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• Wilson's Theorem
  In [[number theory]], '''Wilson's Theorem''' states that if [[integer ]]<math>p > 1</math> , then <math>(p-1) ...e. Consider the [[field]] of integers modulo <math>p</math>. By [[Fermat's Little Theorem]], every nonzero element of this field is a root of the [[po
  4 KB (639 words) - 01:53, 2 February 2023
• Heron's Formula
  '''Heron's Formula''' (sometimes called Hero's formula) is a [[mathematical formula | formula]] for finding the [[area]] o <math>A=\sqrt{s(s-a)(s-b)(s-c)}</math>
  4 KB (675 words) - 00:05, 22 January 2024
• Simon's Favorite Factoring Trick
  Simon's Favorite Factoring Trick (SFFT) (made by AoPS user [https://artofproblemsol ...t 1, then divide the coefficient off of the equation.). According to Simon's Favorite Factoring Trick, this equation can be transformed into: <cmath>(x+
  7 KB (1,107 words) - 07:35, 26 March 2024
• Vieta's Formulas
  #REDIRECT[[Vieta's formulas]]
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• Mill's Constant
  Mill's Constant is defined as the smallest real number <math>\theta</math> such th ...smallest element in that set. If the [[Riemann Hypothesis]] is true, Mill's constant is approximately <math>1.3063778838630806904686144926...</math> an
  794 bytes (105 words) - 01:59, 15 January 2022
• Ceva's Theorem
  #REDIRECT[[Ceva's theorem]]
  27 bytes (3 words) - 16:06, 9 May 2021
• Fermat's Little Theorem
  '''Fermat's Little Theorem''' is highly useful in [[number theory]] for simplifying the A frequently used corollary of Fermat's Little Theorem is <math>a^p \equiv a \pmod {p}</math>. As you can see, it i
  16 KB (2,658 words) - 16:02, 8 May 2024
• Chebyshev's Inequality
  '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if ...nce of the [[Rearrangement inequality]], which gives us that the sum <math>S=a_1b_{i_1}+a_2b_{i_2}+...+a_nb_{i_n} </math> is maximal when <math>i_k=k</m
  1 KB (214 words) - 20:32, 13 March 2022
• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]] Let <math>\phi(n)</math> be [[Euler's totient function]]. If <math>n</math> is a positive integer, <math>\phi{(n)
  3 KB (542 words) - 17:45, 21 March 2023
• Brahmagupta's Formula
  .../math>, <math>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. <cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath>
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's Theorem
  #REDIRECT[[Ptolemy's theorem]]
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• Ptolemy's theorem
  ...[equality condition | equality case]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving i ...ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  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]]
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• Fermat's Last Theorem
  '''Fermat's Last Theorem''' is a recently proven [[theorem]] stating that for positive Fermat's Last Theorem was proposed by [[Pierre de Fermat]] in the <math>1600s</math>
  3 KB (453 words) - 11:13, 9 June 2023
• Pascal's triangle
  '''Pascal's triangle''' is a triangle which contains the values from the [[binomial exp ...oose k}}=2^n</math>, the sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.
  5 KB (838 words) - 17:20, 3 January 2023
• Newton's Sums
  ...sums''' give us a clever and efficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive sev Newton's sums tell us that,
  4 KB (690 words) - 13:11, 20 February 2024
• Euler's number
  '''Euler's number''' is a [[constant]] that appears in a variety of mathematical conte An approximation for Euler's number is <math>e\approx 2.7182818284590452...</math>
  4 KB (764 words) - 21:09, 13 March 2022
• ''See also: [[Hölder's inequality]]'' ...n, \dotsc, z_1, z_2, \dotsc, z_n</math> are [[nonnegative]] [[real number]]s and <math>\lambda_a, \lambda_b, \dotsc, \lambda_z</math> are nonnegative re
  4 KB (774 words) - 12:12, 29 October 2016
• Euler's Polyhedral Formula
  ...>F</math> denote the number of [[vertex|vertices]], [[edge]]s, and [[face]]s, respectively. Then <math>V-E+F=2</math>. Apply Euler's Polyhedral Formula on the following polyhedra:
  1,006 bytes (134 words) - 14:15, 6 March 2022
• Bertrand's Postulate
  '''Bertrand's postulate''' states that for any [[positive integer]] <math>n</math>, there It is similar to the proof of Chebyshev's estimates in the [[Prime Number Theorem|prime number theorem]] article but
  2 KB (309 words) - 21:43, 11 January 2010
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] its [[Laplace transform]] <math>F(s) = \int_0^\infty f(t)e^{-st}dt</math>
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• Euler's identity
  '''Euler's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>. It is na ...ing problems involving [[complex numbers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates
  3 KB (452 words) - 23:17, 4 January 2021
• Holder's Inequality
  33 bytes (4 words) - 14:53, 29 November 2018
• Hölder's inequality
  #REDIRECT [[Hölder's Inequality]]
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• Rolle's Theorem
  '''Rolle's theorem''' is an important theorem among the class of results regarding the <LI>[[Lagrange's mean value theorem]]</LI>
  1 KB (270 words) - 12:13, 30 May 2019
• Pick's Theorem
  '''Pick's Theorem''' expresses the [[area]] of a [[polygon]], all of whose [[vertex |
  2 KB (301 words) - 13:08, 20 February 2024
• Euclid's Lemma
  '''Euclid's Lemma''' is a result in [[number theory]] attributed to [[Euclid]]. It stat ...ies that <math>p \mid a</math> or <math>p\mid b</math>, for all [[integer]]s <math>a</math> and <math>b</math>.
  1 KB (188 words) - 04:18, 20 November 2023
• Euclid's proof of the infinitude of primes
  ...ek mathematician [[Euclid]] that there are infinitely many [[prime number]]s.
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• De Moivre's Theorem
  '''DeMoivre's Theorem''' is a very useful theorem in the mathematical fields of [[complex This is one proof of De Moivre's theorem by [[induction]].
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• Power's of 2 in pascal's triangle
  == Pascal's Triangle == Pascal's Triangle is a triangular array of numbers where each number is the sum of t
  2 KB (341 words) - 16:57, 16 June 2019
• Largange's Interpolation Formula
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• Viete's sums
  #REDIRECT[[Vieta's formulas]]
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• Newton's Inequality
  ...math> in the polynomial <math> \prod_{i=1}^{n}(t+x_i) </math> (see [[Viete's sums]]). We define the ''symmetric average'' <math>d_k </math> to be <math ...have a root between <math>x_i </math> and <math>x_{i+1} </math> by [[Rolle's theorem]] if <math>x_i \neq x_{i+1} </math>, and if <math> x_i = x_{i+1} =
  5 KB (830 words) - 04:05, 28 January 2023
• Maclaurin's Inequality
  ...ymmetric polynomial]]s. For notation and background, we refer to [[Newton's Inequality]]. By the lemma from [[Newton's Inequality]], it suffices to show that for any <math>n </math>,
  992 bytes (146 words) - 16:48, 29 December 2021
• Nesbitt's Inequality
  '''Nesbitt's [[Inequality]]''' is a theorem which, although rarely cited, has many instr If <math> a_1, \ldots a_n </math> are positive and <math> \sum_{i=1}^{n}a_i = s </math>, then
  7 KB (1,224 words) - 16:21, 24 October 2022
• Pascal's Theorem
  '''Pascal's Theorem''' is a result in [[projective geometry]]. It states that if a [[h Since it is a result in the projective plane, it has a dual, [[Brianchon's Theorem]], which states that the diagonals of a hexagon circumscribed about
  4 KB (712 words) - 21:57, 25 April 2020
• Russell's Paradox
  The '''Russell's Paradox''', credited to Bertrand Russell, was one of those which forced the
  580 bytes (100 words) - 17:57, 12 May 2023
• Green's Theorem
  '''Green's Theorem''' is a result in [[real analysis]]. It is continuous [[partial derivative]]s mapping an open set containing
  2 KB (381 words) - 12:12, 30 May 2019
• Cramer's Rule
  '''Cramer's Rule''' is a method of solving systems of equations using [[matrix|matrices Cramer's Rule employs the [http://en.wikipedia.org/wiki/Determinant matrix determina
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• Ceva's Theorem/Problems
  #REDIRECT[[Ceva's theorem/Problems]]
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• Ceva's theorem/Problems
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• L'Hôpital's Rule
  '''L'Hopital's Rule''' is a theorem dealing with [[limit]]s that is very important to [[calculus]]. ...cdot \epsilon(h)}</math>, which would hence prove our lemma for L'Hospital's rule.
  2 KB (475 words) - 15:04, 24 March 2022
• Zorn's Lemma
  '''Zorn's Lemma''' is a [[set theory | set theoretic]] result which is equivalent to We first prove some intermediate results, viz., Bourbaki's Theorem (also known as the Bourbaki-Witt theorem).
  9 KB (1,669 words) - 19:02, 1 August 2018
• Pascal's Identity
  ...coefficients). It can often be used to simplify complicated [[expression]]s involving binomial coefficients. ...is also known as Pascal's Rule, Pascal's Formula, and occasionally Pascal's Theorem.
  3 KB (496 words) - 15:43, 11 April 2024
• Binet's Formula
  '''Binet's formula''' is an explicit formula used to find the <math>n</math>th term of
  3 KB (550 words) - 16:12, 24 February 2024
• Carleman's Inequality
  '''Carleman's Inequality''' states that for [[nonnegative]] [[real numbers]] <math>\{a_n\
  2 KB (278 words) - 16:39, 29 December 2021
• Vieta's formulas
  ...its roots can be easily expressed as a ratio between two of the polynomial's coefficients. It is among the most ubiquitous results to circumvent finding a polynomial's roots in competition math and sees widespread usage in many mathematics con
  3 KB (512 words) - 15:31, 22 February 2024
• Euclid's Elements
  ...ct should have been either common notions or postulates, as some of Euclid's methods of proof were faulty. Euclid's work is split into thirteen volumes. It covers not only geometry, but numbe
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• Euclid's proof on the infitude of primes
  #REDIRECT [[Euclid's proof of the infinitude of primes]]
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• Lagrange's Mean Value Theorem
  '''Lagrange's mean value theorem''' (often called "the mean value theorem," and abbreviat We reduce the problem to [[Rolle's theorem]] by using an auxiliary function.
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• Zassenhaus's Lemma
  ...s proof of the lemma in 1934 to provide a more elegant proof of [[Schreier's Theorem]]. He was a doctorate student under Emil Artin at the time. In th ...up of <math>K' \cdot (H \cap K)</math>; furthermore, the [[quotient group]]s
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• 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}
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• 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>.
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• 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
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• 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>.
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• 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>
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• Bezout's Lemma
  #REDIRECT[[Bézout's Identity]]
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• Carnot's Theorem
  '''Carnot's Theorem''' states that in a [[triangle]] <math>ABC</math>, the signed sum o label("$O_C$",f,S);
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• 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:
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• Aczel's Inequality
  '''Aczél's Inequality''' states that if <math>a_1^2>a_2^2+\cdots +a_n^2</math> or <mat * Popoviciu, T., Sur quelques inégalités, Gaz. Mat. Fiz. Ser. A, 11 (64) (1959) 451–461
  2 KB (428 words) - 16:36, 29 December 2021
• Gauss's Lemma
  * [[Gauss's Lemma (polynomial)]] * [[Quadratic reciprocity|Gauss's Lemma (quadratic reciprocity)]]
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• Gauss's Lemma (polynomial)
  '''Gauss's Lemma for Polynomials''' is a result in [[abstract algebra | algebra]]. The original statement concerns [[polynomial]]s with [[integer]] coefficients. Such a polynomial is called ''primitive'' i
  3 KB (483 words) - 12:23, 30 May 2019
• Fermat's Two Squares Theorem
  '''Fermat's Two Squares Theorem''' states that that a [[prime number]] <math>p</math> c Since 0 and 1 are the only [[quadratic residue]]s mod 4, it follows that if <math>p</math> is a prime number represented as t
  4 KB (612 words) - 12:10, 30 May 2019
• De Morgan's Laws
  '''De Morgan's Laws''' are two very important laws in the fields of [[set theory]] and [[b
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• 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 <
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• Cauchy's Integral Formula
  '''Cauchy's Integral Formula''' is a fundamental result in by application of [[Cauchy's Integral Theorem]].
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• L'Hopital's Rule
  #REDIRECT [[L'Hôpital's Rule]]
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• Liouville's Theorem (complex analysis)
  In [[complex analysis]], '''Liouville's Theorem''' states that a [[Picard's Little Theorem]] is a stronger result.
