Create the page "%s" on this wiki! See also the search results found.

Page title matches

• 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]]
  29 bytes (3 words) - 14:40, 5 November 2021
• 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
  2 KB (398 words) - 16:57, 29 December 2021
• Vieta's
  #REDIRECT[[Vieta's formulas]]
  29 bytes (3 words) - 00:30, 4 June 2022
• 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]]
  34 bytes (4 words) - 13:36, 11 July 2006
• 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.
  1 KB (179 words) - 11:57, 14 August 2022
• 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
  44 bytes (4 words) - 13:38, 3 March 2015
• Viete's sums
  #REDIRECT[[Vieta's formulas]]
  29 bytes (3 words) - 00:31, 4 June 2022
• 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
  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
  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
• 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]]
  31 bytes (4 words) - 13:34, 3 May 2023
• 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)]]
  292 bytes (32 words) - 13:14, 30 September 2020
• 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]]
  31 bytes (4 words) - 21:27, 11 March 2022
• Liouville's Theorem (complex analysis)
  In [[complex analysis]], '''Liouville's Theorem''' states that a [[Picard's Little Theorem]] is a stronger result.
  2 KB (412 words) - 20:30, 16 January 2024
• 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
  406 bytes (67 words) - 20:36, 7 March 2014
• 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]]
  30 bytes (3 words) - 20:11, 14 August 2010
• Hensel's Lemma
  55 bytes (9 words) - 17:43, 31 December 2015
• Newton's sums
  27 bytes (3 words) - 17:21, 23 October 2010
• 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]]
  32 bytes (4 words) - 19:06, 7 February 2023
• ==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
  1 KB (179 words) - 23:24, 5 November 2019
• Dirichlet's Theorem
  805 bytes (138 words) - 12:19, 30 May 2019
• 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
• Viviani's Theorem
  #REDIRECT [[Viviani's theorem]]
  31 bytes (3 words) - 10:11, 3 December 2016
• Women's Technology Program
  The MIT Women's Technology Program (WTP) is a rigorous four-week residential summer program
  798 bytes (119 words) - 13:49, 12 August 2014
• Avogadro's Constant
  Avogadro's Constant: <math>N_a = 6.0221415 * 10^{23}</math>
  59 bytes (6 words) - 12:55, 15 March 2016
• Young's Inequality
  ...cave and we know that <math>\frac{1}{p}+\frac{1}{q}=1</math>, so by Jensen's Inequality, we have Young's Inequality then follows by exponentiation of both sides.
  1 KB (230 words) - 02:48, 16 April 2016
• Euler's theorem
  ...phi(n)} \equiv 1 \quad\mod n</math>, where <math>\phi(n)</math> is [[Euler's Totient Theorem]], and <math>a</math> and <math>n</math> are coprime.
  400 bytes (52 words) - 19:58, 3 August 2023
• Euler's Totient Function
  #REDIRECT[[Euler's totient function]]
  37 bytes (4 words) - 09:53, 2 July 2016
• Wuwang2002's Wiki Games/Go
  38 bytes (7 words) - 18:24, 13 May 2021
• Areteem Institute's Math Zoom Winter Boot Camp (Locations Vary each year including one in Irvine, CA
  1 KB (166 words) - 17:04, 9 March 2017
• Bill's Triangle
  Bill's Triangle is a triangle similar to [[Pascal's Triangle]], except each number is obtained by adding the top three numbers, ==How to Make Bill's Triangle==
  2 KB (307 words) - 20:25, 12 December 2017
• Lagrange's Identity
  In algebra, Lagrange's identity, named after Joseph Louis Lagrange, is:[1][2] The second term on the left side of Lagrange's identity can be expanded as:
  4 KB (914 words) - 17:27, 31 August 2022
• Chen's Theorem
  '''Chen's Theorem''' is a [[theorem]] developed by Chinese [[mathematician]], Chen Ji Chen's Theorem states that any sufficiently large [[even]] number <math>\left(>e^{
  465 bytes (67 words) - 19:03, 28 May 2020
• Titu's Lemma
  Titu's lemma states that: ...r Titu Andreescu and is also known as T2 lemma, Engel's form, or Sedrakyan's inequality.
  5 KB (761 words) - 20:10, 29 April 2024
• Perron's criterion
  Perron's Criterion states: Now, we will prove Perron's Criterion. Let <math>f(x)=g(x)h(x)</math>, where <math>g(x)=x^r+g_{r-1}x^{r
  3 KB (488 words) - 15:23, 14 August 2018
• Cohn's criterion
  3 KB (581 words) - 08:37, 4 March 2021
• Ostrowski's criterion
  Ostrowski's Criterion states that:
  909 bytes (171 words) - 11:23, 15 June 2021
• Dunan's Theorem
  0 bytes (0 words) - 15:19, 21 July 2020
• Karmata's Inequality
  18 bytes (3 words) - 17:44, 20 January 2019
• Euler's formula and It's history
  First, you may have seen the famous "Euler's Identity": e (Euler's Number)
  3 KB (543 words) - 15:24, 13 June 2019
• Pascal's Triangle and its patterns and properties
  = Pascal's Identity = Pascal's Identity states that
  9 KB (1,531 words) - 15:22, 17 June 2019
• Pascal's Triangle and Pythagorean Theorem
  = Pascal's Identity = Pascal's Identity states that
  15 KB (2,425 words) - 09:25, 13 February 2020
• Stewart's theorem
  ...nd his dad put a bomb in the sink.") That is Stewart's Theorem. I know, it's easy to memorize. Setting the two left-hand sides equal and clearing [[denominator]]s, we arrive at the equation: <math> c^{2}n + b^{2}m=m^{2}n +n^{2}m + d^{2}m
  4 KB (738 words) - 12:53, 20 February 2024
• Hotmonkey's current phrase
  s t o n k s
  11 bytes (6 words) - 17:39, 10 January 2020
• Brocard's problem
  Are the known solutions, and it was a conjecture of Paul Erdös, that these are the only solutions.
  626 bytes (104 words) - 19:15, 12 March 2020
• Godel's First Incompleteness Theorem
  Gödel's First Incompleteness Theorem is a [[theorem]] that asserts that any axiomat ...ess Theorem]] (that a theory is consistent iff it has a model) and [[Godel's Second Incompleteness Theorem]] (that a consistent theory cannot prove its
  2 KB (277 words) - 18:17, 31 July 2020
• Lcz's Oly Notes
  0 bytes (0 words) - 20:22, 22 December 2020
• Superagh's Olympiad Notes
  ...is is for my good. (Also this thing is almost the exact same format as Lcz's :P ). (Ok, actually, a LOT of credits to Lcz) If <math>r>s</math>, then <math>P(r) \geq P(s)</math>. Equality occurs if and only if all the <math>a_i</math> are equal.
  4 KB (671 words) - 13:59, 22 July 2020
• Lcz's Mock AMC 10A
  0 bytes (0 words) - 16:10, 16 July 2020
• Lcz's Mock AMC 10A Problems
  0 bytes (0 words) - 16:13, 16 July 2020
• Godel's Second Incompleteness Theorem
  ...its own consistency iff it is consistent. It is closely related to [[Godel's First Incompleteness Theorem]], being in fact a stronger form, and easily d ...nsistency, then it could also prove <math>G_F</math>, contradicting Gödel's First Incompleteness Theorem, and we are done.
