theorem
  #REDIRECT[[Stewart's theorem]]
  30 bytes (3 words) - 16:22, 9 May 2021
• Jensen's Inequality
  '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jen One of the simplest examples of Jensen's inequality is the [[quadratic mean]] - [[arithmetic mean]] inequality. Taki
  3 KB (623 words) - 13:10, 20 February 2024
• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician *[[Ptolemy's Theorem]]
  3 KB (602 words) - 09:01, 7 June 2023
• Muirhead's Inequality
  '''Muirhead's Inequality''' states that if a sequence <math>p</math> [[Majorization|major ...ath> majorizes <math>(4,2)</math> (as <math>5>4, 5+1=4+2</math>), Muirhead's inequality states that for any positive <math>x,y</math>,
  8 KB (1,346 words) - 12:53, 8 October 2023
• Bretschneider's formula
  ...hat order) and [[diagonal]]s of length <math>p, q</math>. '''Bretschneider's formula''' states that the [[area]] [[Lagrange's Identity]] states that <math>|\vec{a}|^2|\vec{b}|^2-(\vec{a}\cdot\vec{b})^2
  3 KB (566 words) - 03:51, 12 February 2021
• Schur's Inequality
  ...hur's Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur. Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</mat
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• Vieta's
  #REDIRECT[[Vieta's formulas]]
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• Fermat's Last Theorem
  '''Fermat's Last Theorem''' is a recently proven [[theorem]] stating that for positive Fermat's Last Theorem was proposed by [[Pierre de Fermat]] in the <math>1600s</math>
  3 KB (453 words) - 11:13, 9 June 2023
• Pascal's triangle
  '''Pascal's triangle''' is a triangle which contains the values from the [[binomial exp ...oose k}}=2^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
• Newton's Sums
  ...sums''' give us a clever and efficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive sev Newton's sums tell us that,
  4 KB (690 words) - 13:11, 20 February 2024
• Euler's number
  '''Euler's number''' is a [[constant]] that appears in a variety of mathematical conte An approximation for Euler's number is <math>e\approx 2.7182818284590452...</math>
  4 KB (764 words) - 21:09, 13 March 2022
• ''See also: [[Hölder's inequality]]'' ...n, \dotsc, z_1, z_2, \dotsc, z_n</math> are [[nonnegative]] [[real number]]s and <math>\lambda_a, \lambda_b, \dotsc, \lambda_z</math> are nonnegative re
  4 KB (774 words) - 12:12, 29 October 2016
• Euler's Polyhedral Formula
  ...>F</math> denote the number of [[vertex|vertices]], [[edge]]s, and [[face]]s, respectively. Then <math>V-E+F=2</math>. Apply Euler's Polyhedral Formula on the following polyhedra:
  1,006 bytes (134 words) - 14:15, 6 March 2022
• Bertrand's Postulate
  '''Bertrand's postulate''' states that for any [[positive integer]] <math>n</math>, there It is similar to the proof of Chebyshev's estimates in the [[Prime Number Theorem|prime number theorem]] article but
  2 KB (309 words) - 21:43, 11 January 2010
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] its [[Laplace transform]] <math>F(s) = \int_0^\infty f(t)e^{-st}dt</math>
  6 KB (1,034 words) - 07:55, 12 August 2019
• Euler's identity
  '''Euler's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>. It is na ...ing problems involving [[complex numbers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates
  3 KB (452 words) - 23:17, 4 January 2021
• Holder's Inequality
  33 bytes (4 words) - 14:53, 29 November 2018
• Hölder's inequality
  #REDIRECT [[Hölder's Inequality]]
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• Rolle's Theorem
  '''Rolle's theorem''' is an important theorem among the class of results regarding the <LI>[[Lagrange's mean value theorem]]</LI>
  1 KB (270 words) - 12:13, 30 May 2019
• Pick's Theorem
  '''Pick's Theorem''' expresses the [[area]] of a [[polygon]], all of whose [[vertex |
  2 KB (301 words) - 13:08, 20 February 2024
• Euclid's Lemma
  '''Euclid's Lemma''' is a result in [[number theory]] attributed to [[Euclid]]. It stat ...ies that <math>p \mid a</math> or <math>p\mid b</math>, for all [[integer]]s <math>a</math> and <math>b</math>.