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• Hilbert's Basis Theorem
  '''Hilbert's Basis Theorem''' is a result concerning [[Noetherian]] [[ring]]s. It states that if <math>A</math> is a (not necessarily [[commutative]])
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• Schonemann's criterion
  See also [[Eisenstein's criterion]].
  1 KB (279 words) - 17:03, 31 October 2023
• Kirchhoff's rules
  Kirchhoff's rules
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• Maine Business & Prof. Women's Club - Rachel E. Lemieux Youth Scholarship
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• KFC Colonel's Scholars Program
  * Plan to pursue a bachelor's degree at a public, in-state college or university
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• 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
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• Carnot's Thereom
  #REDIRECT [[Carnot's Theorem]]
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• Hensel's Lemma
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• Newton's sums
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• Callebaut's Inequality
  '''Callebaut's Inequality''' states that for <math>1\ge x\ge y\ge 0,</math> <cmath> \sum_{
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• ''See also: [[Hölder's Inequality]]'' '''Hölder's Inequality,''' a generalization of the '''Cauchy-Schwarz inequality''', sta
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• 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
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• De Moivre's theorem
  #REDIRECT[[De Moivre's Theorem]]
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• ==Proof of l'Hôpital's rule== ...roof of l'Hôpital's rule uses [[Cauchy's mean value theorem]]. l'Hôpital's rule has many variations depending on whether ''c'' and ''L'' are finite or
  4 KB (764 words) - 23:10, 2 January 2012
• Bézout's Identity
  '''Bézout's Identity''' states that if <math>x</math> and <math>y</math> are nonzero [[ ==Generalization/Extension of Bézout's Identity==
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• 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
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• Euler's Four-Square Identity
  Then Euler's Four-Square Identity simply reads <math>|XY|^2 = |X|^2 |Y|^2</math>; i.e. t
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• Dirichlet's Theorem
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• Viviani's theorem
  '''Viviani's Theorem''' states that for an equilateral triangle, the sum of the altitude ...2}=\dfrac{s}{2}(x+y+z)</math>. Therefore, <math>\dfrac{s}{2}(x+y+z)=\dfrac{s}{2}(a)</math>, so <math>x+y+z=a</math>, which means the sum of the altitude
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• 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
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• 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 <
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• Viviani's Theorem
  #REDIRECT [[Viviani's theorem]]
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• Women's Technology Program
  The MIT Women's Technology Program (WTP) is a rigorous four-week residential summer program
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• Avogadro's Constant
  Avogadro's Constant: <math>N_a = 6.0221415 * 10^{23}</math>
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• 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.
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• 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.
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• Euler's Totient Function
  #REDIRECT[[Euler's totient function]]
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• Wuwang2002's Wiki Games/Go
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• Areteem Institute's Math Zoom Winter Boot Camp (Locations Vary each year including one in Irvine, CA
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• 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==
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• 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:
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• 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^{
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• 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.
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• 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
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• Cohn's criterion
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• Ostrowski's criterion
  Ostrowski's Criterion states that:
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• Dunan's Theorem
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• Karmata's Inequality
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• Euler's formula and It's history
  First, you may have seen the famous "Euler's Identity": e (Euler's Number)
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• 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
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• 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
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• Hotmonkey's current phrase
  s t o n k s
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• Brocard's problem
  Are the known solutions, and it was a conjecture of Paul Erdös, that these are the only solutions.
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• 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
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• Lcz's Oly Notes
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• 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.
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• Lcz's Mock AMC 10A
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• Lcz's Mock AMC 10A Problems
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• 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.
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• 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
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• 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>.
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• 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]]
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• 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 (1,824 words) - 01:56, 2 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]]
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• Menelaus' theorem
  First let's define some masses. * [[Ceva's theorem]]
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• Number theory
  ...cs]] associated with studying the properties and identities of [[ integer]]s. *[[Prime number]]s
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• Number theory/Intermediate
  ** [[Simon's Favorite Factoring Trick]] ** [[Euler's Totient Theorem]]
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• San Diego ARML
  Individually, San Diego Surf had 2 students who scored 7's and went to tiebreakers: In addition, there were multiple students (on both teams) who scored 6's and earned medals as team high scorers:
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• ARML historical results
  | [[New York City ARML]] (New York City S)
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• Fermat's Little Theorem
  '''Fermat's Little Theorem''' is highly useful in [[number theory]] for simplifying the A frequently used corollary of Fermat's Little Theorem is <math>a^p \equiv a \pmod {p}</math>. As you can see, it i
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• Chebyshev's Inequality
  '''Chebyshev's inequality''', named after [[Pafnuty Chebyshev]], states that if ...nce of the [[Rearrangement inequality]], which gives us that the sum <math>S=a_1b_{i_1}+a_2b_{i_2}+...+a_nb_{i_n} </math> is maximal when <math>i_k=k</m
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• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]] Let <math>\phi(n)</math> be [[Euler's totient function]]. If <math>n</math> is a positive integer, <math>\phi{(n)
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• Geometric inequality
  ...[inequality]] involving various measures ([[angle]]s, [[length]]s, [[area]]s, etc.) in [[geometry]]. ...equality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says:
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• Brahmagupta's Formula
  .../math>, <math>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. <cmath>16[ABCD]^2=16(s-a)(s-b)(s-c)(s-d)</cmath>
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• Ptolemy's Theorem
  #REDIRECT[[Ptolemy's theorem]]
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• Ptolemy's theorem
  ...[equality condition | equality case]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving i ...ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
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• Elementary symmetric sum
  .../en.wikipedia.org/wiki/Sums_of_powers sums of powers], combined with Vieta's formulas. Elementary symmetric sums show up in [[Vieta's formulas]]. In a monic polynomial of degree <math>n</math>, the coefficient
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• Cyclic quadrilateral
  * [[Ptolemy's Theorem]] * [[Brahmagupta's formula]]
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• Vieta's Formula
  #REDIRECT[[Vieta's formulas]]
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• Algebra
  ...and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]]. ...ebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common objects of study in higher algebra.
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• Number theory/Advanced
  ...r theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]].
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• Algebra/Intermediate
  ** [[Vieta's Formulas]] ** [[Newton's Sums]]
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• Euler's totient function
  '''Euler's totient function''' <math>\phi(n)</math> applied to a [[positive integer]] ...p_m^{e_m} </math> where the <math>p_i </math> are distinct [[prime number]]s. Now, we can use a [[PIE]] argument to count the number of numbers less th
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• Alabama ARML
  In some years, there are [[Alabama ARML TST]]'s that are written to help decide the team. From 2005-2008, the TST has been
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• Contest strategies
  ...ea to stay up all night solving problems or playing video games because it's easy to get drowsy during the test. Getting a good night sleep can help re
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• Rearrangement Inequality
  ...tation]] of a [[finite]] [[set]] (in fact, [[multiset]]) of [[real number]]s and <math>B=\{b_1,b_2,\cdots,b_n\}</math> is a permutation of another finit ...ath> and <math>a_k</math> with <math>b_j</math> (unless both a's or both b's are equal, in which case either we can choose another pair of products or n
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• Real number
  ...th>, <math>\ 3+2i+2j+k</math>, i.e. [[complex number]]s, and [[quaternion]]s. ...these two classes are best understood as subsets of the [[complex number]]s.
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• Divisibility rules
  ...integer]]s are [[divisibility | divisible]] by particular other [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other b ...s that are relatively prime to the base (and works GREAT in binary). Here's one that works. 12348 - 28 ==> 12320 ==> 1232 +28 ==> 1260 ==> 126 + 14 ==
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• Induction
  ...ombinatorics]] are especially susceptible to induction solutions, but that's not to say that there aren't any problems in other areas, such as [[Inequal * Prove Bernoulli's inequality.
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• Incenter/excenter lemma
  ...math>L</math> be the midpoint of <math>\overarc{BC}</math> on the triangle's circumcenter. Then, the theorem states that <math>L</math> is the center of
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• Triangle
  Triangles are split into six categories; three by their [[angle]]s and three by their side lengths. All the angles of an '''acute''' triangle are [[acute angle]]s.
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• Stewart's Theorem
  #REDIRECT[[Stewart's theorem]]
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• Overcounting
  ...other. Suppose we are given an order to count the number of handshakes. It's not a matter of a great deal if there are less than 10 persons. But assume ...s <math>n</math>.So there will be <math>n*(n-1)</math> handshakes. Now let's try our formula for two people. According to the formula we get 2 handshake
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• Constructive counting
  ...ulate with <math>4</math> <math>A</math>s and <math>3</math> <math>B</math>s. Using constructive counting is an idea, but there are multiple ways one mi ...xes, so we don't have to account for them after choosing the <math>A</math>s. Thus, there are <math>35</math> different permutations of <math>AAAABBB</m
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• Jensen's Inequality
  '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jen One of the simplest examples of Jensen's inequality is the [[quadratic mean]] - [[arithmetic mean]] inequality. Taki
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• Casework
  ...digit can be <math>9</math> digits, with zero included and the first digit's number removed. Then there are <math>9 + 3 \cdot 81 = 252</math> of these n
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• Euclidean algorithm
  ...idean domain]], the most common of which is the [[nonnegative]] [[integer]]s <math>\mathbb{Z}{\geq 0}</math>, without [[factoring]] them. ~The congruence sign above should be replaced by the normal equal sign. It's important to note that <math>a \pmod{b} = r</math><br>is the same as <math>
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• 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
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• 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
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• Combinatorial identity
  == Pascal's Identity == Pascal's Identity states that
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• 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.
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• 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,
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• 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
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• 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
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• 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
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• 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>.
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• 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
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• 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
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• 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>.
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• Scholastic Aptitude Test
  ...tp://www.collegeboard.com/student/testing/sat/about.html The College Board's SAT I Website]
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• 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
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• 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.
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• List of United States college mathematics competitions
  * [https://simiode.org/scudem SIMIODE SCUDEM] -- SIMIODE's (Systemic Initiative for Modeling Investigations and Opportunities using Di
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• 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,
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• 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.
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• 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
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• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician *[[Ptolemy's Theorem]]
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• Cevian
  label("$D$",(70,0),S); label("$D$",(70,0),S);
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• Median of a triangle
  label("$M$", M, S); [[Stewart's Theorem]] applied to the case <math>m=n</math>, gives the length of the med
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• 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]].
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• Phi
  * <math>\phi</math> is also commonly used to represent [[Euler's totient function]].
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• 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]].
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• 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.
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• 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>,
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• 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
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• 1988 AJHSME Problems/Problem 5
  label("B",B,S);
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• 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
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• 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.
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• 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]
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• 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);
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• 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
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• 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
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• 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
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• 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
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• 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,
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• Schur's Inequality
  ...hur's Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur. Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</mat
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• Trigonometric identities
  [[Euler's identity]] states that <math>e^{ix} = \cos (x) + i \sin(x)</math>. We have === Euler's identity ===
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• Iff
  ...emsolving.com/wiki/index.php/Godel%27s_First_Incompleteness_Theorem Gödel's Incompleteness Theorem]
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• Vieta's
  #REDIRECT[[Vieta's formulas]]
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• Vocabulary and spelling competitions
  * [[Reader's Digest National Word Power Challenge | Word Power Challenge]] The premier v
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• 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
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• 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]]
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• 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>.
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• 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
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• Vocabulary competitions
  * [[Reader's Digest National Word Power Challenge | Word Power Challenge]] The premier v
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• 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
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• 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]].
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• Angle bisector
  ...XYZ</math> with [[incenter]] ''I'', [[incircle]] (blue), [[angle bisector]]s (orange), and [[angle bisector|external angle bisectors]] (green)}} * [[Stewart's Theorem]]
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• Mathew Crawford
  Crawford's user page can be found [[user:MCrawford | here]].
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• 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>
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• 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>
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• Fermat's Last Theorem
  '''Fermat's Last Theorem''' is a recently proven [[theorem]] stating that for positive Fermat's Last Theorem was proposed by [[Pierre de Fermat]] in the <math>1600s</math>
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• 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>.
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• 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
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• 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
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• 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);
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• 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
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• 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>.
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• 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,
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• 1985 IMO Problems/Problem 5
  ==Solution 3 (No Miquel's point)== *[[Miquel's point]]
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• 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
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• 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==
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• Physics summer programs
  == U.S. Physics Summer Programs == * [[Art of Problem Solving's 8-month online Olympiad-level physics course (PhysicsWOOT)]] [https://artof
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• 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 /
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• 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>
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• 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
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• 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 ==
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• 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
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• 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
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• 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
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• 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.
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• 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
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• 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</
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• 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
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• 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 ===
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• Majorization
  We say a [[nonincreasing]] [[sequence]] of [[real number]]s <math> a_1, \ldots ,a_n</math> '''majorizes''' another nonincreasing sequen * [[Karamata's Inequality]]
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• 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.