  1 KB (244 words) - 23:47, 22 July 2020
• Mark888's Hangout Corner
  503 bytes (86 words) - 13:05, 19 June 2020
• Jayasharmaramankumarguptareddybavarajugopal's Lemma
  ...opal's Lemma is trivialized by Jayasharmaramankumarguptareddybavarajugopal's Lemma, thus we are done <math>\mathbb{Q.E.D}.</math>
  229 bytes (26 words) - 21:53, 15 June 2021
• Wuwang2002's Wiki Games
  Hello! Welcome to Wuwang2002's Wiki Games! Wuwang2002, I think you should be imporving Piphi's games, not duplicating them, and also, can you link to my wiki projects pag
  673 bytes (100 words) - 14:24, 29 July 2020
• Casey's Theoram
  61 bytes (11 words) - 10:24, 3 December 2020
• The Devil's Triangle
  ==Generalized Wooga Looga Theorem (The Devil's Triangle)== ...}{[ABC]}=1-\frac{r(s+1)+s(t+1)+t(r+1)}{(r+1)(s+1)(t+1)}=\frac{rst+1}{(r+1)(s+1)(t+1)}</math>.
  3 KB (582 words) - 18:05, 23 March 2024
• Bestzack66's AMC10 Study Plan
  1,005 bytes (92 words) - 23:11, 17 January 2021
• FidgetBoss 4000's 2019 AMC 12B Problems/Problem 1
  #redirect [[FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 1]]
  68 bytes (8 words) - 14:56, 19 November 2020
• FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 1
  Use [[Simon's Favorite Factoring Trick]] to deduce <math>xy+x+y=(x+1)(y+1)-1</math>. We k ...getBoss 4000's 2019 Mock|ab=B|before=First problem|after=[[FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 2|Problem 2]]}}
  1 KB (213 words) - 15:26, 19 November 2020
• FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3
  ...ns can be inscribed in a circle, thus any subset of <math>4</math> vertice's from this octagon also all lie on the same circle. It is easy to see that n {{AMC12 box|year=FidgetBoss 4000's 2019 Mock|ab=B|num-b=2|num-a=4}}
  1 KB (157 words) - 16:04, 19 November 2020
• FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 2
  ...s 2019 Mock AMC 12B Problems/Problem 1|Problem 1]]|after=[[FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 3|Problem 3]]}}
  1 KB (201 words) - 15:26, 19 November 2020
• FidgetBoss 4000's 2019 Mock AMC 12B
  '''FidgetBoss 4000's 2019 Mock AMC 12B''' problems and solutions. The first link contains the fu *[[FidgetBoss 4000's 2019 Mock AMC 12B Problems]]
  2 KB (248 words) - 16:14, 19 November 2020
• FidgetBoss 4000's 2019 Mock AMC 12B Problems/Problem 4
  {{AMC12 box|year=FidgetBoss 4000's 2019 Mock|ab=B|num-b=3|num-a=5}}
  726 bytes (98 words) - 23:04, 19 November 2020
• Routh's theorem
  3 bytes (1 word) - 20:01, 1 December 2021
• Euler's Totient Theorem Problem 1 Solution
  This is a direct application of [[Euler's Totient Theorem]]. Since <math> \phi(100)=40 </math>, this reduces to <math
  272 bytes (35 words) - 17:33, 21 March 2023
• Euler's Totient Theorem Problem 2 Solution
  Link back to [[Euler's Totient Theorem]].
  770 bytes (109 words) - 22:14, 15 August 2023
• Ceva's theorem
  '''Ceva's theorem''' is a criterion for the [[concurrence]] of [[cevian]]s in a [[triangle]]. The proof using [[Routh's Theorem]] is extremely trivial, so we will not include it.
  5 KB (934 words) - 13:06, 20 February 2024
• Cavalieri's principle
  1 KB (167 words) - 22:01, 27 August 2021
• Naythan's Theorem
  Naythan's Theorem states: Arithmetic series starting at 1 with difference of 1 ending
  108 bytes (15 words) - 18:53, 15 July 2021
• Pascal's Bomb
  0 bytes (0 words) - 15:20, 30 October 2021
• Xinke's Math Academy
  ...rmer AoPS and MIST Academy student. More information can be found on Xinke's website: [https://xinkesmathacademy.com/]. ...nts interested in trying out for Alabama ARML or taking classes from Xinke's Math Academy can contact him at xinkeguoxue@gmail.com.
  557 bytes (87 words) - 20:51, 28 December 2021
• Cauchy's Criterion
  '''Cauchy's Criterion''' is a result in [[analysis]] that states that a sequence of rea
  2 KB (351 words) - 17:59, 12 June 2022
• Gergonne's Theorem
  349 bytes (64 words) - 20:13, 31 January 2024
• Baye's Theorem
  28 bytes (3 words) - 14:26, 13 July 2022
• Sheldon's Raisins
  58 bytes (10 words) - 15:47, 9 March 2022
• Newton's method
  '''Newton's method''' uses the [[derivative]] of a differentiable [[function]] to appro The sum of the roots is <math>1</math> by [[Vieta's formulas]], so the lesser root is simply <math>1 - 1.6180340 = -0.6180340</
  13 KB (2,298 words) - 23:34, 28 May 2023
• Wolstenholme's Theorem
  '''Wolstenholme's Theorem''' is a result in [[Number Theory]] from English Mathematician Jose
  804 bytes (132 words) - 14:01, 7 April 2022
• Maxwell's equations
  #REDIRECT[[Maxwell's Equations]]
  32 bytes (3 words) - 20:48, 4 May 2022
• Maxwell's Equations
  '''Maxwell's equations''' are a set of four equations that govern electricity and magnet ...\mathbf{E} \cdot d\mathbf{A} = \frac{q_{enc}}{\varepsilon_0}</math> (Gauss's law of electricity),
  626 bytes (103 words) - 12:24, 18 May 2022
• Wildin's Theorem
  ...unknown, but those who know it are able to score perfects on the AMC 10.[/s]
  346 bytes (54 words) - 22:10, 2 October 2023
• Miquel's point
  ==Miquel and Steiner's quadrilateral theorem== ==Analogue of Miquel's point==
  6 KB (976 words) - 23:10, 28 October 2023
• Honest day's work
  Does anyone know how to get the "Honest Day's Work" badge under the Report tab? If you do, please write it below. Thank y
  124 bytes (24 words) - 20:27, 7 December 2022
• Vieta's Formulæ
  #REDIRECT[[Vieta's formulas]]
  29 bytes (3 words) - 00:45, 7 February 2023
• Bézout's Lemma
  ...result in number theory that implies many other theorems, such as [[Euclid's Lemma]] and the [[Chinese Remainder Theorem]]. To see an example of Bézout's Lemma, let <math>a</math> and <math>b</math> be <math>15</math> and <math>6
  3 KB (562 words) - 11:19, 5 March 2023
• Brahmagupta's Identity
  '''Brahmagupta's identity''' states that for integers <math>a, b, c, d, n,</math> Substituting <math>n = -D</math>, the forms involved in Brahmagupta's identity lend themselves to use with solutions to the [[Pell equation]] <cm
  2 KB (274 words) - 20:16, 2 March 2023
• Daniel Zhou's Profile
  6 KB (687 words) - 12:47, 11 May 2024
• Radon's Inequality
  Radon's Inequality states: ...t consequence of [[Hölder's Inequality]], and a generalization of [[Titu's Lemma]] (for p=2, it is just that).