  1 KB (188 words) - 04:18, 20 November 2023
• Euclid's proof of the infinitude of primes
  ...ek mathematician [[Euclid]] that there are infinitely many [[prime number]]s.
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• De Moivre's Theorem
  '''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex This is one proof of De Moivre's theorem by [[induction]].
  2 KB (363 words) - 01:42, 11 January 2024
• Power's of 2 in pascal's triangle
  == Pascal's Triangle == Pascal's Triangle is a triangular array of numbers where each number is the sum of t
  2 KB (341 words) - 16:57, 16 June 2019
• Largange's Interpolation Formula
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• Viete's sums
  #REDIRECT[[Vieta's formulas]]
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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]]
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• 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
  3 KB (550 words) - 16:12, 24 February 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 mathematics con
  3 KB (512 words) - 15:31, 22 February 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
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• Euclid's proof on the infitude of primes
  #REDIRECT [[Euclid's proof of the infinitude of primes]]
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• 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.
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• 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
• Lagrange's Theorem
  '''Lagrange's theorem''' is a result on the indices of [[coset]]s of a [[group]]. so the index and order of <math>H</math> are [[divisor]]s of <math>g</math>.
  2 KB (303 words) - 12:24, 9 April 2019
• Burnside's Lemma
  ...rial]] result in [[group theory]] that is useful for counting the [[orbit]]s of a [[set]] on which a [[group]] [[group action|acts]]. ...led the '''Cauchy-Frobenius Lemma''', or '''the lemma that is not Burnside's'''. The lemma was (mistakenly) attributed to Burnside because he quoted an
  5 KB (757 words) - 18:11, 23 October 2023
• Cayley's Theorem
  ...characterizes group structure as the structure of a family of [[bijection]]s.
  1 KB (214 words) - 16:56, 19 February 2024
• Legendre's Formula
  '''Legendre's Formula''' states that ...of <math>n!</math> and <math>S_p(n)</math> is the [[sum]] of the [[digit]]s of <math>n</math> when written in [[base]] <math>p</math>.
  4 KB (699 words) - 17:55, 5 August 2023
• Routh's Theorem
  ...be the intersection of <math>CF</math> and <math>AD</math>. Then, '''Routh's Theorem''' states that <cmath>[GHI]=\dfrac{(rst-1)^2}{(rs+r+1)(st+s+1)(tr+t+1)}[ABC]</cmath>
  2 KB (267 words) - 00:02, 24 March 2021
• Bezout's Lemma
  #REDIRECT[[Bézout's Identity]]
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• Carnot's Theorem
  '''Carnot's Theorem''' states that in a [[triangle]] <math>ABC</math>, the signed sum o label("$O_C$",f,S);
  4 KB (723 words) - 01:45, 18 February 2021
• Karamata's Inequality