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• Prime Number Theorem
  the exception of [[Fermat's Last Theorem]]. (Fortunately, the proof asymptotic formula for the distribution of the [[prime number]]s;
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• ''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
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• 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.
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• 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:
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• 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]].
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• Rectangle
  ...angle''' is a [[quadrilateral]] in which all [[angle]]s are [[right angle]]s. ...n addition, rectangles have [[congruent (geometry)|congruent]] [[diagonal]]s.
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• 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'''
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• Mock AIME
  ...oblemsolving.com/community/c5t183f5h1074599_trumpeters_mock_aime Trumpeter's Mock AIME] ** [https://artofproblemsolving.com/community/c594864h2441992 Treemath's Mock AIME]
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• 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>.
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• 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
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• 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
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• 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
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• Andrew Wiles
  ...sity]]. He is most widely known as the mathematician who proved [[Fermat's Last Theorem]].
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• 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>
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• 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
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• Convex function
  * [[Jensen's Inequality]] * [[Karamata's Inequality]]
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• 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
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• 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]]
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• 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
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• Subset
  ...or <math>2^S</math>. The number of subsets of <math>S</math> is <math>2^{|S|}</math>.
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• 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.
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• 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!
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• 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);
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• 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!
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• 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
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• 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
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• 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
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• 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);
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• 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);
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• 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
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• 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!
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• 2006 AIME I Problems/Problem 3
  Here's another way to finish using this solution. From the above, you have <cmath>
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• 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>.
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• MediaWiki
  * [http://meta.wikipedia.org/wiki/MediaWiki_User%27s_Guide User's Guide]
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• 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?
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• 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?
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• 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
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• 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>?
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• 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?
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• 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);
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• 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?
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• 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);
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• 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
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• 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,
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• 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);
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• 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
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• 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>.
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• 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
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• 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?
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• 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>.
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• 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>
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• 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
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• 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
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• 2006 AMC 12B Problems/Problem 22
  To see how we can do better, let's rearrange the terms as follows:
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• 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);
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• 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);
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• 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
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• 2006 AMC 12A Problems/Problem 14
  Bezout's Lemma:
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• 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;
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• 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>?
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• 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:
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• 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>.
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• 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);
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• 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
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• 2005 AMC 12B Problems/Problem 6
  == Solution 3 (Stewart's Theorem) == Let <math>BD=k</math>. Then, by [[Stewart's Theorem]],
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• 2005 AMC 12B Problems/Problem 7
  label("$x$",(5,0),S); label("4",(4,0),S);
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• 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]]
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• 2005 AMC 12B Problems/Problem 14
  label("$x$",(Xmax+0.25,0),S);
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• 2005 AMC 12B Problems/Problem 15
  <math>221</math> can be written as the sum of four two-digit numbers, let's say <math>\overline{ae}</math>, <math>\overline{bf}</math>, <math>\overline
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• 2005 AMC 12B Problems/Problem 18
  ...ath> as the hypotenuse. Note <math>C</math> cannot be inside this triangle's circumscribed circle or else <math>\angle C > 90^\circ</math>. Hence, the a
  2 KB (262 words) - 21:20, 21 December 2020
• 2005 AMC 12B Problems/Problem 23
  Let <math>S</math> be the set of ordered triples <math>(x,y,z)</math> of real numbers f ...h>b</math> such that for all ordered triples <math>(x,y.z)</math> in <math>S</math> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math>
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• 2005 AMC 12B Problems/Problem 24
  dot("$A(a,a^2)$", A, S);
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• Power of a Point Theorem/Introductory Problem 4
  label("3", E--A, S); ...em]], <math> AE = 3 </math> (or by just knowing your [[Pythagorean Triple]]s). Applying the Power of a Point Theorem gives <math> AE\cdot BE = CE\cdot
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• Root (polynomial)
  ...x)</math> whose range is the [[real number | real]], the [[complex number]]s or any abstract [[field]] is a value <math>a</math> in the [[domain]] of th ...rivative <math>P'(x)</math> has at least one root in the interval <math>(r,s)</math>. In particular, if <math>P'(x)</math> has no roots in an interval <
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• 2006 AMC 10A Problems
  Sandwiches at Joe's Fast Food cost <math> \textdollar 3 </math> each and sodas cost <math> \tex The ratio of Mary's age to Alice's age is <math>3:5</math>. Alice is <math>30</math> years old. How many years
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• 1995 AHSME Problems/Problem 6
  ...A</math> upwards, and thus <math>x</math> is touching <math>B</math> on it's left side. We now fold <math>B</math> up, and we realize that <math>x</math
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• 2006 AMC 10A Problems/Problem 12
  label('Rope', (20,-8),S); label("Rope", (20,-24), S);
  3 KB (424 words) - 10:14, 17 December 2021
• 2006 AMC 10A Problems/Problem 15
  We first find the amount of minutes, <math>k</math>, until Odell and Kershaw's next meeting. Let <math>a</math> be the angle in [[radians]] between their Since Odell's rate is <math>5/6</math> that of Kershaw, but Kershaw's lap distance is <math>6/5</math> that of Odell, they each run a lap in the
  3 KB (532 words) - 17:49, 13 August 2023
• 2006 AMC 10A Problems/Problem 17
  D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W); D('W',W,1.6*N); D('X',X,1.6*plain.E); D('Y',Y,1.6*S); D('Z',Z,1.6*plain.W);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2006 AMC 10A Problems/Problem 21
  ...btracting off those which do not have any <math>2</math>s or <math>3</math>s as digits. Case <math>2</math> : There are two <math>2</math>s OR two <math>3</math>s. If the <math>2</math> or <math>3</math> is occupying the first digit, we h
  3 KB (525 words) - 20:25, 30 April 2024
• Set
  ...all it an [[infinite]] set. The objects in a set are called the [[element]]s of the set. A common misconception is that a set can have multiple indistin ...ect <math>x</math>, we have <math>x\in S'</math> if and only if <math>x\in S''</math>.
  11 KB (2,021 words) - 00:00, 17 July 2011
• 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
• 2006 USAMO Problems
  .../math> be a prime number and let <math>s</math> be an integer with <math>0<s<p</math>. Prove that there exist integers <math>m</math> and <math>n</math> ...brace\frac{sm}{p}\right\rbrace<\left\lbrace\frac{sn}{p}\right\rbrace<\frac{s}{p}</cmath>
  3 KB (520 words) - 09:24, 14 May 2021
• 1991 AJHSME Problems
  ...$1'$",(.5,0),S); label("$3'$",(1,1.5),E); label("$9'$",(1+9*sqrt(3)/4,9/4),S); ...,S); label("$1'$",(1+5*sqrt(3)/2,5),E);label("$1'$",(1/2+5*sqrt(3)/2,11/2),S);
  17 KB (2,246 words) - 13:37, 19 February 2020
• 3D Geometry
  '''3D Geometry''' deals with objects in 3 [[dimension]]s. For example, a drawing on a piece of paper is 2-dimensional since it has ...lems is to make it 2D. We can do this by looking at certain cross-section(s) of the diagram one at a time.
  2 KB (263 words) - 12:29, 30 December 2023
• 2005 AIME II Problems/Problem 2
  ...<math> m </math> and <math> n </math> are [[relatively prime]] [[integer]]s, find <math> m+n. </math> We then label three "zones" corresponding to the three people's meals as the first, second, and third objects in the line corresponding to
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 3
  ...<math> m </math> and <math> n </math> are [[relatively prime]] [[integer]]s. Find <math> m+n. </math> Let's call the first term of the original [[geometric series]] <math>a</math> and
  3 KB (581 words) - 07:54, 4 November 2022
• 2005 AIME II Problems/Problem 13
  ...8a-10=c-3</math>. Thus <math>P(n)=an^2-(41a)n+(408a-10)=0</math>. By Vieta's formulas, we know that the sum of the roots(<math>n</math>) is equal to 41
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems/Problem 9
  ...convert our given expression into a form from which we can apply De Moivre's Theorem. ...hus, the question is equivalent to asking for how many [[positive integer]]s <math>n \leq 1000</math> we have that <math>\left(\sin\left(\frac\pi2 - u\r
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ...where <math> p,q, </math> and <math> r </math> are [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], f ...E); label("$D$",D,SW); label("$E$",X,N); label("$F$",Y,NNE); label("$O$",O,S); label("$M$",M,N);
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  ...rac {s^3}{3\sqrt2}</math> and the whole octahedron has volume <math>\frac {s^3\sqrt2}3</math>. ...olumes is then <math>\frac{\left(\frac{2s^3\sqrt2}{27}\right)}{\left(\frac{s^3\sqrt2}{3}\right)} = \frac29</math> and so the answer is <math>\boxed{011}
  3 KB (436 words) - 03:10, 23 September 2020
• Line
  ...<math>(x_1,y_1)</math> <math>(x_2,y_2)</math> one can solve for the line's [[equation]].
  674 bytes (106 words) - 18:40, 9 May 2024
• 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
• Research Science Institute
  ...changed to Research Science Institute in 1987, following Admiral Rickover's death, and after a nomadic existence in the Washington, DC area, RSI moved
  2 KB (284 words) - 16:37, 6 July 2023
• 2005 AIME I Problems
  ...ote the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> even. Find <math> |a-b|. </math> # There are no right angle turns in the particle's path.
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 4
  ...>. The [[quadratic formula]] yields <math>r = \frac{7 \pm \sqrt{49 - 4(1)(-s^2 - 5)}}{2} = \frac{7 \pm \sqrt{4s^2 + 69}}{2}</math>. <math>\sqrt{4s^2 + 6 ...are, we have an <math>(n + 3)</math> x <math>(n + 3)</math> formation that's missing a <math>3</math> x <math>3</math> corner. For the remaining row of
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  ...<math>1</math>, the numbers following <math>1</math> must be <math>1\text{'s}</math>; therefore, the number of possible permutations if all the coins ar
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 6
  ...\sqrt{2006}</math>. Now transforming the original function and using Vieta's formula, <math>x^4-4x^3+6x^2-4x-2005=0</math> so <math>x_1 x_2 x_3 x_4 = \f ==Solution 6 (De Moivre's Theorem)==
  4 KB (686 words) - 01:55, 5 December 2022
• 2005 AIME I Problems/Problem 7
  ...h> where <math> p </math> and <math> q </math> are [[positive]] [[integer]]s, find <math> p+q. </math> label("$P$",(5,0),S);
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 8
  ...</math> and <math> n </math> are [[relatively prime]] [[positive integer]]s, find <math> m+n. </math> ...equation has roots <math>r_1, r_2</math> and <math>r_3</math>, by [[Vieta's formulas]] we have <math>r_1\cdot r_2\cdot r_3 = 4</math>. Let the corresp
  1 KB (161 words) - 19:50, 2 January 2022
• 2005 AIME I Problems/Problem 11
  ...the points where the square is tangent to the semicircle. Then the square's area is the area of the small square +2* the area of the trapezoids on the
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME I Problems/Problem 12
  ...ote the number of positive integers <math> n \leq 2005 </math> with <math> S(n) </math> [[even integer | even]]. Find <math> |a-b|. </math> ...le <math>S(4), S(5), \ldots, S(8)</math> are even, and <math>S(9), \ldots, S(15)</math> are odd, and so on.
  4 KB (647 words) - 02:29, 4 May 2021
• 2005 AIME I Problems/Problem 13
  # There are no right angle turns in the particle's path. ...two <math>R</math>'s or <math>U</math>'s are adjacent. The <math>D</math>'s split the string into three sections (<math>-D-D-</math>): by the [[Pigeonh
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME I Problems/Problem 14
  ...erent side of <math> S. </math> Let <math> K </math> be the area of <math> S. </math> Find the remainder when <math> 10K </math> is divided by <math>100
  3 KB (561 words) - 14:11, 18 February 2018
• 2005 AIME I Problems/Problem 15
  ...s <math> BC=20. </math> The [[incircle]] of the triangle evenly [[trisect]]s the [[median of a triangle | median]] <math> AD. </math> If the area of the size(300); pointpen=black;pathpen=black+linewidth(0.65); pen s = fontsize(10);
  5 KB (906 words) - 23:15, 6 January 2024
• 2005 AIME II Problems/Problem 14
  label("$E$",E,S); ...om E and D to AB and AC respectively. Call the feet of the altitudes R and S respectively.