  1 KB (175 words) - 16:30, 14 March 2023
• Fermat's theorem
  [[Fermat's Last Theorem]] [[Fermat's Little Theorem]]
  102 bytes (12 words) - 17:11, 21 March 2023
• Fermat's Theorem
  [[Fermat's theorem]]
  20 bytes (2 words) - 17:10, 21 March 2023
• Pell's equation (simple solutions)
  Pell's equation is any Diophantine equation of the form <math>x^2 – Dy^2 = 1,</m It is the form of Pell's equation, therefore
  15 KB (2,351 words) - 08:28, 20 April 2023
• SANSKAR'S OG PROBLEMS
  10 KB (1,710 words) - 08:57, 4 March 2024
• Happyshark's User Page
  Hello! This is HappyShark's User Page.
  38 bytes (6 words) - 23:15, 27 January 2024

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
  894 bytes (123 words) - 15:48, 13 June 2022
• 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]].
  3 KB (362 words) - 17:06, 2 April 2024
• 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
  10 KB (1,497 words) - 11:42, 10 March 2024
• Colorado mathematics competitions
  * [[St. Mary's Academy Math Team Challenge]] -- Contests for middle school students. [http
  2 KB (224 words) - 00:51, 29 January 2019
• Georgia mathematics competitions
  *[[Soli Deo Gloria Fall Tournament, aka Billy's Math Meet]] [[User:Solafidefarms/MathMeet]]
  4 KB (565 words) - 13:24, 13 September 2019
• Kansas mathematics competitions
  * [[K-State S. Thomas Parker Mathematics Competition]] [http://www.math.ksu.edu/~zlin/mat
  3 KB (473 words) - 16:11, 16 June 2020
• Kentucky mathematics competitions
  * Governor's Cup [http://www.kaac.com/governors-cup/] * Governor's Cup [http://www.kaac.com/governors-cup/]
  1 KB (129 words) - 14:36, 25 October 2018
• 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
  2 KB (279 words) - 19:41, 23 March 2017
• 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
  2 KB (231 words) - 13:07, 14 December 2019
• 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].
  16 KB (2,152 words) - 21:46, 6 May 2024
• 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].
  4 KB (516 words) - 03:01, 13 April 2023
• 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
  24 KB (3,177 words) - 12:53, 20 February 2024
• 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
  2 KB (215 words) - 02:54, 18 November 2020
• 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
  7 KB (901 words) - 14:11, 6 January 2022
• 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
  4 KB (623 words) - 13:11, 20 February 2024
• 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
  2 KB (258 words) - 19:31, 8 March 2023
• Computer science books
  ...os/ASIN/0387948600/artofproblems-20 The Algorithm Design Manual] by Steve S. Skiena.
  2 KB (251 words) - 00:45, 17 November 2023
• 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.
  10 KB (1,410 words) - 13:07, 20 February 2024
• 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.
  9 KB (1,355 words) - 07:29, 29 September 2021
• 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
  665 bytes (96 words) - 23:17, 2 February 2021
• 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.
  6 KB (902 words) - 12:53, 3 September 2019
• 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>:
  12 KB (1,798 words) - 16:20, 14 March 2023
• 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
  4 KB (613 words) - 13:08, 18 July 2023
• 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
  3 KB (337 words) - 03:35, 7 September 2020
• Math Jam
  ...AMC]] -- AoPS hosts sessions for discussion of the problems from each year's [[AMC 10]], [[AMC 12]], and [[AIME]] exams.
  989 bytes (130 words) - 16:20, 20 August 2020
• 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.
  5 KB (696 words) - 03:47, 24 December 2019
• 1954 AHSME Problems/Problem 46
  label("$E$",(0,0),S);
  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> ...any two different 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
  ...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
  21 KB (3,500 words) - 18:41, 23 April 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 (1,107 words) - 07:35, 26 March 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
• 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);
  2 KB (282 words) - 22:04, 11 July 2008
• 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>
  9 KB (1,703 words) - 07:25, 24 March 2024
• 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
• Combination
  ...ly that you choose the rest. This identity is also the reason why [[Pascal's Triangle]] is symmetrical. * [[Pascal's Triangle]]
  4 KB (615 words) - 11:43, 21 May 2021
• 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
  13 KB (2,048 words) - 15:28, 22 February 2024
• 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
  4 KB (734 words) - 19:19, 10 October 2023
• Ceva's Theorem
  #REDIRECT[[Ceva's theorem]]
  27 bytes (3 words) - 16:06, 9 May 2021
• Menelaus' theorem
  First let's define some masses. * [[Ceva's theorem]]
  5 KB (804 words) - 03:01, 12 June 2023
• Number theory
  ...cs]] associated with studying the properties and identities of [[ integer]]s. *[[Prime number]]s
  3 KB (399 words) - 23:08, 8 January 2024
• Number theory/Intermediate
  ** [[Simon's Favorite Factoring Trick]] ** [[Euler's Totient Theorem]]
  1,016 bytes (108 words) - 21:05, 26 January 2016
• 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:
  2 KB (378 words) - 16:34, 5 January 2010
• ARML historical results
  | [[New York City ARML]] (New York City S)
  19 KB (2,632 words) - 14:31, 12 June 2022
• 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
• 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:
  7 KB (1,296 words) - 14:22, 22 October 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
• 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
  2 KB (275 words) - 12:51, 26 July 2023
• Cyclic quadrilateral
  * [[Ptolemy's Theorem]] * [[Brahmagupta's formula]]
  1 KB (162 words) - 20:39, 9 March 2024
• Vieta's Formula
  #REDIRECT[[Vieta's formulas]]
  29 bytes (3 words) - 00:31, 4 June 2022
• 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.
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]].
  5 KB (849 words) - 16:14, 18 May 2021
• Algebra/Intermediate
  ** [[Vieta's Formulas]] ** [[Newton's Sums]]
  2 KB (198 words) - 17:47, 3 November 2021
• 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
• 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
  2 KB (258 words) - 00:50, 28 December 2021
• 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
  3 KB (538 words) - 13:13, 16 January 2021
• 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
  5 KB (804 words) - 13:54, 26 January 2023
• 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.
  3 KB (496 words) - 23:22, 5 January 2022
• 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 ==
  8 KB (1,315 words) - 18:18, 2 March 2024
• 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.
  5 KB (768 words) - 20:45, 1 September 2022
• 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
  2 KB (291 words) - 16:31, 18 May 2021
• 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.