  '''Karamata's Inequality''' states that if <math>(a_i)</math> [[Majorization|majorizes]] ...ming <math>a_i\geq a_{i+1}</math> and similarily with the <math>b_i</math>'s, we get that <math>c_i\geq c_{i+1}</math>. Now, we know:
  2 KB (370 words) - 03:39, 28 March 2024
• Aczel's Inequality
  '''Aczél's Inequality''' states that if <math>a_1^2>a_2^2+\cdots +a_n^2</math> or <mat * Popoviciu, T., Sur quelques inégalités, Gaz. Mat. Fiz. Ser. A, 11 (64) (1959) 451–461
  2 KB (428 words) - 16:36, 29 December 2021
• Gauss's Lemma
  * [[Gauss's Lemma (polynomial)]] * [[Quadratic reciprocity|Gauss's Lemma (quadratic reciprocity)]]
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• Gauss's Lemma (polynomial)
  '''Gauss's Lemma for Polynomials''' is a result in [[abstract algebra | algebra]]. The original statement concerns [[polynomial]]s with [[integer]] coefficients. Such a polynomial is called ''primitive'' i
  3 KB (483 words) - 12:23, 30 May 2019
• Fermat's Two Squares Theorem
  '''Fermat's Two Squares Theorem''' states that that a [[prime number]] <math>p</math> c Since 0 and 1 are the only [[quadratic residue]]s mod 4, it follows that if <math>p</math> is a prime number represented as t
  4 KB (612 words) - 12:10, 30 May 2019
• De Morgan's Laws
  '''De Morgan's Laws''' are two very important laws in the fields of [[set theory]] and [[b
  3 KB (448 words) - 19:53, 19 February 2022
• Van Aubel's Theorem
  ...; yielding centers <math>P_{AB}, P_{BC}, P_{CD}, P_{DA}</math>. Van Aubel's Theorem states that the two line segments connecting opposite centers are p dot("$q$",Q,S);
  2 KB (410 words) - 14:01, 4 March 2023
• Bolzano's theorem
  '''Bolzano's Theorem''' is a special case of the [[Intermediate Value Theorem]], where <
  225 bytes (34 words) - 12:18, 30 May 2019
• Cauchy's Integral Formula
  '''Cauchy's Integral Formula''' is a fundamental result in by application of [[Cauchy's Integral Theorem]].
  4 KB (689 words) - 17:19, 18 January 2024
• L'Hopital's Rule
  #REDIRECT [[L'Hôpital's Rule]]
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• Liouville's Theorem (complex analysis)
  In [[complex analysis]], '''Liouville's Theorem''' states that a [[Picard's Little Theorem]] is a stronger result.
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• Hilbert's Basis Theorem
  '''Hilbert's Basis Theorem''' is a result concerning [[Noetherian]] [[ring]]s. It states that if <math>A</math> is a (not necessarily [[commutative]])
  4 KB (617 words) - 19:59, 23 April 2023
• Schonemann's criterion
  See also [[Eisenstein's criterion]].
  1 KB (279 words) - 17:03, 31 October 2023
• Kirchhoff's rules
  Kirchhoff's rules
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• Maine Business & Prof. Women's Club - Rachel E. Lemieux Youth Scholarship
  391 bytes (56 words) - 20:10, 2 December 2015
• KFC Colonel's Scholars Program
  * Plan to pursue a bachelor's degree at a public, in-state college or university
  721 bytes (101 words) - 17:36, 29 June 2022
• The Lowe's Scholarship
  ...for high school seniors who plan to enroll in undergraduate study in the U.S. Students within the U.S., including Lowe's employees and their children, are eligible.