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 15
  ...ed for that scenario above, so our string of <math>10</math> <math>A</math>s must be preceded by a <math>B</math>. There are no other restrictions on th ...> ways to select the operations for the <math>7</math> <math>\square</math>s, and <math>8</math> places to place our <math>BA^{[10]}</math> block. Thus,
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  ...here <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> label("\(E\)",E,S); dot(E);
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 12
  ...}\right] </math> are both even. Given that the area of the graph of <math> S </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> Graphing this yields a series of [[rectangle]]s which become smaller as you move toward the [[origin]]. The <math>x</math>
  2 KB (303 words) - 22:28, 11 September 2020
• 2004 AIME I Problems/Problem 9
  ...h>U_1</math>, and hence <math>U_2</math>, are <math>3-4-5\,\triangle</math>s. label("\(U_2\)",(1,3)); label("\(V_2\)",(5,3)); MP("7",(D+G)/2,S); MP("6",(D+E)/2,W); MP("9/2",(E+H)/2,N);
  4 KB (618 words) - 20:01, 4 July 2013
• 2004 AIME I Problems/Problem 2
  Let us give the [[element]]s of our sets names: ...the two two sequences is <math>99</math> (forget about absolute value, it's insignificant here since we can just assume both sets end with positive las
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME I Problems/Problem 4
  ...sides of the square. The [[midpoint]]s of the line segments in set <math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 5
  ...by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was 300/500 = 3/5. The largest possible two-day succe This makes Beta's success ratio <math>\frac{349}{500}</math>. Thus, the answer is <math>m+n =
  3 KB (436 words) - 18:31, 9 January 2024
• 2004 AIME I Problems/Problem 6
  ...in the first solution. And, let's just pick the digits from 0-9. This get's a total count of <math>5\cdot{10 \choose 4}</math> But, this over-counts si
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME I Problems/Problem 7
  Let <math>S</math> be the [[set]] of integers <math>\{-1,2,-3,\ldots,14,-15\}</math>. T ...ath>S</math> all size 2 subsets of this set. Replace each element of <math>S</math> by the product of the elements. Now, the quantity we seek is the sum
  5 KB (833 words) - 19:43, 1 October 2023
• 2004 AIME I Problems/Problem 8
  ...S \mid i\, \textrm{ divides }\,1000\}</math>. The number of <math>m</math>'s that are not relatively prime to <math>1000</math> is: ...f the <math>1</math>-step case), where <math>\phi(n)</math> is the [[Euler's totient function]]. It is well-known that <math>\phi(n) = n\left(1-\frac{1}
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  ...cent sides of the square. The midpoints of the line segments in set <math> S </math> enclose a region whose area to the nearest hundredth is <math> k. < ...by points attempted) on each day was less than Alpha's on that day. Alpha's two-day success ratio was <math>\frac{300}{500} = \frac{3}{5}</math>. The l
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 15
  ...d. Each square <math>s</math> is now paired with the square <math>2^{k} - s - 1</math>. Now, imagine that we relabel these pairs with the indices <mat Therefore, we can keep track of the square's location in the following table.
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  ...th> 7+77+777+7+7=875 </math> could be obtained from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to inse ...ath>s, <math>b</math> <math>77</math>s, and <math>c</math> <math>777</math>s to sum up to <math>7000</math> (<math>a,b,c \ge 0</math>). Then <math>7a +
  11 KB (1,857 words) - 21:55, 19 June 2023
• 2004 AIME II Problems/Problem 12
  ...where <math> k, m, n, </math> and <math> p </math> are [[positive integer]]s, <math> n </math> is not [[divisibility | divisible]] by the [[perfect squa D(O); D((3,0)--(3,D.y),d); D(A--O--D,d); MP("3",(3/2,0),S,f);MP("2",(2,D.y),N,f);
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 11
  ...umference of the base is <math>C=1200\pi</math>. The sector's radius (cone's sweep) is <math>R=\sqrt{r^2+h^2}=\sqrt{600^2+(200\sqrt{7})^2}=\sqrt{360000+ ...math> then we can take the end point <math>B</math> on <math>\theta</math>'s bisector at <math>\frac{3\pi}{4}</math> radians along the <math>y=-x</math>
  2 KB (268 words) - 22:20, 23 March 2023
• 2004 AIME II Problems/Problem 10
  ...exactly two <math>1</math>'s. If a number is chosen at random from <math> S, </math> the [[probability]] that it is divisible by <math>9</math> is <mat ...if necessary. Therefore the number of sets where there are exactly two 1’s in this binary representation is just <math>\binom {40}{2}</math> because w
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems/Problem 8
  ...and <math>c+1</math>. So let's partition the 2's first. There are two 2's so this is equivalent to partitioning two items in three containers. We ca
  2 KB (353 words) - 18:08, 25 November 2023
• 2004 AIME II Problems/Problem 6
  ...>3x+6y+12z</math> from all, we get <math>9x+10z = 21z = 3x+12y</math>. Let's split this into 3 equations. Let's look at the first equation. Rearranging, it gets us <math>9x = 11z</math>
  6 KB (950 words) - 14:18, 15 January 2024
• 2004 AIME II Problems/Problem 5
  Let each worker's speed be <math>w</math>, the entire time be <math>t</math>, and the total w
  4 KB (592 words) - 19:02, 26 September 2020
• 2004 AIME II Problems/Problem 4
  ...[[positive integer]]s less than 10,000 have at most two different [[digit]]s? First, let's count numbers with only a single digit. We have nine of these for each len
  3 KB (508 words) - 01:16, 19 January 2024
• 2004 AIME II Problems
  ...pansions have exactly two 1's. If a number is chosen at random from <math> S, </math> the probability that it is divisible by 9 is <math> p/q, </math> w ...th> 7+77+777+7+7=875 </math> could be obtained from eight <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to inse
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1983 AIME Problems
  label("$B$",B,S); Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen
  7 KB (1,104 words) - 12:53, 6 July 2022
• 2019 AIME I Problems/Problem 2
  ~Shreyas S
  5 KB (830 words) - 22:15, 28 December 2023
• 1984 AIME Problems
  ...ps to <math>55</math>. What is the largest number that can appear in <math>S</math>? ...ems and that one's score, <math>s</math>, is computed by the formula <math>s=30+4c-w</math>, where <math>c</math> is the number of correct answers and <
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  ...per divisors of <math>1000000</math>. What is the integer nearest to <math>S</math>? ...16}+a_{17}y^{17}</math>, where <math>y=x+1</math> and the <math>a_i</math>'s are constants. Find the value of <math>a_2</math>.
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  ...te the least common multiple of positive integers <math>r</math> and <math>s</math>. Find the number of ordered triples <math>(a,b,c)</math> of positiv ...teps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time?
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...to type, each time putting the letter on top of the pile in the secretary's in-box. When there is time, the secretary takes the top letter off the pile ...t letter 8 has already been typed, but says nothing else about the morning's typing. The colleague wonders which of the nine letters remain to be typed
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  </asy></center><!-- Minsoen's image: [[Image:AIME_1989_Problem_6.png]] --> One of Euler's conjectures was disproved in the 1960s by three American mathematicians whe
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1990 AIME Problems
  ...]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are neither the [[perfect square | square]] nor the [[perfect cube | c ...9}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>?
  6 KB (870 words) - 10:14, 19 June 2021
• 1991 AIME Problems
  label("$1$", (0.5,0), S); ...math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{}</math> are interior points on sides <math>\overline{AB}</math>, <math
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  ...end of the weekend, her win ratio is greater than <math>.503</math>. What's the largest number of matches she could've won before the weekend began? In Pascal's Triangle, each entry is the sum of the two entries above it. The first few
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught <math ...subsets of <math>S\,</math> so that the union of the two subsets is <math>S\,</math>? The order of selection does not matter; for example, the pair of
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  <center><math>S=p(1)+p(2)+p(3)+\cdots+p(999)</math></center>. What is the largest prime factor of <math>S\,</math>?
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1997 AIME Problems
  ...</math> rectangles, of which <math>s</math> are squares. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> ...y different <math>4\times 4</math> arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1999 AIME Problems
  ...)</math> be <math>|S(x+2)-S(x)|.</math> For example, <math>T(199)=|S(201)-S(199)|=|3-19|=16.</math> How many values of <math>T(x)</math> do not exceed ...left(\frac 12,\frac 13,\frac 16\right).</math> The area of <math>\mathcal{S}</math> divided by the area of <math>\mathcal{T}</math> is <math>m/n,</math
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  ...1000.</math> What is the greatest integer that does not exceed <math>\frac{S}{10}</math>?
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  ...than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>. ...lity that the midpoint of the segment they determine also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are rel
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  ...ined by interchanging the digits of Dick's age. Let <math>d</math> be Dick's present age. How many ordered pairs of positive integers <math>(d,n)</math> ...cal{S},</math> what is the greatest number of elements that <math>\mathcal{S}</math> can have?
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  ...<math> \mathcal{S}, </math> she writes on her list the greater of the set's two elements. Find the sum of the numbers on the list. ...um of all possible perimeters of <math> \triangle ACD. </math> Find <math> s. </math>
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  A deck of forty cards consists of four 1's, four 2's,..., and four 10's. A matching pair (two cards with the same number) is removed from the deck
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...> be the inscribed circle. Construct <math>\overline{ST}</math> with <math>S</math> on <math>\overline{PR}</math> and <math>T</math> on <math>\overline{ ...line{JK}</math>, and <math>\overline{KI}</math>. The surface area of <math>S</math>, including the walls of the tunnel, is <math>m + n\sqrt {p}</math>,
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  ...h>a</math>, <math>b</math>, and <math>c</math> are [[positive]] [[integer]]s that form an increasing [[geometric sequence]] and <math>b - a</math> is th ...be the number of sets of two non-empty disjoint subsets of <math>\mathcal{S}</math>. (Disjoint sets are defined as sets that have no common elements.)
  7 KB (1,177 words) - 15:42, 11 August 2023
• 1983 AIME Problems/Problem 3
  What is the product of the [[real]] [[root]]s of the [[equation]] <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? ...18^2 - 4 \cdot 1 \cdot 20 = 244</math>, which is positive. Thus by [[Vieta's formulas]], the product of the real roots is simply <math>\boxed{020}</math
  3 KB (532 words) - 05:18, 21 July 2022
• 1983 AIME Problems/Problem 4
  label("$B$",B,S); label("$B$",B,S);
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 5
  Because we want the largest possible <math>w</math>, let's find an expression for <math>z</math> in terms of <math>w</math>. ...of some polynomial of the form <math>w^2+bw+c</math>, such that by [[Vieta's Formulæ]] and some algebra (left as an exercise to the reader), <math>b^2-
  4 KB (672 words) - 10:17, 17 March 2023
• 1983 AIME Problems/Problem 6
  Since <math>\phi(49) = 42</math> (see [[Euler's totient function]]), [[Euler's Totient Theorem]] tells us that <math>a^{42} \equiv 1 \pmod{49}</math> wher
  3 KB (361 words) - 20:20, 14 January 2023
• 1983 AIME Problems/Problem 7
  Twenty five of King Arthur's knights are seated at their customary round table. Three of them are chosen
  9 KB (1,392 words) - 20:37, 19 January 2024
• 1983 AIME Problems/Problem 11
  ...th>2s</math>. All other edges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? real s = 6 * 2^.5;
  5 KB (865 words) - 21:11, 6 February 2023
• 1983 AIME Problems/Problem 13
  Let <math>S</math> be a non-[[empty set | empty]] [[subset]] of <math>\{1,2,3,4,5,6\}</ ...ath> ends up with the opposite sign of each corresponding element of <math>S\cup \{7\}</math>.