  4 KB (628 words) - 17:17, 17 May 2018
• Stewart's Theorem
  #REDIRECT[[Stewart's theorem]]
  30 bytes (3 words) - 16:22, 9 May 2021
• 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
  4 KB (635 words) - 12:19, 2 January 2022
• 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
  12 KB (1,896 words) - 23:55, 27 December 2023
• 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
• 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
  5 KB (709 words) - 10:28, 19 February 2024
• 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>
  6 KB (924 words) - 21:50, 8 May 2022
• Binomial Theorem
  ...lected are the same as the entries in the <math>n</math>th row of [[Pascal's Triangle]]. .../math> will be the entries of the <math>n^\text{th}</math> row of [[Pascal's Triangle]]. This is explained further in the Counting and Probability textb
  5 KB (935 words) - 13:11, 20 February 2024
• Complementary counting
  ...10,..., 95</math> or <math>1 \cdot 5, 2 \cdot 5,..., 19 \cdot 5</math>; it's easy to see that there are <math>19</math> of them. Thus, our answer is is Next, we find the possibilities where every house's next-door neighbor is a different color. Using a constructive approach, she
  8 KB (1,192 words) - 17:20, 16 June 2023
• Combinatorial identity
  == Pascal's Identity == Pascal's Identity states that
  12 KB (1,993 words) - 23:49, 19 April 2024
• Circle
  A circle is defined as the [[set]] (or [[locus]]) of [[point]]s in a [[plane]] with an equal distance from a fixed point. The fixed point '''Case 1:''' The circle's area is greater than the triangle's area.
  9 KB (1,581 words) - 18:59, 9 May 2024
• Ellipse
  Equivalently, it is defined as the [[locus]], or [[set]], of all [[point]]s <math>P</math> such that the sum of the distances from <math>P</math> to tw ...They occur in nature as well as in mathematics: as was proven in [[Kepler's Laws]], the planets all revolve about the sun in elliptical, not circular,
  5 KB (892 words) - 21:52, 1 May 2021
• Base numbers
  ...^2</math>'s, and the fourth digit tells us there are two <math>10^3</math>'s. ...M=1000). Imagine how difficult it would be to multiply LXV by MDII! That's why the introduction of the '''Arabic numeral system''', base-10, revolutio
  4 KB (547 words) - 17:23, 30 December 2020
• Base numbers/Common bases
  [[Binary]] is base 2. It's a favorite among computer programmers. It has just two digits: <math>0</mat ...ically count in base 10 with partial conversions to base 8 on the way. Let's multiply <math>12345_8</math> by <math>7_8</math>. <math>5\cdot 7=35_{10}=4
  2 KB (351 words) - 10:39, 1 October 2015
• Henstock-Kurzweil integral
  ...ion of an interval|tagged partition]] on <math>[a,b]</math>, then <math>|L-S(f,\mathcal{\dot{P}})|<\epsilon</math> Here, <math>S(f,\mathcal{\dot{P}})</math> is the [[Riemann sum]] of <math>f</math> on <ma
  2 KB (401 words) - 09:46, 31 January 2018
• 2001 AMC 12 Problems/Problem 1
  The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then Suppose the two numbers are <math>a</math> and <math>b</math>, with <math>a+b=S</math>.
  788 bytes (120 words) - 10:32, 8 November 2021
• 2001 AMC 12 Problems/Problem 2
  Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits of the integer <math>n</math>. For example, <math>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
  1,007 bytes (165 words) - 00:28, 30 December 2023
• 2001 AMC 12 Problems/Problem 3
  Let <math>A</math>, <math>T</math> be Kristin's annual income and the income tax total, respectively. Notice that
  1 KB (203 words) - 01:25, 1 March 2024
• Square (geometry)
  ...ge|sides]] have equal length and all [[angle | angles]] are [[right angle]]s. ...a <math>A</math> of a square with side length <math>s</math> is <math> A = s^2 </math>.
  1 KB (169 words) - 01:12, 13 June 2022
• Scholastic Aptitude Test
  ...tp://www.collegeboard.com/student/testing/sat/about.html The College Board's SAT I Website]
  908 bytes (146 words) - 14:44, 12 December 2023
• Mathematical Kangaroo
  ...d 4 students will take their test (which is distinct from levels 1 and 2’s test), and similarly, up to levels 11 and 12. Levels 1 through 4 tests is ...answered or incorrectly answered questions, so it will be to the student’s advantage to guess questions he or she cannot solve. Thus, the maximum sco
  6 KB (936 words) - 15:38, 22 February 2024
• American Scholastic Mathematics Association
  ...mpetition with the top eight scorers of each team counted towards the team's total. The test is 35 minutes long and assumes the use of a calculator.
  1 KB (153 words) - 13:11, 14 May 2019
• List of United States college mathematics competitions
  * [https://simiode.org/scudem SIMIODE SCUDEM] -- SIMIODE's (Systemic Initiative for Modeling Investigations and Opportunities using Di
  1 KB (149 words) - 22:58, 15 January 2024
• Least common multiple
  ...ite]] [[set]] of integers has an [[infinite]] number of [[common multiple]]s, but only one LCM. The LCM of a set of numbers <math>\{a_1,a_2,\cdots,a_n\} Let's use our first example. The GCD of 4 and 6 is 2. Using the above equation,
  2 KB (383 words) - 10:49, 4 September 2022
• Common factorizations
  == Vieta's/Newton Factorizations == ...e a polynomial, and ask a question about the roots. Combined with [[Vieta's formulas]], these are excellent, useful factorizations.
  2 KB (327 words) - 02:06, 28 April 2024
• Math Day at the Beach
  ...eam works together on this round, and submits one set of answers. The team's score is 10 times the number of correct answers, for a maximum of 80 points ...ly. The winner adds 10 points to his or her individual score (and his team's team score) and the 2nd place individual adds 5 points. While these point t
  4 KB (644 words) - 12:56, 29 March 2017
• 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
• Cevian
  label("$D$",(70,0),S); label("$D$",(70,0),S);
  1 KB (194 words) - 01:35, 19 June 2018
• Median of a triangle
  label("$M$", M, S); [[Stewart's Theorem]] applied to the case <math>m=n</math>, gives the length of the med
  1 KB (185 words) - 20:24, 6 March 2024
• Pi
  ...al to 3.141592653. The number pi is one of the most important [[constant]]s in all of mathematics and appears in some of the most surprising places, su ...results in mathematics since it involves five of the greatest [[constant]]s: [[e]], pi, [[i]], [[unity | 1]], and [[zero (constant)| 0]].
  8 KB (1,469 words) - 21:11, 16 September 2022
• Phi
  * <math>\phi</math> is also commonly used to represent [[Euler's totient function]].
  2 KB (302 words) - 14:04, 1 January 2024
• Fibonacci sequence
  The '''Fibonacci sequence''' is a [[sequence]] of [[integer]]s in which the first and second terms are both equal to 1 and each subsequent ...]] with constant coefficients. There is also an explicit formula [[#Binet's formula|below]].
  6 KB (957 words) - 23:49, 7 March 2024
• Chain Rule
  There's another way to look at it: The following is a proof of the multi-variable Chain Rule. It's a "rigorized" version of the intuitive argument given above.
  12 KB (2,377 words) - 11:48, 22 July 2009
• 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
• Cardinality
  ...t]]s. For [[finite]] sets, the cardinality of is the number of [[element]]s in that set, i.e. the size of the set. The cardinality of <math>\{3, 4\}</ ...can reasonably talk about the least cardinal in bijection with a set <math>S</math>. In the absence of <math>\sf{AC}</math>, one can define cardinals us
  2 KB (263 words) - 00:54, 17 November 2019
• 1988 AJHSME Problems/Problem 5
  label("B",B,S);
  1 KB (160 words) - 16:53, 17 December 2020
• Fundamental Theorem of Calculus
  This section is for people who know what [[integral]]s are but don't know the Fundamental Theorem of Calculus yet, and would like ...ight line, and its velocity at time <math>{t}</math> is <math>t^3</math> m/s. Exactly how far does the object go between times <math>t=2</math> sec and
  11 KB (2,082 words) - 15:23, 2 January 2022
• Writing scholarships
  *[https://lawyerbound.com/scholarship LawyerBound’s “Spokane Community Scholarship 2023] applicants must be planning on atten ...tion to reduce the occurrence of fatal accidents for tourists on Florida’s roadways.