  652 bytes (87 words) - 10:55, 23 July 2010
• Harry S. Truman Scholarship
  * a full-time junior-level student pursuing a bachelor's degree at a four-year institution (see website for a more specific definiti * nominated by the institution's Truman Scholarship Faculty Representative
  2 KB (239 words) - 11:15, 23 July 2010
• Carnot's Thereom
  #REDIRECT [[Carnot's Theorem]]
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• Hensel's Lemma
  55 bytes (9 words) - 17:43, 31 December 2015
• Newton's sums
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• Callebaut's Inequality
  '''Callebaut's Inequality''' states that for <math>1\ge x\ge y\ge 0,</math> <cmath> \sum_{
  1 KB (236 words) - 17:30, 4 December 2010
• ''See also: [[Hölder's Inequality]]'' '''Hölder's Inequality,''' a generalization of the '''Cauchy-Schwarz inequality''', sta
  1 KB (197 words) - 12:12, 29 October 2016
• Hölder's Inequality
  ...n, \dotsc, z_1, z_2, \dotsc, z_n</math> are [[nonnegative]] [[real number]]s and <math>\lambda_a, \lambda_b, \dotsc, \lambda_z</math> are nonnegative re ...||_p^p}, b=\frac{|g(x)|^q}{||g||_q^q},\alpha=1/p,\beta=1/q</math>. [[Young's Inequality]] gives us
  4 KB (762 words) - 00:11, 18 June 2023
• De Moivre's theorem
  #REDIRECT[[De Moivre's Theorem]]
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• ==Proof of l'Hôpital's rule== ...roof of l'Hôpital's rule uses [[Cauchy's mean value theorem]]. l'Hôpital's rule has many variations depending on whether ''c'' and ''L'' are finite or
  4 KB (764 words) - 23:10, 2 January 2012
• Bézout's Identity
  '''Bézout's Identity''' states that if <math>x</math> and <math>y</math> are nonzero [[ ==Generalization/Extension of Bézout's Identity==
  4 KB (768 words) - 16:50, 6 September 2023
• Steiner's Theorem
  '''Steiner's Theorem''' states that in a [[trapezoid]] <math>ABCD</math> with <math>AB\p ...t's not hard to see that <math>\triangle HAB \sim \triangle HCD</math>. It's also not hard to show that <math>\triangle HBE \sim \triangle HDF</math> by
  3 KB (444 words) - 03:02, 23 March 2013
• Euler's Four-Square Identity
  Then Euler's Four-Square Identity simply reads <math>|XY|^2 = |X|^2 |Y|^2</math>; i.e. t
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• Dirichlet's Theorem
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• Viviani's theorem
  '''Viviani's Theorem''' states that for an equilateral triangle, the sum of the altitude ...2}=\dfrac{s}{2}(x+y+z)</math>. Therefore, <math>\dfrac{s}{2}(x+y+z)=\dfrac{s}{2}(a)</math>, so <math>x+y+z=a</math>, which means the sum of the altitude
  4 KB (420 words) - 14:13, 4 June 2021
• Euler's inequality
  Euler's Inequality states that <cmath>R \ge 2r</cmath> where R is the circumradius ...he incenter. Then <cmath>d=\sqrt{R(R-2r)}</cmath> From this formula, Euler's Inequality follows as <cmath>d^2=R(R-2r)</cmath> By the [[Trivial Inequalit
  597 bytes (106 words) - 00:44, 25 June 2015
• Eisenstein's criterion
  Let <math>a_0, a_1, ... ,a_n</math> be integers. Then, '''Eisenstein's Criterion''' states that the polynomial ...g_rx^r+g_{r-1}x^{r-1}+\cdots+ g_1x+g_0</math> and <math>h=h_sx^s+h_{s-1}x^{s-1}+\cdots+ h_1x+h_0</math>. Since <math>a_0</math> has only one factor of <
  2 KB (475 words) - 14:47, 14 August 2018

Page text matches

• Pythagorean Theorem
  ...theorems in [[geometry]], and is one of the many tools in a good geometer's arsenal. A very large number of geometry problems can be solved by building label("$G$", G, S);
  5 KB (886 words) - 21:12, 22 January 2024
• 2005 AMC 12A Problems/Problem 16
  ...tangent to both axes and to the second and third circles. What is <math>r/s</math>? label("$s$",(-1.5,4));
  2 KB (307 words) - 15:30, 30 March 2024
• 2016 AMC 10A Problems/Problem 6
  ...ers and Emilio adds his numbers. How much larger is Star's sum than Emilio's?