  5 KB (894 words) - 22:02, 5 April 2024
• 1983 AIME Problems/Problem 14
  ...l("$P$",P,1.5*dir(80)); label("$R$",R,NE); label("12",waypoint(O1--O2,0.4),S);</asy> ...th>AP=8</math>, we can find the third side of the triangle using [[Stewart's Theorem]] or similar approaches. We get <math>AC = \sqrt{56}</math>. Now we
  13 KB (2,149 words) - 18:44, 5 February 2024
• 1983 AIME Problems/Problem 15
  dot("$P$", P, S); pair Q = (1.5,0); dot(Q); label("$Q$",Q,S);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 4
  ...ps to <math>55</math>. What is the largest number that can appear in <math>S</math>? ...bers other than the <math>68,</math> and the sum of these numbers is <math>s.</math>
  2 KB (319 words) - 03:38, 16 January 2023
• 1984 AIME Problems/Problem 6
  ...ine passes through the center of the bottom circle; hence it is the circle's [[diameter]] and splits the circle into two equal areas. For the rest of th ...), and call it <math>M</math>. If we draw the feet of the [[perpendicular]]s from <math>A,C</math> to the line (call <math>E,F</math>), we see that <mat
  6 KB (1,022 words) - 19:29, 22 January 2024
• 1984 AIME Problems/Problem 8
  So we can use [[De Moivre's theorem]] (which I would suggest looking at if you never heard of it before ...he angles of the previous solutions to get new solutions because De Moivre's formula says that <math>(\cos \theta + i \sin \theta)^n = \cos n\theta + i
  3 KB (430 words) - 19:05, 7 February 2023
• 1984 AIME Problems/Problem 9
  path3 rightanglemark(triple A, triple B, triple C, real s=8) P=s*markscalefactor*unit(A-B)+B;
  6 KB (947 words) - 20:44, 26 November 2021
• 1984 AIME Problems/Problem 10
  ...ems and that one's score, <math>s</math>, is computed by the formula <math>s=30+4c-w</math>, where <math>c</math> is the number of correct answers and < Let Mary's score, number correct, and number wrong be <math>s,c,w</math> respectively. Then
  7 KB (1,163 words) - 23:53, 28 March 2022
• 1984 AIME Problems/Problem 11
  Then let's take out one tree between each pair of birch trees. So you would remove <ma ...math>4</math> <math>n</math>s to separate the <math>5</math> <math>b</math>s. Specifically,
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1984 AIME Problems/Problem 14
  For example, let's say we assume our end digit of the number is 4. If we have 5 as one of our
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1984 AIME Problems/Problem 15
  ...+ 3^2 + 5^2 + 7^2 + x^2 + y^2 + z^2 + w^2)t^3 \dots = 0.</cmath> By Vieta's, we know that the sum of the roots of this equation is <cmath>1^2 + 3^2 + 5 ...e <math>a^3</math> terms to the left hand side, then we have that by Vieta's formulas, the sum of the roots is <math>-84-x^2-y^2-z^2-w^2=-(2^2+4^2+6^2+8
  6 KB (1,051 words) - 04:52, 8 May 2024
• 1995 IMO Problems/Problem 2
  ...z/(x+y)\bigr)</math> are similarly sorted sequences. Then by [[Chebyshev's Inequality]], By AM-GM, <math>\frac{x+y+z}{3} \ge \sqrt[3]{xyz}=1</math>, and by [[Nesbitt's Inequality]],
  6 KB (1,122 words) - 12:23, 6 January 2022
• 1985 AIME Problems/Problem 15
  ...h>B</math>, as shown in the first figure below, by joining the [[midpoint]]s of two adjacent sides. These six pieces are then attached to a [[regular po ...th the plane XYZ that contains the regular hexagon, we form a pyramid with S the top vertex and the base being an equilateral triangle with side length
  2 KB (245 words) - 22:44, 4 March 2024
• 1985 AIME Problems/Problem 13
  ...<math>d_n</math> as <math>n</math> ranges through the [[positive integer]]s. ...common divisor. Since the <math>n^2</math> term is quite restrictive, let's multiply by <math>4</math> so that we can get a <math>(2n+1)^2</math> in th
  4 KB (671 words) - 20:04, 6 March 2024
• 1985 AIME Problems/Problem 12
  ...>A</math> to <math>A.</math> So we need to find the number of [[partition]]s of <math>7</math> using only <math>2,3,4,5,</math> and <math>7.</math> Thes We will now proceed with casework on the number of <math>2</math>s in the septuple.
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 10
  How many of the first 1000 [[positive integer]]s can be expressed in the form Recall from Hermite's Identity that <math>\sum_{k = 0}^{n - 1}\left\lfloor x + \frac kn\right\rfl
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 9
  ...lpha</math>, <math>\beta</math>, and <math>\alpha + \beta</math> [[radian]]s, respectively, where <math>\alpha + \beta < \pi</math>. If <math>\cos \alph ...angle has [[semiperimeter]] <math>\frac{2 + 3 + 4}{2}</math> so by [[Heron's formula]] it has [[area]] <math>K = \sqrt{\frac92 \cdot \frac52 \cdot \frac
  5 KB (763 words) - 16:20, 28 September 2019
• 1985 AIME Problems/Problem 8
  .../math>, <math>1\le i \le 7</math>, so that the sum of the <math>A_i</math>'s is also 19 and so that <math>M</math>, the [[maximum]] of the "errors" <mat For the question. Let's say that you have determined 7-tuple <math>(A_1,A_2,A_3,A_4,A_5,A_6,A_7)</m
  2 KB (377 words) - 02:17, 16 February 2021
• 1985 AIME Problems/Problem 7
  ...<math>b</math>, <math>c</math>, and <math>d</math> are [[positive integer]]s such that <math>a^5 = b^4</math>, <math>c^3 = d^2</math>, and <math>c - a = ...math>s - t^2 = 1</math>. Then <math>s = 10, t = 3</math> and so <math>d = s^3 = 1000</math>, <math>b = t^5 = 243</math> and <math>d-b=\boxed{757}</math
  1 KB (222 words) - 11:04, 4 November 2022
• 1985 AIME Problems/Problem 6
  ...rom the [[vertex | vertices]] through a common interior point. The [[area]]s of four of these triangles are as indicated. Find the area of triangle <mat ...ectively. Also, let <math>[APE]=x,[CPD]=y.</math> Then notice that by Ceva's, <math>\frac{FB\cdot DC\cdot EA}{DB\cdot CE\cdot AF}=1.</math> However, we
  5 KB (789 words) - 03:09, 23 January 2023
• 1985 AIME Problems/Problem 5
  A [[sequence]] of [[integer]]s <math>a_1, a_2, a_3, \ldots</math> is chosen so that <math>a_n = a_{n - 1} ...his <math>n</math> times, <math>a_{j + 6n} = a_j</math> for all [[integer]]s <math>n</math> and <math>j</math>.
  2 KB (410 words) - 13:37, 1 May 2022
• 1986 AIME Problems/Problem 15
  ...[[hypotenuse]] <math>AB</math> is <math>60</math>, and that the [[median]]s through <math>A</math> and <math>B</math> lie along the lines <math>y=x+3</ label("$A$", A, S);
  11 KB (1,722 words) - 09:49, 13 September 2023
• 1986 AIME Problems/Problem 14
  ...his would give us the guess that the sides are of the ratio 1:2:3, but let's provide the complete solution.)
  2 KB (346 words) - 13:13, 22 July 2020
• 1986 AIME Problems/Problem 13
  Let's consider each of the sequences of two coin tosses as an [[operation]] inste ...>{{5+3}\choose3} = 56</math> combinations. We do the same with 2 <tt>H</tt>s to get <math>{{2+3}\choose3} = 10</math> combinations; thus there are <math
  4 KB (772 words) - 21:09, 7 May 2024
• 1986 AIME Problems/Problem 12
  ...s of <math>S</math> have the same sum. What is the largest sum a set <math>S</math> with these properties can have? ...sum greater than 61. Suppose such a set <math>S</math> existed. Then <math>S</math> must have more than 4 elements, otherwise its sum would be at most <
  2 KB (364 words) - 19:41, 1 September 2020
• 1986 AIME Problems/Problem 11
  ...math>, where <math>y=x+1</math> and the <math>a_i</math>'s are [[constant]]s. Find the value of <math>a_2</math>.
  6 KB (872 words) - 16:51, 9 June 2023
• 1986 AIME Problems/Problem 10
  When we test <math>a+b+c=7, 10b+11c=260</math>, well, it's impossible
  3 KB (565 words) - 16:51, 1 October 2023
• 1986 AIME Problems/Problem 9
  .../math>. An interior [[point]] <math>P</math> is then drawn, and [[segment]]s are drawn through <math>P</math> [[parallel]] to the sides of the [[triangl pathpen = black; pointpen = black +linewidth(0.6); pen s = fontsize(10);
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1986 AIME Problems/Problem 8
  ...ding itself) of <math>1000000</math>. What is the integer nearest to <math>S</math>? ...so each power is actually <math>141</math> times. The answer is thus <math>S = \log 2^{141}5^{141} = \log 10^{141} = \boxed{141}</math>.
  3 KB (487 words) - 20:52, 16 September 2020
• 1986 AIME Problems/Problem 7
  ...th>1,3,4,9,10,12,13\cdots</math> consists of all those positive [[integer]]s which are [[exponent|powers]] of 3 or sums of distinct powers of 3. Find th ...of <math>64</math> terms which are of the form <math>729</math> + <math>'''S'''</math>, <math>32</math> of them include <math>243</math> and <math>32</m
  5 KB (866 words) - 00:00, 22 December 2022
• 1986 AIME Problems/Problem 3
  ...eed not be solved. The value of <math>ab</math> can be found through Vieta's.
  3 KB (545 words) - 23:44, 12 October 2023
• 1986 AIME Problems/Problem 2
  ...e-lengths <math>2\sqrt5,2\sqrt6,</math> and <math>2\sqrt7,</math> by Heron's Formula, the area is the square root of the original expression.
  3 KB (460 words) - 00:44, 5 February 2022
• 1987 AIME Problems/Problem 15
  ...triangle]]s in the figure are [[similar]] to triangle <math>ABC</math>, it's a good idea to use [[area ratios]]. In the diagram above, <math>\frac {T_1} label("$F$",F,S);
  5 KB (838 words) - 18:05, 19 February 2022
• 1987 AIME Problems/Problem 14
  By <b>De Moivre's Theorem</b>, we have <cmath>N^4=r^4\operatorname{cis}(4\theta)=18^2(-1),</c ...ouple of multiplicands in the numerator and denominator. By Sophie Germain's: <cmath>(10^4 + 324) = (10^4 + \cdot 4 \cdot 3^4) = (10^2 + 2 \cdot 10 \cdo
  7 KB (965 words) - 10:42, 12 April 2024
• 1987 AIME Problems/Problem 10
  ...teps. If Al's speed of walking (in steps per unit time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? ...that if we could cancel out the <math>b</math>'s, then the <math>e</math>'s in the numerator and denominator would cancel out, resulting in an equation
  7 KB (1,187 words) - 16:21, 27 January 2024
• 1987 AIME Problems/Problem 7
  ...[least common multiple]] of [[positive integer]]s <math>r</math> and <math>s</math>. Find the number of [[ordered tuple | ordered triples]] <math>(a,b, ...5^n</math> and <math>c = 2^p5^q</math> for some [[nonnegative]] [[integer]]s <math>j, k, m, n, p, q</math>. Dealing first with the powers of 2: from th
  3 KB (547 words) - 22:54, 4 April 2016
• 1987 AIME Problems/Problem 6
  ...XY = WZ</math>, <math>PQ = PQ</math> and the [[area]]s of the [[trapezoid]]s <math>PQZW</math> and <math>PQYX</math> are the same, then the heights of t ...= WZ</math> and <math>PQ = PQ</math> and the [[area]]s of the [[trapezoid]]s <math>PQZW</math> and <math>PQYX</math> are the same, the heights of the tr
  3 KB (530 words) - 07:46, 1 June 2018
• 1987 AIME Problems/Problem 5
  ...<math>3x^2 y^2</math> if <math>x</math> and <math>y</math> are [[integer]]s such that <math>y^2 + 3x^2 y^2 = 30x^2 + 517</math>. ...<math>x^2</math> term to the left side, it is factorable with [[SFFT|Simon's Favorite Factoring Trick]]:
  1 KB (160 words) - 04:44, 21 January 2023
• 1988 AIME Problems/Problem 15
  ...to type, each time putting the letter on top of the pile in the secretary's inbox. When there is time, the secretary takes the top letter off the pile ...th>8</math> has already been typed but says nothing else about the morning's typing. The colleague wonders which of the nine letters remain to be typed
  7 KB (1,186 words) - 10:16, 4 June 2023
• 1988 AIME Problems/Problem 13
  ...} + bx^{16} + 1</math> by <math>x^2-x-1</math> would be very tough, so let's try to divide using smaller degrees of <math>x</math>. Doing <math>\frac{ax There is somewhat of a pattern showing up, so let's try <math>\frac{ax^5+bx^4+1}{x^2-x-1}</math> to verify. We get:
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1988 AIME Problems/Problem 12
  Call the [[cevian]]s AD, BE, and CF. Using area ratios (<math>\triangle PBC</math> and <math>\tr ...+ d} + \frac {d}{b + d} + \frac {d}{c + d} = 1</math> is a form of [[Ceva's Theorem]].