  7 KB (1,039 words) - 18:45, 18 January 2024
• National Scholarships
  ...niors gain admission and full four-year scholarships to some of the nation's most selective colleges. [http://www.questbridge.org/ website] * [[KFC Colonel's Scholars Program]] [http://www.kfcscholars.org/ website]
  4 KB (538 words) - 00:48, 28 January 2024
• 2011 AMC 10B Problems/Problem 22
  ...th>s\sqrt{2}</math> because the longer length was the diagonal of the cube's base and the shorter length was a side of the cube. label("$1$",(A--C),NW); label("$1$",(B--C),NE); label("$\sqrt{2}$",(A--B),S);
  4 KB (691 words) - 18:38, 19 September 2021
• Semiperimeter
  ...</math> is the total perimeter of a figure. It is typically denoted <math>s</math>. ...math> is the [[area]] of a [[triangle]] and <math>r</math> is the triangle's [[inradius]] (that is, the [[radius]] of the [[circle]] [[inscribed]] in th
  641 bytes (97 words) - 00:28, 31 December 2020
• 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
• Vector
  MC(-10,"\vec{v}+\vec{w}",D((0,0)--(5,-1),red+p,Arrow),S); where <math>\hat{i},\hat{j},\hat{k}</math> are [[unit vector]]s along the coordinate axes, or equivalently, <math>\bold{a}\times\bold{b}=\l
  7 KB (1,265 words) - 13:22, 14 July 2021
• Logarithm
  ...iscrete logarithm, used in [[cryptography]] via [[modular arithmetic]]. It's the lowest value <math>c</math> such that <math>a^c=mx+b</math> for given < It's related to the usual logarithm by the fact that if <math>b</math> isn't an
  4 KB (680 words) - 12:54, 16 October 2023
• Law of Cosines
  ...of Cosines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle]]. It can be derived in several different ways, the most co ...length <math>a</math>, <math>b</math> and <math>c</math> opposite [[angle]]s of measure <math>A</math>, <math>B</math> and <math>C</math>, respectively,
  6 KB (1,003 words) - 09:11, 7 June 2023
• 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
  2 KB (398 words) - 16:57, 29 December 2021
• Trigonometric identities
  [[Euler's identity]] states that <math>e^{ix} = \cos (x) + i \sin(x)</math>. We have === Euler's identity ===
  8 KB (1,397 words) - 21:55, 20 January 2024
• Iff
  ...emsolving.com/wiki/index.php/Godel%27s_First_Incompleteness_Theorem Gödel's Incompleteness Theorem]
  903 bytes (140 words) - 02:13, 24 December 2020
• Vieta's
  #REDIRECT[[Vieta's formulas]]
  29 bytes (3 words) - 00:30, 4 June 2022
• Vocabulary and spelling competitions
  * [[Reader's Digest National Word Power Challenge | Word Power Challenge]] The premier v
  786 bytes (99 words) - 17:53, 22 June 2006
• Irrational number
  ...[real number]] that cannot be expressed as the [[ratio]] of two [[integer]]s. Equivalently, an irrational number, when expressed in [[decimal notation] Because the [[rational number]]s are [[countable]] while the reals are [[uncountable]], one can say that the
  3 KB (368 words) - 19:26, 6 June 2015
• Complex plane
  The '''complex plane''' is one representation of the [[complex number]]s. It is a [[coordinate plane]] with two perpendicular axes, the real axis ( * [[De Moivre's Theorem]]
  892 bytes (134 words) - 16:52, 3 September 2017
• Complex number
  The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...lex numbers contains the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.
  5 KB (860 words) - 15:36, 10 December 2023
• Cis
  '''Cis''' notation is a [[polar form | polar]] notation for [[complex number]]s. For all complex numbers <math>z</math>, we can write <math>z=r\mathrm{cis ...\theta}</math> rather than <math>r\mathrm{cis }(\theta)</math>, as [[Euler's formula]] states that
  1 KB (171 words) - 20:59, 11 July 2023
• Vocabulary competitions
  * [[Reader's Digest National Word Power Challenge | Word Power Challenge]] The premier v
  392 bytes (50 words) - 02:36, 29 November 2018
• Georgia ARML
  ...everal years of strong performances by the second team, and the third team's seventh place finish in the B division in 2011. Historically, in 1992 Georg
  1 KB (186 words) - 11:52, 4 June 2012
• Roots of unity
  ...ts of unity come up when we examine the [[complex number|complex]] [[root]]s of the [[polynomial]] <math> x^n=1 </math>. ** This is an immediate result of [[Vieta's formulas]] on the polynomial <math> x^n-1 = 0 </math> and [[Newton sums]].
  3 KB (558 words) - 21:36, 11 December 2011
• Angle bisector
  ...XYZ</math> with [[incenter]] ''I'', [[incircle]] (blue), [[angle bisector]]s (orange), and [[angle bisector|external angle bisectors]] (green)}} * [[Stewart's Theorem]]
  3 KB (575 words) - 15:27, 19 March 2023
• Mathew Crawford
  Crawford's user page can be found [[user:MCrawford | here]].
  2 KB (360 words) - 02:20, 2 December 2010
• Geometric sequence
  ...= a_1(r^n-1).</cmath> Dividing both sides by <math>r-1</math> yields <math>S=\frac{a_1(r^n-1)}{r-1}</math>, as desired. <math>\square</math> ...a_1r^2 + \cdots = S.</cmath> Thus, <math>rS + a_1 = S</math>, and so <math>S = \frac{a_1}{1-r}</math>. <math>\square</math>
  4 KB (644 words) - 12:55, 7 March 2022
• Arithmetic sequence
  ...+ (a_1 + a_n) + \cdots + (a_1 + a_n) = n(a_1 + a_n),</cmath> and so <math>S = \frac{n(a_1 + a_n)}{2}</math>, as required. <math>\square</math>
  4 KB (736 words) - 02:00, 7 March 2024
• 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
• 2005 USAMO Problems/Problem 4
  ...s <math>L_2</math>, <math>L_3</math>, and <math>L_4</math> iff the problem's condition is met. ...math>A_2=\dfrac{k_3-k_4}{s(k_2+k_3-k_4)}</math>, <math>B_2=\dfrac{k_2-k_4}{s(k_2+k_3-k_4)}</math>, <math>C_2=\dfrac{1}{k_2+k_3-k_4}</math>.