  967 bytes (143 words) - 03:18, 27 June 2023
• 2016 AMC 10A Problems/Problem 8
  ...lish Fox to double Fox's money every time Fox crosses the bridge by Rabbit's house, as long as Fox pays <math>40</math> coins in toll to Rabbit after ea
  1 KB (169 words) - 14:59, 8 August 2021
• 2016 AMC 10A Problems/Problem 11
  The second line's equation can be found in a similar fashion. Its slope is <math>m = \frac{0-
  8 KB (1,016 words) - 00:17, 31 December 2023
• List of United States elementary school mathematics competitions
  * [[Math Kangaroo]] may be the world's largest [[mathematics competition]]. [http://www.mathkangaroo.org/mk/defau
  4 KB (473 words) - 16:14, 1 May 2024
• List of United States high school mathematics competitions
  ...tions in the United States. These exams lead up to the selection of the U.S. team for the [[International Mathematical Olympiad]]. ([http://www.maa.org * [https://simiode.org/scudem SIMIODE SCUDEM] -- SIMIODE's (Systemic Initiative for Modeling Investigations and Opportunities using Di
  5 KB (583 words) - 17:26, 21 February 2024
• List of mathematics competitions
  There's also some exams with the part of mathematics being as hard as an olympiad,
  4 KB (406 words) - 11:20, 26 February 2021
• Alabama mathematics competitions
  Alabama's math competition community is fairly large and active and there are a numbe ...Math Tournament]]-- Grades 7-12. The VHHS tournament is one of the nation's largest local math tournaments, drawing around 1700 students annually.
  3 KB (411 words) - 21:32, 8 December 2014
• Mathematics summer program
  * [[Women's Technology Program]] at [[MIT]], [http://wtp.mit.edu/ Website]
  14 KB (1,851 words) - 23:51, 13 May 2024
• MathILy (serious mathematics infused with levity)
  ...shift, six days a week. The lead instructors are mathematicians with Ph.D.s and the apprentice instructors are graduate or undergraduate math students. Here's what the {MathILy, MathILy-Er} application process looks like:
  5 KB (706 words) - 23:49, 29 January 2024
• Mathematics scholarships
  * [[Moody's Mega Math Challenge]] [http://m3challenge.siam.org/ website]
  3 KB (409 words) - 15:02, 28 January 2022
• Mathematics, science, and technology scholarships
  * National Science Foundation's Robert Noyce Scholarship. <dollar/>10,000 for undergraduate or graduate stu * American Meteorological Society's undergraduate scholarships and graduate fellowships for students in STEM ma
  7 KB (851 words) - 10:54, 29 January 2022
• Proton Energy Scholarship
  * Current high school senior enrolled in a U.S. high school * U.S Citizen or permanent resident
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• List of international mathematics competitions
  * [[Mathematical Kangaroo]] is perhaps the world's largest [[mathematics competition]]. * [[Mathematical Kangaroo]] is perhaps the world's largest [[mathematics competition]].
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• MATHCOUNTS
  An individual's score is their total number of correct sprint round answers plus 2 times th ...ided by 4 plus 2 points for every correct team round answer, making a team's maximum possible score 66 points. Therefore, it is possible to win with a r
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• Colorado mathematics competitions
  * [[St. Mary's Academy Math Team Challenge]] -- Contests for middle school students. [http
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• Georgia mathematics competitions
  *[[Soli Deo Gloria Fall Tournament, aka Billy's Math Meet]] [[User:Solafidefarms/MathMeet]]
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• Kansas mathematics competitions
  * [[K-State S. Thomas Parker Mathematics Competition]] [http://www.math.ksu.edu/~zlin/mat
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• Kentucky mathematics competitions
  * Governor's Cup [http://www.kaac.com/governors-cup/] * Governor's Cup [http://www.kaac.com/governors-cup/]
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• Oklahoma mathematics competitions
  ...he top 5 (alternate national participant included) students from that year's MathCounts state competition will compete as a mixed team. Oklahoma does n
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• Utah mathematics competitions
  ...a/intermountain/index.html] is an college proof based math competition. It's format is similar to the Putnam, but the problems are much more approachabl
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• Resources for mathematics competitions
  Ritvik Rustagi's [https://www.tmasacademy.com/ace-the-amc10-12-free-book ACE The AMC 10 and * Elias Saab's [[MathCounts]] [http://mathcounts.saab.org/ Drills page].
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• Mathematics websites
  * [http://www.geometer.org/mathcircles/ Tom Davis's] site for [[math circles]] topics. * [[AoPS]] -- That's where you are now! [http://www.artofproblemsolving.com Home].