  4 KB (727 words) - 23:37, 7 March 2024
• 1988 AIME Problems/Problem 11
  .../math> in the [[complex plane]] is called a mean [[line]] for the [[point]]s <math>w_1, w_2, \dots, w_n</math> if <math>L</math> contains points (comple And because the sum of the 5 <math>z</math>'s must cancel this out,
  2 KB (422 words) - 00:22, 6 September 2020
• 1988 AIME Problems/Problem 8
  Since all of the function's properties contain a recursive definition except for the first one, we know
  4 KB (538 words) - 13:24, 12 October 2021
• 1988 AIME Problems/Problem 3
  Raise both as [[exponent]]s with base 8: ...anation of the steps: On the 1st step, we use the property of [[logarithm]]s that <math>a^{\log_a x} = x</math>. On the 2nd step, we use the fact that <
  3 KB (481 words) - 21:52, 18 November 2020
• 1989 AIME Problems/Problem 15
  === Solution 1 (Ceva's Theorem, Stewart's Theorem) === label("$Y$", Y, S);
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 14
  ...lex number]] of the form <math>r+si</math>, where <math>r</math> and <math>s</math> are integers, can be uniquely expressed in the base <math>-n+i</math
  2 KB (408 words) - 17:28, 16 September 2023
• 1989 AIME Problems/Problem 13
  .../math> or <math>7</math>. What is the largest number of [[element]]s <math>S</math> can have? ...e we can put the last <math>1969</math> numbers into groups of 11. Now let's examine <math>\{1, 2, \ldots , 20\}</math>. If we pick <math>1, 3, 4, 6, 9<
  2 KB (274 words) - 04:07, 17 December 2023
• 1989 AIME Problems/Problem 12
  ...e. Let <math>d</math> be the distance between the [[midpoint]]s of [[edge]]s <math>AB</math> and <math>CD</math>. Find <math>d^{2}</math>. ...of Cosines]] applied twice with the angles formed by the median ([[Stewart's Theorem]]). We can also get this formula from the parallelogram law, that t
  2 KB (376 words) - 13:49, 1 August 2022
• 1989 AIME Problems/Problem 11
  A sample of 121 [[integer]]s is given, each between 1 and 1000 inclusive, with repetitions allowed. The ...be <math>M</math>, and the sum of the numbers <math>\neq x</math> be <math>S</math>. Then
  5 KB (851 words) - 18:01, 28 December 2022
• 1989 AIME Problems/Problem 10
  ...[altitude]] <math>h</math> to <math>c</math>, to get two [[right triangle]]s. label("$A$", A, S);
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1989 AIME Problems/Problem 9
  One of Euler's conjectures was disproved in the 1960s by three American mathematicians whe By either <b>Fermat's Little Theorem (FLT)</b> or inspection, we get
  6 KB (874 words) - 15:50, 20 January 2024
• 1989 AIME Problems/Problem 8
  ...math> in that order. Let <cmath>16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7=S. \hspace{29.5mm}(4)</cmath> 7x_1 + 9x_2 + 11x_3 + 13x_4 + 15x_5 + 17x_6 + 19x_7 &=S-123. &(7) \\
  8 KB (1,146 words) - 04:15, 20 November 2023
• 1989 AIME Problems/Problem 6
  Two skaters, Allie and Billie, are at [[point]]s <math>A</math> and <math>B</math>, respectively, on a flat, frozen lake. Th </asy></center><!-- Minsoen's image: [[Image:AIME_1989_Problem_6.png]] -->
  5 KB (864 words) - 19:55, 2 July 2023
• 1989 AIME Problems/Problem 4
  If <math>a<b<c<d<e</math> are [[consecutive]] [[positive]] [[integer]]s such that <math>b+c+d</math> is a [[perfect square]] and <math>a+b+c+d+e</m ...ath>y</math> denoting an [[integer]]s. <math>c</math> is minimized if it’s [[prime factorization]] contains only <math>3,5</math>, and since there is
  3 KB (552 words) - 12:41, 3 March 2024
• 1989 AIME Problems/Problem 3
  Repeating decimals represent [[rational number]]s. To figure out which rational number, we sum an [[infinite]] [[geometric s Since <math>37</math> is prime, we can apply Euclid's Lemma (which states that if <math>p</math> is a prime and if <math>a</math>
  3 KB (499 words) - 22:17, 29 March 2024
• 1989 AIME Problems/Problem 1
  ...he inside is a perfect square trinomial, since <math>b^2 = 4ac</math>. It's equal to <math>\sqrt{\left(n^2 - \frac{5}{4}\right)^2}</math>, which simpli
  4 KB (523 words) - 00:12, 8 October 2021
• 1990 AIME Problems/Problem 15
  Set <math>S = (x + y)</math> and <math>P = xy</math>. Then the relationship Consequently, <math>S = - 14</math> and <math>P = - 38</math>. Finally:
  4 KB (644 words) - 16:24, 28 May 2023
• 1990 AIME Problems/Problem 14
  ...[[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s. Find the volume of this pyramid. label("$13\sqrt{3}$", A--D, S);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1990 AIME Problems/Problem 13
  ...7 [[digit]]s and that its first (leftmost) digit is 9, how many [[element]]s of <math>T_{}^{}</math> have 9 as their leftmost digit? '''Lemma''': For all positive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 10
  The 18 and 48th roots of <math>1</math> can be found by [[De Moivre's Theorem]]. They are <math>\text{cis}\,\left(\frac{2\pi k_1}{18}\right)</mat
  3 KB (564 words) - 04:47, 4 August 2023
• 1990 AIME Problems/Problem 8
  ...</math>'s, we have counted each distinct arrangement of the <math>L</math>'s for <math>3!</math> times. </li><p> ...</math>'s, we have counted each distinct arrangement of the <math>M</math>'s for <math>2!</math> times. </li><p>
  3 KB (491 words) - 04:24, 4 November 2022
• 1990 AIME Problems/Problem 7
  pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17); pair P=(-8,5),Q=(-15,-19),R=(1,-7),S=(7,-15),T=(-4,-17),U=IP(P--T,Q--R);
  8 KB (1,319 words) - 11:34, 22 November 2023
• 1990 AIME Problems/Problem 3
  ...9}{58}</math> as large as each interior angle of <math>P_2^{}</math>. What's the largest possible value of <math>s_{}^{}</math>? ...rs</math>. Solving for <math>r</math>, we get <math>r = \frac{116s}{118 - s}</math>.
  3 KB (516 words) - 19:18, 16 April 2024
• 1990 AIME Problems/Problem 2
  ...d <math>b</math> equal one of <math>\pm3, \pm1</math>. The possible [[set]]s are <math>(3,1)</math> and <math>(-3,-1)</math>; the latter can be discarde ...qrt{43} + 3)^3 - (\sqrt{43} - 3)^3</math>. Using the difference of [[cube]]s, we get that <math>[\sqrt{43} + 3\ - \sqrt{43} + 3]\ [(43 + 6\sqrt{43} + 9)
  5 KB (765 words) - 23:00, 26 August 2023
• 1990 AIME Problems/Problem 1
  ...]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are neither the [[perfect square | square]] nor the [[perfect cube | c Because there aren't that many perfect squares or cubes, let's look for the smallest perfect square greater than <math>500</math>. This ha
  2 KB (283 words) - 23:11, 25 June 2023
• 2006 AMC 10B Problems/Problem 8
  real r=sqrt(50), s=sqrt(10); draw((-s,0)--(s,0)--(s,2*s)--(-s,2*s)--cycle);
  1 KB (176 words) - 13:49, 26 January 2022
• 2006 AMC 10B Problems/Problem 9
  The calorie to gram ratio of Francesca's lemonade is <math>\frac{25+386+0}{100+100+400}=\frac{411\textrm{ calories}} So in <math>200\textrm{ grams}</math> of Francesca's lemonade there are <math>200\textrm{ grams}\cdot\frac{137\textrm{ calories}
  1 KB (140 words) - 00:54, 2 April 2023
• 2006 AMC 10B Problems/Problem 13
  ...he resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? After drinking and adding cream, Joe's cup has <math>2</math> ounces of cream.
  1 KB (182 words) - 14:11, 26 January 2022
• 2006 AMC 10B Problems/Problem 14
  ...x + c = 0 </math>, the product of the [[root]]s is <math>c</math> ([[Vieta's Formulas]]). ...nd <math>a+\frac1b=\frac32</math> and <math>b+\frac1a=3</math>. By [[Vieta's formulas]], <math>q</math> is the product of the roots of <math>x^2-px+q=0<
  2 KB (264 words) - 21:10, 19 September 2023
• 1991 AIME Problems/Problem 15
  ...fference between squares, we get that <math>k^2-s=289</math> and <math>k^2+s=-1</math>. Therfore <math>2k^2</math> is <math>288</math> and <math>k=n=12<
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 14
  [[Ptolemy's Theorem]] on <math>ABCD</math> gives <math>81y+31\cdot 81=xz</math>, and Pt
  2 KB (284 words) - 03:56, 23 January 2023
• 2006 AMC 10B Problems/Problem 15
  .../math>. Let the lengths of the sides of rhombus <math>ABCD</math> be <math>s</math>. ...a side length of <math>s</math>, <math> AC = 2 \cdot \frac{s\sqrt{3}}{2} = s\sqrt{3} </math>.
  3 KB (445 words) - 22:01, 20 August 2022
• 1991 AIME Problems/Problem 12
  ...math>P^{}_{}</math>, <math>Q^{}_{}</math>, <math>R^{}_{}</math>, and <math>S^{}_{}</math> are interior points on sides <math>\overline{AB}</math>, <math ...); label("\(P\)",P,N);label("\(Q\)",Q,E);label("\(R\)",R,SW);label("\(S\)",S,W); label("\(15\)",B/2+P/2,N);label("\(20\)",B/2+Q/2,E);label("\(O\)",O,SW)
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  label("$1$", (0.5,0), S);
  4 KB (740 words) - 19:33, 28 December 2022
• 1991 AIME Problems/Problem 10
  Let <math>S(a,n)</math> be the <math>n</math>th letter of string <math>S(a)</math>. ...er of the string <math>S(a)</math> to the first letter of the string <math>S(b)</math>.
  5 KB (813 words) - 06:10, 25 February 2024
• 2006 AMC 10B Problems/Problem 17
  ...ag. Bob then randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the tw ...n't matter. [[Without loss of generality]], let the ball Alice puts in Bob's bag be red.
  1 KB (211 words) - 04:32, 4 November 2022
• 1991 AIME Problems/Problem 8
  By [[Vieta's formulas]], <math>x_1 + x_2 = -a</math> where <math>x_1, x_2</math> are the Let <math>x^2 + ax + 6a = (x - s)(x - r)</math>. Vieta's yields <math>s + r = - a, sr = 6a</math>.
  2 KB (310 words) - 11:25, 13 June 2023
• 1991 AIME Problems/Problem 6
  Recall by Hermite's Identity that <math>\lfloor x\rfloor +\lfloor x+\frac{1}{n}\rfloor +...+\lf
  3 KB (447 words) - 17:02, 24 November 2023
• 1991 AIME Problems/Problem 3
  ...ing this fraction by <math>\frac{995}{6}</math>. Notice that this fraction's numerator plus denominator is equal to <math>1001</math>. Calling the numer
  5 KB (865 words) - 12:13, 21 May 2020
• 1991 AIME Problems/Problem 1
  ...ic equation|quadratic]]: <math>a^2 - 71a + 880 = 0</math>, which [[factor]]s to <math>(a - 16)(a - 55) = 0</math>. Either <math>a = 16</math> and <math> ...1)(y + 1) = 72</math>. As <math>x</math> and <math>y</math> are [[integer]]s, the possible sets for <math>(x,y)</math> (ignoring cases where <math>x > y
  4 KB (628 words) - 22:05, 7 June 2021
• 1992 AIME Problems/Problem 14
  == Solution 4 (Ceva's Theorem) == A consequence of Ceva's theorem sometimes attributed to Gergonne is that <math>\frac{AO}{OA'}=\frac
  4 KB (667 words) - 01:26, 16 August 2023
• 1992 AIME Problems/Problem 13
  ...math>ABC</math> has <math>AB=9</math> and <math>BC: AC=40: 41</math>. What's the largest area that this triangle can have? ...the area is <math>\frac14\sqrt {(81^2 - 81x^2)(81x^2 - 1)}</math> by Heron's formula. By AM-GM, <math>\sqrt {(81^2 - 81x^2)(81x^2 - 1)}\le\frac {81^2 -
  4 KB (703 words) - 02:40, 29 December 2023
• 1992 AIME Problems/Problem 12
  ...>, two players alternately take bites from a 5-by-7 grid of [[unit square]]s. To take a bite, a player chooses one of the remaining [[square (geometry) ...s opponent take the last bite. The diagram shows one of the many [[subset]]s of the [[set]] of 35 unit squares that can occur during the game of Chomp.