  7 KB (1,276 words) - 20:51, 6 January 2024
• Diophantine equation
  See also: [https://en.wikipedia.org/wiki/B%C3%A9zout%27s_identity Bézout's identity]. A Pythagorean triple is a set of three [[integer]]s that satisfy the [[Pythagorean Theorem]], <math>a^2+b^2=c^2</math>. There a
  9 KB (1,434 words) - 13:10, 20 February 2024
• Area
  ...d the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elementary means. One can find the area of even more complex regio [[Rectangle]]s are the most basic figures whose area we can study. It makes sense that th
  6 KB (1,181 words) - 22:37, 22 January 2023
• Exponential form
  ...where <math>r=|z| = \sqrt{a^2 + b^2}</math>. By [[Euler's identity|Euler's formula]], which states that <math>e^{i\theta}=\cos\theta+i\sin\theta</math label("Re",2*C/3,S);
  1 KB (238 words) - 22:51, 20 February 2022
• Floor function
  * [[Hermite's Identity]]: <cmath>\lfloor na\rfloor = \left\lfloor a\right\rfloor+\left\lf * How many of the first 1000 [[positive integer]]s can be expressed in the form
  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 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
• 1985 IMO Problems/Problem 5
  ==Solution 3 (No Miquel's point)== *[[Miquel's point]]
  3 KB (496 words) - 13:35, 18 January 2023
• Exponentiation
  ...efinition) defines the operation of "multiplication by [[positive integer]]s." We can then extend the notion of multiplication to non-integers. ...ch as <math>2^{-4}</math>? How do we multiply 2 by itself -4 times!? Let's think about what a negative sign means a little more. When we append a neg
  5 KB (803 words) - 16:25, 10 August 2020
• Rational approximation
  ...n [[real number]] <math>x</math> can be approximated by [[rational number]]s. Of course, since the rationals are dense on the real line, we, surely, can ==Dirichlet's theorem==
  7 KB (1,290 words) - 12:18, 30 May 2019
• Physics summer programs
  == U.S. Physics Summer Programs == * [[Art of Problem Solving's 8-month online Olympiad-level physics course (PhysicsWOOT)]] [https://artof
  3 KB (441 words) - 05:22, 3 March 2024
• Trigonometry
  ...[[Law of Sines]] and the [[Law of Cosines]]; many more, such as [[Stewart's Theorem]], are most easily proven using trigonometry. In algebra, expressio A common mnemonic to remember this is '''SOH-CAH-TOA''', where '''S'''ine = '''O'''pposite / '''H'''ypotenuse, '''C'''osine = '''A'''djacent /
  8 KB (1,217 words) - 20:15, 7 September 2023
• 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
• Mathematical Olympiad Summer Program
  ...with the first three kinds of invitations. Black MOP consists of that year's USAMO winners and contains the IMO team members and alternates. Blue MOP is ...ng practice test for an average of roughly 9 hours a day of math- and that's before time spent doing problem sets and working on the team contest outsid
  6 KB (936 words) - 10:37, 27 November 2023
• Complex analysis
  ...f complex numbers would be quite similar to the calculus of [[real number]]s, but, amazingly, this turns out to be not the case. There are many patholog == Liouville's Theorem ==
  2 KB (271 words) - 22:06, 12 April 2022
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...ric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says: ...g radicals because the 0th root of any number is undefined when the number's absolute value is greater than or equal to 1. This creates the indeterminat
  5 KB (912 words) - 20:06, 14 March 2023
• Goldbach Conjecture
  ...every [[even integer]] greater than two is the sum of two [[prime number]]s. The conjecture has been tested up to 400,000,000,000,000. Goldbach's conjecture is one of the oldest unsolved problems in [[number theory]] and
  7 KB (1,201 words) - 16:59, 19 February 2024
• Twin Prime Conjecture
  ...m]]) that states that there are [[infinite]]ly many pairs of [[twin prime]]s, i.e. pairs of primes that differ by <math>2</math>. ...infinitude of twin primes is an idea adopted from the proof of [[Dirichlet's Theorem]]. If one can show that the sum
  2 KB (308 words) - 02:27, 1 May 2024
• Fallacy
  === Polya's Proof That All horses Are the Same Color === ...es in a group of 1 horse have the same color to be true. Of course, there's only 1 horse in the group so certainly our base case holds.
  2 KB (429 words) - 08:27, 5 June 2013
• Riemann Hypothesis
  ...e zeta function, it is easy to see that <math>\zeta(s)=0</math> when <math>s=-2,-4,-6,\ldots</math>. These are called the trivial zeros. This hypothesi ...Riemann Hypothesis is an important problem in the study of [[prime number]]s. Let <math>\pi(x)</math> denote the number of primes less than or equal to
  2 KB (425 words) - 12:01, 20 October 2016
• Quadratic residues
  Let <math>a</math> and <math>m</math> be [[integer]]s, with <math>m\neq 0</math>. We say that <math>a</math> is a '''quadratic re Euler's criterion: <math>\left(\frac{a}{p}\right) \equiv a^{\frac{p-1}{2}} \mod p</
  5 KB (778 words) - 13:10, 29 November 2017
• Power of a Point Theorem
  ...holds between the lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...re. Let two arbitrary lines passing through <math>P</math> intersect <math>S</math> at <math>A_1,B_1;A_2,B_2</math>, respectively. Then
  5 KB (827 words) - 17:30, 21 February 2024
• Fundamental Theorem of Algebra
  ...onconstant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves === Proof by Liouville's Theorem ===
  5 KB (832 words) - 14:22, 11 January 2024
• Majorization
  We say a [[nonincreasing]] [[sequence]] of [[real number]]s <math> a_1, \ldots ,a_n</math> '''majorizes''' another nonincreasing sequen * [[Karamata's Inequality]]
  2 KB (288 words) - 22:48, 5 July 2023
• International Mathematical Olympiad
  ...Olympiad''' is the pinnacle of all high school [[mathematics competition]]s and the oldest of all international scientific competitions. Each year, co ...ally, however, the scores of each team are compared each year where a team's score is the sum of their individual scores.
  3 KB (490 words) - 03:32, 23 July 2023
• Prime Number Theorem
  the exception of [[Fermat's Last Theorem]]. (Fortunately, the proof asymptotic formula for the distribution of the [[prime number]]s;
  10 KB (1,729 words) - 19:52, 21 October 2023
• ''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
• Infinite
  ...surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. ...inite if it can be put into [[bijection]] with one of its proper [[subset]]s.
  1 KB (186 words) - 23:19, 16 August 2013
• 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
• Well-Ordering theorem
  ...y ordered set]] <math>(S,\prec)</math> for which each set <math>A\subseteq S</math> has a [[minimum]] element. ...Well-Ordering theorem is equivalent to the [[Axiom of choice]] and [[Zorn's Lemma]].
  381 bytes (59 words) - 12:40, 2 June 2019
• Rectangle
  ...angle''' is a [[quadrilateral]] in which all [[angle]]s are [[right angle]]s. ...n addition, rectangles have [[congruent (geometry)|congruent]] [[diagonal]]s.
  782 bytes (108 words) - 13:36, 6 March 2022
• Mock AMC
  * Look at famous theorems and formulas and see if there's any way you can make a good problem out of them. ! scope="row" | '''Mock AMC S'''
  51 KB (6,175 words) - 20:58, 6 December 2023
• Mock AIME
  ...oblemsolving.com/community/c5t183f5h1074599_trumpeters_mock_aime Trumpeter's Mock AIME] ** [https://artofproblemsolving.com/community/c594864h2441992 Treemath's Mock AIME]
  8 KB (906 words) - 17:30, 26 April 2024
• Commutative property
  ...lex number]]s and its [[subset]]s such as the [[real number]]s, [[integer]]s, etc.) because <math>\displaystyle a + b = b + a</math>. However, the oper ...ath>G: S \to S</math> is commutative if and only if <math>\forall a, b \in S, G(a, b) = G(b, a)</math>.