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• Math books
  ...publishes [[Richard Rusczyk]]'s, [[David Patrick]]'s, and [[Ravi Boppana]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=prealgebra Prea * [[AoPS]] publishes [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Store/viewitem.php?item=intro:algebra I
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• Math Olympiads for Elementary and Middle Schools
  MOEMS Executive Director [[Richard Kalman]] and many talented MOEMS PICO's run online [[Math Jams]] at [[Art of Problem Solving]]. These Math Jams ar
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• Math textbooks
  This '''Math textbooks''' page is for compiling a list of [[textbook]]s for mathematics -- not problem books, contest books, or general interest bo * [[AoPS]] publishes Dr. [[Richard Rusczyk]]'s [http://www.artofproblemsolving.com/Books/AoPS_B_Item.php?item_id=200 Intro
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• William Lowell Putnam Mathematical Competition
  ...may have as many students as are interested sit for the exam. Each school's team score is determined by adding the ranks (not the scores) of its top th
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• Chemistry competitions
  ...ms are administered annually in March and students are nominated for the U.S. National Chemistry Olympiad competition based on their performance. The U.S. National Chemistry Olympiad national exam (USNCO) is a 3-part, 4 hour and
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• Computer science books
  ...os/ASIN/0387948600/artofproblems-20 The Algorithm Design Manual] by Steve S. Skiena.
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• Physics books
  * [https://www.amazon.com/s?k=conceptual+physics&ref=nb_sb_noss Conceptual Physics] by Paul Hewitt ...artofproblems-20 Warped Passages: Unraveling the Mysteries of the Universe's Hidden Dimensions] by Lisa Randall.
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• Physics
  What is now considered Newton's most famous achievement is the formal statement of three basic, almost triv #If the net force on any amount of matter is [[Zero]], then the object's velocity will not change if viewing from a constant reference point.
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• Newton
  ...] squared, or <math>\mathrm{N}=\mathrm{kg}\times \frac{\mathrm{m}}{\mathrm{s}^2}</math>. This is because <math>F = ma</math>, which means force (Newtons
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• Mathematics
  ...asy image|<math>1\,2\,3\,4\,5\,6\,7\,8\,9\,0</math>|right|The ten [[digit]]s making up <br /> the base ten number system.}} ...ematics, as shown by the [http://www.ams.org American Mathematical Society's] [http://www.ams.org/msc/ Mathematics Subject Classification] scheme.
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• Inequality
  ...slightly to [[number theory]]. They deal with [[relations]] of [[variable]]s denoted by four signs: <math>>,<,\ge,\le</math>. For two [[number]]s <math>a</math> and <math>b</math>:
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• United States of America Mathematical Talent Search
  ...t-shirt, along with other prizes like books or software of the participant's choices (with first priority to the top scorers, and then down the ranks). ...ublished. KöMaL is a popular abbreviation of Középiskolai Matematikai és Fizikai Lapok (KMaL), which means "High School Mathematics and Physics Jour
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• Academic scholarships
  [[AoPSWiki]] includes one of the internet's most comprehensive guides to '''academic scholarships'''. Get started by c * Your high school's website or those of other area high schools
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• Math Jam
  ...AMC]] -- AoPS hosts sessions for discussion of the problems from each year's [[AMC 10]], [[AMC 12]], and [[AIME]] exams.
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• American Mathematics Competitions
  ...d Summer Program]], where students train for possible inclusion on the [[U.S. IMO]] team. ...ed to only 25 questions, and 2 years later, the A and B version of the AMC's were introduced.
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• 1954 AHSME Problems/Problem 46
  label("$E$",(0,0),S);
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• 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
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• 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
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• 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
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• 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
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• Community Service Scholarships
  * Kohl's Kids Who Care Scholarship Program [http://www.kohlscorporation.com/communit
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• Mu Alpha Theta
  #10 The set S is {#, !, @, *, $, %}. How many different proper subsets are possible?