  2 KB (443 words) - 22:41, 22 December 2021
• 1992 AIME Problems/Problem 8
  Note that the <math>\Delta</math>s are reminiscent of differentiation; from the condition <math>\Delta(\Delta{
  5 KB (778 words) - 21:36, 3 December 2022
• 1992 AIME Problems/Problem 5
  ...^{}_{}</math> are not necessarily distinct. To write the elements of <math>S^{}_{}</math> as fractions in lowest terms, how many different numerators ar
  2 KB (277 words) - 20:45, 4 March 2024
• 1992 AIME Problems/Problem 4
  In Pascal's Triangle, each entry is the sum of the two entries above it. The first few In which row of [[Pascal's Triangle]] do three consecutive entries occur that are in the ratio <math>3
  3 KB (476 words) - 14:13, 20 April 2024
• 1992 AIME Problems/Problem 3
  ...end of the weekend, her win ratio is greater than <math>.503</math>. What's the largest number of matches she could've won before the weekend began?
  2 KB (251 words) - 08:05, 2 January 2024
• 1992 AIME Problems/Problem 1
  Find the sum of all [[positive number |positive]] [[rational number]]s that are less than 10 and that have [[denominator]] 30 when written in [[re There are 8 [[fraction]]s which fit the conditions between 0 and 1: <math>\frac{1}{30},\frac{7}{30},\
  1 KB (190 words) - 20:02, 23 February 2022
• 2006 AMC 10B Problems/Problem 21
  ...are <math>6</math> ways to get the sum of <math>7</math> of the dice. Let's do case by case.
  3 KB (484 words) - 19:09, 15 October 2023
• 2006 AMC 10B Problems/Problem 23
  label("$F$",F,S); Since [[triangle]]s <math>AFB</math> and <math>DFB</math> share an [[altitude]] from <math>B</m
  5 KB (861 words) - 00:53, 25 November 2023
• 2006 AMC 10B Problems/Problem 25
  ...." Which of the following is <b><i>not</i></b> the age of one of Mr. Jones's children? ...nce at least one of <math>4</math> or <math>8</math> is contained in <math>S</math>, we have <math>4 | m</math>.
  5 KB (878 words) - 14:39, 3 December 2023
• 1993 AIME Problems/Problem 15
  ...an altitude of <math>\triangle ABC</math>. Let <math>R\,</math> and <math>S\,</math> be the points where the circles inscribed in the triangles <math>A dot(x[0]); label("$S$",x[0],SW);
  3 KB (449 words) - 21:39, 21 September 2023
• 1993 AIME Problems/Problem 13
  Let <math>A</math> and <math>B</math> be Kenny's initial and final points respectively and define <math>C</math> and <math>D ...W); label("$B$",B,NNW); label("$C$",C,S); label("$D$",D,NE); label("$P$",P,S); label("$Q$",Q,NE); label("$O$",O,W); label("$X$",X,ESE);
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1993 AIME Problems/Problem 12
  ...h two <math>A\,</math>'s, two <math>B\,</math>'s, and two <math>C\,</math>'s. Point <math>P_1 = (k,m)\,</math> is chosen in the interior of <math>\tria ...and <math>(r,s)</math> and we want to find <math>(u,v)</math> so <math>(r,s)</math> is the midpoint of <math>(u,v)</math> and <math>(p,q)</math>, then
  4 KB (611 words) - 13:59, 15 July 2023
• 1993 AIME Problems/Problem 11
  Therefore Alfred’s chance of winning on his second flip is <math>\frac{1}{4} \times \frac{1}{2 From this, we can see that Alfred’s (who goes first) chance of winning the first round is: <math>\frac{1}{2} +
  7 KB (1,058 words) - 20:57, 22 December 2020
• 1993 AIME Problems/Problem 10
  ...[[vertex|vertices]], <math>E</math> [[edge]]s, and <math>F</math> [[face]]s, <math>V-E+F=2</math>. A particular convex polyhedron has 32 faces, each of ...acent faces share one edge). Thus, <math>E=60</math>. Finally, using Euler's formula we have <math>V=E-30=30</math>.
  4 KB (716 words) - 20:50, 17 April 2022
• 1993 AIME Problems/Problem 9
  ...{5}</math>, one of them is divisible by <math>5</math>, which indicates it's divisible by <math>125</math>.
  3 KB (488 words) - 02:06, 22 September 2023
• 1993 AIME Problems/Problem 8
  ...subsets of <math>S\,</math> so that the union of the two subsets is <math>S\,</math>? The order of selection does not matter; for example, the pair of ...h>m</math> and <math>n</math> contain all <math>6</math> elements of <math>S.</math> So our final answer is then <math>\frac {3^6 - 1}{2} + 1 = \boxed{3
  9 KB (1,400 words) - 14:09, 12 January 2024
• 1993 AIME Problems/Problem 7
  ...<math>a_1,a_2,</math> and <math>a_3.</math> Similarly, the <math>b</math>'s are different from each other, as there are <math>3!=6</math> ways to permu
  5 KB (772 words) - 09:04, 7 January 2022
• 1955 AHSME Problems/Problem 27
  ...nd <math>s</math> are the roots of <math>x^2-px+q=0</math>, then <math>r^2+s^2</math> equals: We can write <math>r^2+s^2</math> in terms of the sum of the roots <math>(r+s)</math> and the products of the roots <math>(rs):</math>
  600 bytes (108 words) - 10:47, 15 February 2021
• 1993 AIME Problems/Problem 5
  ...nclude>Let <math>P_0(x) = x^3 + 313x^2 - 77x - 8\,</math>. For [[integer]]s <math>n \ge 1\,</math>, define <math>P_n(x) = P_{n - 1}(x - n)\,</math>. W ...r_2+r_2r_3+r_3r_1+210\cdot 2(r_1+r_2+r_3)+3\cdot 210^2.</math> Using Vieta's (again) and plugging stuff in yields <math>-77+210\cdot 2\cdot -313+3\cdot
  2 KB (355 words) - 13:25, 31 December 2018
• 1993 AIME Problems/Problem 3
  The table below displays some of the results of last summer's Frostbite Falls Fishing Festival, showing how many contestants caught <math
  2 KB (364 words) - 00:05, 9 July 2022
• 1993 AIME Problems/Problem 2
  The N/S displacement is
  2 KB (241 words) - 11:56, 13 March 2015
• 1994 AIME Problems/Problem 15
  ...s\,</math> is not divisible by the square of any prime. What is <math>q+r+s\,</math>? ...<math>O_{AB}, O_{BC}, O_{CA}</math> are, respectively, the [[circumcenter]]s of <math>\triangle PAB, PBC, PCA</math>. According to the problem statement
  4 KB (717 words) - 22:20, 3 June 2021
• 1994 AIME Problems/Problem 11
  ...ges we can get from <math>0</math>'s, <math>2</math>'s, and <math>5</math>'s. ...orm of <math>2m + 5n</math> for <math>m,n</math> being [[positive integer]]s is <math>5 \times 2 - 5 - 2=3</math>.
  4 KB (645 words) - 15:12, 15 July 2019
• 1994 AIME Problems/Problem 9
  ...ept that matching tiles are put aside as soon as they appear in the player's hand. The game ends if the player ever holds three tiles, no two of which ...bability]] of emptying the bag when it has <math>k</math> pairs in it. Let's consider the possible draws for the first three cards:
  3 KB (589 words) - 14:18, 21 July 2019
• 1994 AIME Problems/Problem 8
  '''Note''': This solution was also present in Titu Andreescu and Zuming Feng's "103 Trigonometry Problems". ...o, you really have two equations. Multiply out the equation given by Vieta's Formulas and isolate the ones with imaginary components. Simplify that equa
  5 KB (788 words) - 13:53, 8 July 2023
• 1994 AIME Problems/Problem 5
  <center><math>S=p(1)+p(2)+p(3)+\cdots+p(999)</math></center>. What is the largest prime factor of <math>S\,</math>?
  2 KB (275 words) - 19:27, 4 July 2013
• 1995 AIME Problems/Problem 15
  ...could be <tt>T</tt> or the sequence could start with a block of <tt>H</tt>'s, the total probability is that <math>3/2</math> of it has to start with an ...</math> to <math>\infty</math>, since that's how many blocks of <tt>H</tt>'s there can be before the final five. This is an infinite geometric series wh
  6 KB (979 words) - 13:20, 11 April 2022
• 1995 AIME Problems/Problem 14
  In a [[circle]] of [[radius]] <math>42</math>, two [[chord]]s of length <math>78</math> intersect at a point whose distance from the cent ...arc <math>\stackrel{\frown}{BC}</math>. The former can be found by [[Heron's formula]] to be <math>[BCE] = \sqrt{60(60-48)(60-42)(60-30)} = 360\sqrt{3}<
  3 KB (484 words) - 13:11, 14 January 2023
• 1995 AIME Problems/Problem 12
  label("$A$", A, S); The angle <math>\theta</math> is the angle formed by two [[perpendicular]]s drawn to <math>BO</math>, one on the plane determined by <math>OAB</math> a
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 9
  ...W); dot((2,0),ds); label("$C$",(2,0),SE); dot((1,0),ds); label("$M$",(1,0),S); dot((1,0.7),ds); label("$D$",(1,0.7),NE);
  7 KB (1,181 words) - 13:47, 3 February 2023
• 1995 AIME Problems/Problem 5
  Since the [[coefficient]]s of the [[polynomial]] are real, it follows that the non-real roots must com By [[Vieta's formulas]], we have that <math>b = mm' + nn' + mn' + nm' + mn + m'n' = (m +
  3 KB (451 words) - 15:02, 6 September 2021
• 1995 AIME Problems/Problem 4
  ...Q</math>. Let <math>A_3,A_6,A_9</math> be the feet of the [[perpendicular]]s from <math>O_3,O_6,O_9</math> to <math>\overline{PQ}</math> (so <math>A_3,A ...of this circle is <math>(x-2\sqrt{2})^2+(y+5)^2=81</math>. Since the chord's equation is <math>y=0</math>, we must find all values of <math>x</math> sat
  3 KB (605 words) - 11:30, 5 May 2024
• 1995 AIME Problems/Problem 3
  ...ncel out, either <tt>N/S</tt> or <tt>W/E</tt>. The sequences <tt>N,N,N,E,E,S</tt> can be permuted in <math>\frac{6!}{3!2!1!} = 60</math> ways. However, Let's let the object wander for 6 steps so we get a constant denominator of <math
  3 KB (602 words) - 23:15, 16 June 2019
• 1996 AIME Problems/Problem 13
  ...e is a point <math>D</math> for which <math>\overline{AD}</math> [[bisect]]s <math>\overline{BC}</math>, and <math>\angle ADB</math> is a right angle. T By [[Stewart's Theorem]], <math>AE = \frac{\sqrt{2(AB^2 + AC^2) - BC^2}}2 = \frac{\sqrt {5
  3 KB (521 words) - 01:18, 25 February 2016
• 1996 AIME Problems/Problem 12
  So now we only have to form the sum <math>S= (a_1 + a_3 + a_5 + a_7 + a_9) - (a_2 + a_4 + a_6 + a_8 + a_{10})</math>. D \overline{S} &= 10 + \left(\frac{8}{9} \cdot 9\right) - \left(\frac{1}{9} \cdot 9\right
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 9
  ...ll take about 25 minutes (if you don't start to see the pattern), but that's okay, eventually, you will get <math>\boxed{342}</math>.