  2 KB (301 words) - 17:46, 16 March 2012
• Permutation
  ...ermutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself. ...<math>r</math>-element [[subset]] of a set with <math>n</math> [[element]]s, where order matters. To find how many ways we can do this, note that for
  3 KB (422 words) - 11:01, 25 December 2020
• Algebraic topology
  ...we can associate various algebraic objects, such as [[group]]s and [[ring]]s. ...ath> that start and end at <math>x</math>, i.e. all [[continuous function]]s <math>f:[0,1]\to X</math> with <math>f(0)=f(1)=x</math>. Call this collecti
  3 KB (479 words) - 15:35, 1 December 2015
• 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
• Andrew Wiles
  ...sity]]. He is most widely known as the mathematician who proved [[Fermat's Last Theorem]].
  312 bytes (42 words) - 21:59, 2 May 2018
• Riemann zeta function
  <cmath>\zeta (s)=\sum_{n=1}^{\infty}\frac{1}{n^s}= 1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\cdots</cmath>
  9 KB (1,547 words) - 03:04, 13 January 2021
• Modular arithmetic/Introduction
  Let's use a clock as an example, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>. Now let's look back at this solution, using modular arithmetic from the start. Note
  15 KB (2,396 words) - 20:24, 21 February 2024
• Convex function
  * [[Jensen's Inequality]] * [[Karamata's Inequality]]
  2 KB (417 words) - 00:10, 20 February 2016
• Uncountable
  ...The most common example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. ...)|</math>, where <math>\mathcal{P}(S)</math> is the [[power set]] of <math>S</math>. First, we note that the [[Cantor set]] <math>\mathcal{C}</math> has
  2 KB (403 words) - 20:53, 13 October 2019
• Modular arithmetic/Intermediate
  ...ers seem to fail in <math>\mathbb{Z}_{21}</math>? To understand this, let's take a closer look at the congruence * [[Fermat's Little Theorem]]
  14 KB (2,317 words) - 19:01, 29 October 2021
• Right triangle
  label("$a$", midpoint(C--B), S); ...re very useful in [[geometry]] and for finding the [[area]]s of [[polygon]]s. The most important relationship for right triangles is the [[Pythagorean
  3 KB (499 words) - 23:41, 11 June 2022
• Subset
  ...or <math>2^S</math>. The number of subsets of <math>S</math> is <math>2^{|S|}</math>.
  1 KB (217 words) - 09:32, 13 August 2011
• Limit
  ...rty. For example, consider the function <math>f(x)</math> over the [[real]]s defined as follows: <cmath>f(x) = \begin{cases} 0 & \text{if } x\neq 0,\\ 1 ...>. Here <math>d_A</math> and <math>d_B</math> are the [[distance function]]s of <math>A</math> and <math>B</math>, respectively.
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2006 AIME I Problems
  ...f <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math> Let <math> N </math> be the number of consecutive 0's at the right end of the decimal representation of the product <math> 1!2!3!
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 14
  triple M=(B+C)/2,S=(4*A+T)/5; draw(T--S--B--T--C--B--S--C);draw(B--A--C--A--S,ddash);draw(T--O--M,ddash);
  6 KB (980 words) - 21:45, 31 March 2020
• 2006 AIME I Problems/Problem 13
  ...ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math> but not by <math>2^2</math>. We are certainly not going to expand all of this out... so let's look for patterns from these <math>4</math> values!
  10 KB (1,702 words) - 00:45, 16 November 2023
• LaTeX:Symbols
  |<math>\S</math>||\S||<math>\P</math>||\P||<math>\Vdash</math>||\Vdash Thx for taking the time to scroll through and read all this... Here's a treat: https://t3.ftcdn.net/jpg/01/65/99/48/360_F_165994815_BzUNaOCtcf2jV
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2006 AIME I Problems/Problem 10
  ...\mathcal{R} </math> into two regions of equal area. Line <math> l </math>'s equation can be expressed in the form <math> ax=by+c, </math> where <math> Assume that if unit [[square]]s are drawn circumscribing the circles, then the line will divide the area of
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 9
  ...log_8 (a^{12}r^{66})=2006</math> for <math>a, r</math> [[positive integer]]s. <math>a^{12}r^{66}=8^{2006} = (2^3)^{2006} = (2^6)^{1003}</math> so <math> ...h>x=1</math> because <math>1001=91*11</math>. Because only [[even integer]]s are being subtracted from <math>1003</math>, the numerator never equals an
  4 KB (651 words) - 18:27, 22 May 2021
• 2006 AIME I Problems/Problem 8
  ...as shown. Rhombuses <math> \mathcal{P, Q, R,} </math> and <math> \mathcal{S} </math> are [[congruent (geometry) | congruent]], and each has [[area]] <m label("$\mathcal{S}$",(4.2,-2.2),SW);
  5 KB (730 words) - 15:05, 15 January 2024
• 2006 AIME I Problems/Problem 7
  An [[angle]] is drawn on a set of equally spaced [[parallel]] [[line]]s as shown. The [[ratio]] of the [[area]] of shaded [[region]] <math> C </mat label("$\mathcal{A}$", A+0.2*dir(-17), S);
  4 KB (709 words) - 01:50, 10 January 2022
• 2006 AIME I Problems/Problem 6
  ...> are distinct [[digit]]s. Find the sum of the elements of <math> \mathcal{S}. </math> ...solution can be determined by dividing the total number of [[permutation]]s by 2. The answer is <math>\frac{10 \cdot 9 \cdot 8}{2} = \frac{720}{2}= \bo
  2 KB (237 words) - 19:14, 20 November 2023
• 2006 AIME I Problems/Problem 4
  Let <math> N </math> be the number of consecutive <math>0</math>'s at the right end of the decimal representation of the product <math> 1!2!3!
  2 KB (278 words) - 08:33, 4 November 2022
• 2006 AIME I Problems/Problem 3
  Here's another way to finish using this solution. From the above, you have <cmath>
  4 KB (622 words) - 03:53, 10 December 2022
• 2006 AIME I Problems/Problem 2
  ...f <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math> ...>4995</math> are possible values of S, so the number of possible values of S is <math>4995-4095+1=901</math>.
  1 KB (189 words) - 20:05, 4 July 2013
• MediaWiki
  * [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide]
  505 bytes (73 words) - 01:18, 10 November 2023
• 2021 JMC 10
  ...'binary'' if all of the digits are either <math>0</math>s or <math>1</math>s with leading zeros allowed. How many days in a year are binary? ...they are both at the same point, not necessarily a vertex. What is the ant's expected lifespan in seconds?
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12B Problems
  ...wo sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? ...he resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2006 AMC 12A Problems
  Sandwiches at Joe's Fast Food cost <math>3</math> dollars each and sodas cost <math>2</math> do The ratio of Mary's age to Alice's age is <math>3:5</math>. Alice is <math>30</math> years old. How old is Mar
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  ...had ridden for twice the length of time as Mike and at four-fifths of Mike's rate. How many miles had Mike ridden when they met? ...tangent to both axes and to the second and third circles. What is <math>r/s</math>?