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• 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
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• 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
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• 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_
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• 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>
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• 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
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• 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+
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• 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 ==
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• Proof by contradiction
  ===[[Euclid's proof of the infinitude of primes]]===
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• Mathematical problem solving
  ...king. Mathematical [[problem solving]] involves using all the tools at one's disposal to attack a problem in a new way.
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• 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>.''
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• 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.
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• 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
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• Vieta's Formulas
  #REDIRECT[[Vieta's formulas]]
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• Minnesota MathCounts
  * 2015 - Frank Han (11th written, S)
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• Geometric mean
  ...tric mean''' of a collection of <math>n</math> [[positive]] [[real number]]s is the <math>n</math>th [[root]] of the product of the numbers. Note that MC("a",D((-5,-0.3)--(3,-0.3),black,Arrows),S);
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• Principle of Inclusion-Exclusion
  ...set]]s, the size of each set, and the size of all possible [[intersection]]s among the sets. Now, for <math>|A\cap B|</math>, that's just putting four guys in order. By the same logic as above, this is <math>
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• Mill's Constant
  Mill's Constant is defined as the smallest real number <math>\theta</math> such th ...smallest element in that set. If the [[Riemann Hypothesis]] is true, Mill's constant is approximately <math>1.3063778838630806904686144926...</math> an
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• Combination
  ...ly that you choose the rest. This identity is also the reason why [[Pascal's Triangle]] is symmetrical. * [[Pascal's Triangle]]
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• Cauchy-Schwarz Inequality
  Its elementary algebraic formulation is often referred to as '''Cauchy's Inequality''' and states that for any list of reals <math>a_1, a_2, \ldots, ...as Sedrakyan's Inequality, Bergström's Inequality, Engel's Form or Titu's Lemma the following inequality is a direct result of Cauchy-Schwarz inequal
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• Discriminant
  ...use if the discriminant is positive, the equation has two [[real]] [[root]]s; if the discriminant is negative, the equation has two [[nonreal]] roots; a ...s a polynomial of degree 3, which also makes possible to us to use Cardano's formula, by doing the substitution <math>x=z-\frac{a}{3}</math> on the poly
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• Ceva's Theorem
  #REDIRECT[[Ceva's theorem]]
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• Menelaus' theorem
  First let's define some masses. * [[Ceva's theorem]]
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• Number theory
  ...cs]] associated with studying the properties and identities of [[ integer]]s. *[[Prime number]]s
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• Number theory/Intermediate
  ** [[Simon's Favorite Factoring Trick]] ** [[Euler's Totient Theorem]]
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• San Diego ARML
  Individually, San Diego Surf had 2 students who scored 7's and went to tiebreakers: In addition, there were multiple students (on both teams) who scored 6's and earned medals as team high scorers:
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• ARML historical results
  | [[New York City ARML]] (New York City S)
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• Fermat's Little Theorem
  '''Fermat's Little Theorem''' is highly useful in [[number theory]] for simplifying the A frequently used corollary of Fermat's Little Theorem is <math>a^p \equiv a \pmod {p}</math>. As you can see, it i
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• Chebyshev's Inequality
  '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if ...nce of the [[Rearrangement inequality]], which gives us that the sum <math>S=a_1b_{i_1}+a_2b_{i_2}+...+a_nb_{i_n} </math> is maximal when <math>i_k=k</m
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• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]] Let <math>\phi(n)</math> be [[Euler's totient function]]. If <math>n</math> is a positive integer, <math>\phi{(n)
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• Geometric inequality
  ...[inequality]] involving various measures ([[angle]]s, [[length]]s, [[area]]s, etc.) in [[geometry]]. ...equality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says:
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• Brahmagupta's Formula
  .../math>, <math>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. <cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath>
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• Ptolemy's Theorem
  #REDIRECT[[Ptolemy's theorem]]
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• Ptolemy's theorem
  ...[equality condition | equality case]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving i ...ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
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• Elementary symmetric sum
  .../en.wikipedia.org/wiki/Sums_of_powers sums of powers], combined with Vieta's formulas. Elementary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial of degree <math>n</math>, the coefficient
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• Cyclic quadrilateral
  * [[Ptolemy's Theorem]] * [[Brahmagupta's formula]]
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• Vieta's Formula
  #REDIRECT[[Vieta's formulas]]
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• Algebra
  ...and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]]. ...ebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common objects of study in higher algebra.