  3 KB (525 words) - 23:51, 6 September 2023
• 1996 AIME Problems/Problem 6
  Now, computing this probability directly seems a little hard, so let's compute the complement -- the probability that there is an undefeated team,
  3 KB (461 words) - 01:00, 19 June 2019
• 1996 AIME Problems/Problem 5
  Suppose that the [[root]]s of <math>x^3+3x^2+4x-11=0</math> are <math>a</math>, <math>b</math>, and <m ...) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0</math>, we have <math>a + b + c = s = -3</math>, <math>ab + bc + ca = 4</math>, and <math>abc = 11</math>. Then
  3 KB (585 words) - 22:08, 19 November 2022
• 1996 AIME Problems/Problem 3
  ...s <math>[(x+7)(y-3)]^n = (x+7)^n(y-3)^n</math>. Both [[binomial expansion]]s will contain <math>n+1</math> non-like terms; their product will contain <m
  3 KB (515 words) - 04:29, 27 November 2023
• 1996 AIME Problems/Problem 1
  Let's make a table. The <math>a</math>'s cancel! We now have
  2 KB (332 words) - 11:28, 4 August 2021
• 1997 AIME Problems/Problem 15
  ...spectively. Let <math>E (11,a)</math> and <math>F (b, 10)</math>. Since it's equilateral, then <math>E\cdot\text{cis}60^{\circ} = F</math>, so <math>(11 ...a^2 + AB^2 = 884 - 440\sqrt{3}</math>. Using the area formula <math>\frac{s^2\sqrt{3}}{4}</math>, the area of the equilateral triangle is <math>\frac{(
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 14
  .../math> and <math>n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n</math>. Define <math>\theta = 2\pi/1997</math>. By [[De Moivre's Theorem]] the roots are given by
  5 KB (874 words) - 22:30, 1 April 2022
• 1997 AIME Problems/Problem 13
  ...-1\Big|+\Big|\big||y|-2\big|-1\Big|=1.</math></center> If a model of <math>S</math> were built from wire of negligible thickness, then the total length ...e <math>f(x) = f(-x)</math>, there's a symmetry about all four [[quadrant]]s, so just consider the first quadrant. We now gather some points:
  7 KB (1,225 words) - 19:56, 4 August 2021
• 1997 AIME Problems/Problem 12
  ...frac {(a^2 + bc)x + b(a + d)}{c(a + d)x + (bc + d^2)} =\frac {px + q}{rx + s} = x. </cmath> ...educe to <math>x</math>, we must have <math>q = r = 0</math> and <math>p = s\not = 0</math>. From <math>c(a + d) = b(a + d) = 0</math>, we get <math>a =
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1997 AIME Problems/Problem 11
  ...{\text{cis } 45^\circ - 1}{\text{cis } 1^\circ - 1}</math> (by [[De Moivre's Theorem]] with geometric series) ...4} \cos n^\circ = C</math> and <math>\sum\limits_{n=1}^{44} \sin n^\circ = S</math>. Then, we have
  10 KB (1,514 words) - 14:35, 29 March 2024
• 1997 AIME Problems/Problem 10
  Let's say we have picked two cards. We now compare their attributes to decide how
  3 KB (585 words) - 19:37, 25 April 2022
• 1997 AIME Problems/Problem 8
  ...y different <math>4\times 4</math> arrays whose entries are all 1's and -1's have the property that the sum of the entries in each row is 0 and the sum ...for all configurations of <math>4\times 4</math> grids with 2 1's and 2 -1's in each row and column. We do casework upon the first two columns:
  4 KB (638 words) - 16:41, 22 January 2024
• 1997 AIME Problems/Problem 7
  ...n <math>\left(\frac{1}{2}t, 110 - \frac{1}{2}t\right)</math> for the storm's center. This circle with radius 51 yields <math>\left(x - \frac{1}{2}t\righ Now substitute the car's coordinates into the circle's:
  4 KB (617 words) - 18:47, 17 July 2022
• 1997 AIME Problems/Problem 6
  Using [[Simon's Favorite Factoring Trick|SFFT]],
  3 KB (497 words) - 00:39, 22 December 2018
• 1997 AIME Problems/Problem 3
  ...have <math>1000x+y=9xy \Longrightarrow 9xy-1000x-y=0</math>. Using [[Simon's Favorite Factoring Trick|SFFT]], this factorizes to <math>(9x-1)\left(y-\df
  2 KB (375 words) - 19:34, 4 August 2021
• 1997 AIME Problems/Problem 2
  ...[[rectangle]]s, of which <math>s</math> are [[square]]s. The number <math>s/r</math> can be written in the form <math>m/n,</math> where <math>m</math> ...es 8</math> squares. Using the sum of squares formula, that gives us <math>s=1^2+2^2+\cdots+8^2=\dfrac{(8)(8+1)(2\cdot8+1)}{6}=12*17=204</math>.
  3 KB (416 words) - 21:09, 27 October 2022
• 1998 AIME Problems/Problem 15
  You need to have all even number of [[segment]]s coming from each point except 0 or 2 which have an odd number of segments c ...of pairs in <math>D</math> in order for one domino's end to match another's beginning. Then, in order to maximize the length we want to minimize the nu
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 13
  ...>S_n</math> be the sum of the complex power sums of all nonempty [[subset]]s of <math>\{1,2,\ldots,n\}.</math> Given that <math>S_8 = - 176 - 64i</math ...ing a <math>9</math> to a subset doesn't change anything about that subset's complex power sum besides adding an additional term, we have that <math>S_9
  2 KB (384 words) - 19:02, 20 October 2023
• 1998 AIME Problems/Problem 11
  label("\(B\,(20,0,0)\)",B,S); ...10b - 4d = d \Longrightarrow d = 10b</cmath>Thus plane <math>PQR</math>’s [[equation]] reduces to <cmath>\frac{d}{5}x + \frac{d}{10}y - \frac{d}{5}z
  7 KB (1,084 words) - 11:48, 13 August 2023
• 1998 AIME Problems/Problem 10
  ...h> where <math>a, b,</math> and <math>c</math> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]]. Find ...is to realize the significance that the figures are spheres, not [[circle]]s. The 2D analogue of the diagram onto the flat surface will not contain 8 ci
  3 KB (496 words) - 13:02, 5 August 2019
• 1998 AIME Problems/Problem 9
  ...h> where <math>a, b,</math> and <math>c</math> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]]. Find label("$m$",(m,0),S);
  4 KB (624 words) - 18:34, 18 February 2018
• 1998 AIME Problems/Problem 4
  ...h>m</math> and <math>n</math> are [[relatively prime]] [[positive integer]]s. Find <math>m+n.</math> ...e two <math>E</math>'s and an <math>O</math> in them, respectively.) . Let's do the case <math>OOO</math>, <math>OEE</math>, <math>OEE</math>.
  5 KB (917 words) - 02:37, 12 December 2022
• 1998 AIME Problems/Problem 2
  Find the number of [[ordered pair]]s <math>(x,y)</math> of positive integers that satisfy <math>x \le 2y \le 60< [[Pick's theorem]] states that:
  6 KB (913 words) - 16:34, 6 August 2020
• 1999 AIME Problems/Problem 15
  ...(16,24).</math> The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle a ...6,12,12); draw(A--B--C--A); draw(P--Q--R--P); draw(S--P..S--Q..S--R); draw(S--(16,12,0)); </asy></center><!-- Asymptote renderings of Image:AIME_1999_So
  7 KB (1,169 words) - 15:28, 13 May 2024
• 1999 AIME Problems/Problem 14
  ...{}</math> is located inside [[triangle]] <math>ABC</math> so that [[angle]]s <math>PAB, PBC,</math> and <math>PCA</math> are all congruent. The sides o MP("13",(A+B)/2,S);MP("15",(A+C)/2,NW);MP("14",(C+B)/2,NE);
  7 KB (1,184 words) - 13:25, 22 December 2022
• 1999 AIME Problems/Problem 12
  ...1)(50 + x) = A</math>. By [[Heron's formula]], <math>A = \sqrt{s(s-a)(s-b)(s-c)} = \sqrt{(50+x)(x)(23)(27)}</math>. Equating and squaring both sides, ...des of triangles are <math>a,b,c</math>, the area of the triangle is <math>S</math>.
  3 KB (472 words) - 15:59, 25 February 2022
• 1999 AIME Problems/Problem 11
  ...a \sin b = \frac 12(\cos (a-b) - \cos (a+b))</math>, we can rewrite <math>s</math> as s \cdot \sin 5 = \sum_{k=1}^{35} \sin 5k \sin 5 &= \sum_{k=1}^{35} \frac{1}{2
  4 KB (614 words) - 04:38, 8 December 2023
• 1999 AIME Problems/Problem 10
  ...th> and <math>n_{}</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m+n.</math> Instead of working with the four segments, let's focus on their endpoints. When we select these segments, we are working wit
  3 KB (524 words) - 17:25, 17 July 2023
• 1999 AIME Problems/Problem 9
  A function <math>f</math> is defined on the [[complex number]]s by <math>f(z)=(a+bi)z,</math> where <math>a_{}</math> and <math>b_{}</math> Let's start canceling. <math>2ad^2+2bcd=-2ac^2+2bcd+c^2+d^2</math>. Subtracting,
  6 KB (1,010 words) - 19:01, 24 May 2023
• 1999 AIME Problems/Problem 7
  *Case of no 2's: ...mathrm{odd})(\mathrm{odd})</math>. There are <math>5</math> [[odd integer]]s in <math>0</math> to <math>9</math>, so we have <math>5 \times 5 \times 5 =
  3 KB (475 words) - 13:33, 4 July 2016
• 1999 AIME Problems/Problem 5
  ...)</math> be <math>|S(x+2)-S(x)|.</math> For example, <math>T(199)=|S(201)-S(199)|=|3-19|=16.</math> How many values of <math>T(x)</math> do not exceed
  1 KB (170 words) - 19:40, 4 July 2013
• 1999 AIME Problems/Problem 8
  ...left(\frac 12,\frac 13,\frac 16\right).</math> The area of <math>\mathcal{S}</math> divided by the area of <math>\mathcal{T}</math> is <math>m/n,</math ...rt{3}}{18}, \frac{\sqrt{3}}{72}</math>. The [[ratio]] <math>\frac{\mathcal{S}}{\mathcal{T}} = \frac{\frac{9\sqrt{3} + 4\sqrt{3} + \sqrt{3}}{72}}{\frac{\
  3 KB (445 words) - 19:40, 4 July 2013
• 2000 AIME I Problems/Problem 15
  Next, let's think about how we can construct a stack of <math>n+1</math> cards from a s
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 14
  ...\)",P,(1,1)); label("\(Q\)",Q,(-1,1));label("\(R\)",R,(1,0));label("\(S\)",S,(-1,0)); Let <math>S</math> be the intersection of <math>QC</math> and <math>BP</math>. Since <m
  8 KB (1,275 words) - 03:04, 27 February 2022
• 2000 AIME I Problems/Problem 11
  ...What is the [[floor function|greatest integer]] that does not exceed <math>S/10</math>? ...8124}{4000} = 2480 + \frac{437}{1000}</math>, and <math>\left\lfloor \frac{S}{10} \right\rfloor = \boxed{248}</math>.
  4 KB (667 words) - 13:58, 31 July 2020
• 2000 AIME I Problems/Problem 10
  ...teger <math>k</math>, <math>x_k = \mathbb{S}-x_k-k \Longrightarrow \mathbb{S} - 2x_k = k</math>. Summing this up for all <math>k</math> from <math>1, 2, <cmath>\begin{align*}100\mathbb{S}-2(x_1 + x_2 + \cdots + x_{100}) &= 1 + 2 + \cdots + 100\\
  2 KB (319 words) - 22:26, 29 December 2022
• 2000 AIME I Problems/Problem 9
  ...</math> be <math>\log x, \log y, \log z</math> respectively. Using [[Simon's Favorite Factoring Trick|SFFT]], the above equations become (*)
  4 KB (623 words) - 15:56, 8 May 2021
• 2000 AIME I Problems/Problem 1
  ...e case when the two factors have the <math>2</math>s and the <math>5</math>s separated, so we need to find the first power of 2 or 5 that contains a 0.
  1 KB (163 words) - 17:44, 16 December 2020
• 2001 AIME I Problems/Problem 14
  ...o two consecutive <math>1</math>'s and no three consecutive <math>0</math>'s. ..._{n-3}</math> ( the cases are 01 and 001). We can refer back to solution 1's table and only keep track of <math>b_n</math>, ignoring both <math>a_n</mat
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2001 AIME I Problems/Problem 13
  Using Ptolemy's theorem,
  3 KB (561 words) - 19:25, 27 November 2022
• 2001 AIME I Problems/Problem 12
  label("$A$",A,S); label("$B$",B,S);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2014 USAJMO Problems
  ...ateral <math>BPQC</math>. Determine the range of possible values for <math>s/t</math>.
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 10
  ...that the [[midpoint]] of the segment they determine also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are rel ...nts' coordinates must be the same in order for the midpoint to be in <math>S</math>.
  8 KB (1,187 words) - 02:40, 28 November 2020
• 2001 AIME I Problems/Problem 8
  ...1}\cdots a_0)_{10}</cmath> (This is because the digits in <math>N</math> ' s base 7 representation make a number with the same digits in base 10 when mu Since the <math>a_i</math>s are base-<math>7</math> digits, it follows that <math>a_i < 7</math>, and t
  3 KB (502 words) - 11:28, 9 December 2023
• 2001 AIME I Problems/Problem 7
  ...</math>, so that <math>BI</math> and <math>CI</math> are [[angle bisector]]s of <math>\angle ABC</math> and <math>\angle ACB</math> respectively. Then, ...a <math>A = rs</math>, we find that the [[inradius]] is <math>r = \frac{A}{s} = \frac{\sqrt{1311}}6</math>. Since <math>\triangle ADE \sim \triangle ABC
  9 KB (1,540 words) - 08:31, 1 December 2022
• 2001 AIME I Problems/Problem 6
  .../math> and <math>n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <math>m + n</math>. ...</math>s, <math>e</math> <math>5</math>s, and <math>f</math> <math>6</math>s. Now we see that we must have <cmath>a + b + c + d + e + f = 4</cmath> with
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2001 AIME I Problems/Problem 4
  label("$12\sqrt 3$", D--B, S); label("$12$", A--D, S);
  3 KB (534 words) - 03:22, 23 January 2023

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