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  ...</math> is deducted to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes? ...ghters and granddaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  ...math> is in <math>S</math>, what is the smallest number of points in <math>S</math>? label("$M$",M,S);
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  ...</math> arc of circle B. What is the ratio of circle A's area and circle B's area? ...set <math>\{1, 2, \ldots, 10\}</math>. What is the probability that Sergio's number is larger than the sum of the two numbers chosen by Tina?
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2000 AMC 12 Problems
  ...ence <math>1,1,2,3,5,8,13,21,\ldots </math> starts with two <math>1</math>'s, and each term afterwards is the sum of its two predecessors. Which one of label("Figure",(0.5,-1),S);
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2001 AMC 12 Problems
  The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then Let <math>P(n)</math> and <math>S(n)</math> denote the product and the sum, respectively, of the digits
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2003 AMC 12B Problems
  ...y for two weeks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one gree Penniless Pete's piggy bank has no pennies in it, but it has 100 coins, all nickels, dimes,
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2004 AMC 12B Problems
  ...border she exchanged them all, receiving 10 Canadian dollars for every 7 U.S. dollars. After spending 60 Canadian dollars, she had <math>d</math> Canadi label("$N$",N,S);
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  At the beginning of the school year, Lisa's goal was to earn an A on at least <math>80\%</math> of her <math>50</math> Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers f
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 3
  ...rgin of 14 points, then the Panthers' score would be half of (34-14). That's 10 <math>\Rightarrow \boxed{\text{(A)}}</math>.
  910 bytes (136 words) - 13:39, 13 February 2016
• 2006 AMC 12B Problems/Problem 7
  ...wo sit in the back. Either Mr. Lopez or Mrs. Lopez must sit in the driver's seat. How many seating arrangements are possible? There are only two possible occupants for the driver's seat. After the driver is chosen, any of the remaining three people can sit
  1 KB (213 words) - 15:33, 9 April 2024
• 2006 AMC 12B Problems/Problem 9
  Let's set the middle (tens) digit first. The middle digit can be anything from 2-
  3 KB (409 words) - 17:10, 30 April 2024
• 2006 AMC 12B Problems/Problem 11
  ...he resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee?
  927 bytes (137 words) - 10:45, 4 July 2013
• 2006 AMC 12B Problems/Problem 15
  ...ath> and <math>HP</math> has a length of <math>2</math>, so by pythagorean's, <math>OH</math> is <math>\sqrt{32}</math>.
  3 KB (458 words) - 16:40, 6 October 2019
• 2006 AMC 12B Problems/Problem 16
  ...tead, we do not need to know the value of the apothem. We could just apply s, which is the side length in this problem, <math>\frac{5\sqrt{6}}{3}</math>
  1 KB (203 words) - 16:36, 18 September 2023
• 2006 AMC 12B Problems/Problem 19
  ...n to be my age." Which of the following is not the age of one of Mr. Jones's children? If <math>b=1</math>, the number is not divisible by <math>2</math> (unless it's <math>1818</math>, which is not divisible by <math>4</math>), which means t
  4 KB (696 words) - 09:47, 10 August 2015
• 2006 AMC 12B Problems/Problem 21
  ...> and because <math>a=\frac{l+w}{2}</math>, or one-fourth of the rectangle's perimeter, we multiply by four to get an answer of <math>\boxed{8\sqrt{1003
  2 KB (339 words) - 13:15, 12 July 2015
• 2006 AMC 12B Problems/Problem 22
  To see how we can do better, let's rearrange the terms as follows:
  5 KB (881 words) - 15:52, 23 June 2021
• 2006 AMC 12B Problems/Problem 23
  ...math>\overline{AC}</math> and <math>\overline{BC}</math> have length <math>s=\sqrt{a+b\sqrt{2}}</math>, where <math>a</math> and <math>b</math> are posi MP("s",(A+C)/2,plain.S,f);
  7 KB (1,169 words) - 14:04, 10 June 2022
• 2006 AMC 12B Problems/Problem 24
  ...0 \le y \le \frac{\pi}{2}</math>. What is the area of the subset of <math>S</math> for which <cmath> MP("\frac{\pi}{6}", (1,0), plain.S);
  3 KB (563 words) - 22:45, 24 October 2021
• 2006 AMC 12B Problems/Problem 25
  Let's keep in mind that <math>2006 \equiv 2 \pmod 3</math> and that <math>a_1 = 9
  5 KB (924 words) - 12:02, 15 June 2022
• 2006 AMC 12A Problems/Problem 14
  Bezout's Lemma:
  3 KB (442 words) - 03:13, 8 August 2022
• 2006 AMC 12A Problems/Problem 17
  ...ent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>? real s = 90;
  6 KB (958 words) - 23:29, 28 September 2023
• 2006 AMC 12A Problems/Problem 23
  ...S=(a_1,a_2,\ldots ,a_n)</math> of <math>n</math> real numbers, let <math>A(S)</math> be the sequence .../math>, and let <math>S=(1,x,x^2,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>?
  3 KB (466 words) - 22:40, 29 September 2023
• 2006 AMC 12A Problems/Problem 24
  ...math> is even or <math>b</math> is even and <math>c</math> is odd. Now let's do some casework to see how many terms fit this criteria:
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  How many non-[[empty set | empty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> have the following two properti <math>(1)</math> No two consecutive [[integer]]s belong to <math>S</math>.
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 25
  ...ow many [[equilateral]] [[triangle]]s all have their [[vertices]] in <math>S</math>? label("$x=2$",(1,0,0),S);
  4 KB (498 words) - 00:46, 4 August 2023
• 2005 AMC 12B Problems/Problem 3
  Let <math>m = </math> Brianna's money. We have <math>\frac15 m = \frac13 (\mbox{CDs}) \Rightarrow \frac35
  978 bytes (156 words) - 14:14, 14 December 2021
• 2005 AMC 12B Problems/Problem 4
  At the beginning of the school year, Lisa's goal was to earn an <math>A</math> on at least <math>80\%</math> of her <ma ...>A</math>'s on <math>22</math> quizzes, so she needs to get <math>A</math>'s on <math>40-22=18</math> more. There are <math>50-30=20</math> quizzes lef
  1 KB (197 words) - 14:16, 14 December 2021
• 2005 AMC 12B Problems/Problem 6
  == Solution 3 (Stewart's Theorem) == Let <math>BD=k</math>. Then, by [[Stewart's Theorem]],
  2 KB (299 words) - 15:29, 5 July 2022
• 2005 AMC 12B Problems/Problem 7
  label("$x$",(5,0),S); label("4",(4,0),S);
  2 KB (357 words) - 20:15, 27 December 2020
• 2005 AMC 12B Problems/Problem 12
  ...px + m = 0</math> are <math>a</math> and <math>b</math>, then using Vieta's formulas, * [[Vieta's Formulas]]
  2 KB (317 words) - 12:27, 16 December 2021
• 2005 AMC 12B Problems/Problem 14
  label("$x$",(Xmax+0.25,0),S);
  2 KB (278 words) - 21:12, 24 December 2020

View (previous 250 | next 250) (20 | 50 | 100 | 250 | 500)