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• Number theory/Advanced
  ...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]].
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• Algebra/Intermediate
  ** [[Vieta's Formulas]] ** [[Newton's Sums]]
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• Euler's totient function
  '''Euler's totient function''' <math>\phi(n)</math> applied to a [[positive integer]] ...p_m^{e_m} </math> where the <math>p_i </math> are distinct [[prime number]]s. Now, we can use a [[PIE]] argument to count the number of numbers less th
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• Alabama ARML
  In some years, there are [[Alabama ARML TST]]'s that are written to help decide the team. From 2005-2008, the TST has been
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• Contest strategies
  ...ea to stay up all night solving problems or playing video games because it's easy to get drowsy during the test. Getting a good night sleep can help re
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• Rearrangement Inequality
  ...tation]] of a [[finite]] [[set]] (in fact, [[multiset]]) of [[real number]]s and <math>B=\{b_1,b_2,\cdots,b_n\}</math> is a permutation of another finit ...ath> and <math>a_k</math> with <math>b_j</math> (unless both a's or both b's are equal, in which case either we can choose another pair of products or n
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• Real number
  ...th>, <math>\ 3+2i+2j+k</math>, i.e. [[complex number]]s, and [[quaternion]]s. ...these two classes are best understood as subsets of the [[complex number]]s.
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• Divisibility rules
  ...integer]]s are [[divisibility | divisible]] by particular other [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other b ...s that are relatively prime to the base (and works GREAT in binary). Here's one that works. 12348 - 28 ==> 12320 ==> 1232 +28 ==> 1260 ==> 126 + 14 ==
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• Induction
  ...ombinatorics]] are especially susceptible to induction solutions, but that's not to say that there aren't any problems in other areas, such as [[Inequal * Prove Bernoulli's inequality.
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• Incenter/excenter lemma
  ...math>L</math> be the midpoint of <math>\overarc{BC}</math> on the triangle's circumcenter. Then, the theorem states that <math>L</math> is the center of
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• Triangle
  Triangles are split into six categories; three by their [[angle]]s and three by their side lengths. All the angles of an '''acute''' triangle are [[acute angle]]s.
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• Stewart's Theorem
  #REDIRECT[[Stewart's theorem]]
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• Overcounting
  ...other. Suppose we are given an order to count the number of handshakes. It's not a matter of a great deal if there are less than 10 persons. But assume ...s <math>n</math>.So there will be <math>n*(n-1)</math> handshakes. Now let's try our formula for two people. According to the formula we get 2 handshake
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• Constructive counting
  ...ulate with <math>4</math> <math>A</math>s and <math>3</math> <math>B</math>s. Using constructive counting is an idea, but there are multiple ways one mi ...xes, so we don't have to account for them after choosing the <math>A</math>s. Thus, there are <math>35</math> different permutations of <math>AAAABBB</m
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• Jensen's Inequality
  '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jen One of the simplest examples of Jensen's inequality is the [[quadratic mean]] - [[arithmetic mean]] inequality. Taki
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• Casework
  ...digit can be <math>9</math> digits, with zero included and the first digit's number removed. Then there are <math>9 + 3 \cdot 81 = 252</math> of these n
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• Euclidean algorithm
  ...idean domain]], the most common of which is the [[nonnegative]] [[integer]]s <math>\mathbb{Z}{\geq 0}</math>, without [[factoring]] them. ~The congruence sign above should be replaced by the normal equal sign. It's important to note that <math>a \pmod{b} = r</math><br>is the same as <math>
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