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Page title matches

• Median of a triangle
  A '''median''' of a [[triangle]] is a [[cevian]] of the triangle that joins one [[vertex]] to the [[midpoint]] of In the following figure, <math>AM</math> is a median of triangle <math>ABC</math>.
  1 KB (185 words) - 20:24, 6 March 2024
• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =
  5 KB (827 words) - 17:30, 21 February 2024
• Power of a Point Theorem/Introductory Problem 1
  Applying the [[Power of a Point Theorem]], we get <math> 3\cdot(3+5) = x (x+10) \rightarrow x^2 + 10x ''Back to the [[Power of a Point Theorem]].''
  448 bytes (67 words) - 15:15, 23 March 2020
• Power of a Point Theorem/Introductory Problem 2
  Applying the Power of a Point Theorem gives <math> 6\cdot x = 4\cdot 1 </math>, so <math> x = \frac ''Back to the [[Power of a Point Theorem]].''
  289 bytes (45 words) - 13:14, 16 July 2017
• Power of a Point Theorem/Introductory Problem 4
  ...</math> of a given circle are perpendicular to each other and intersect at a right angle at <math>E</math> . Given that <math> BE = 16, DE = 4, </math> pair A = (3,0);
  1 KB (177 words) - 02:14, 26 November 2020
• 2006 AIME A Problems/Problem 3
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• 2006 AIME A Problems/Problem 4
  44 bytes (5 words) - 12:03, 28 June 2009
• 2006 AIME A Problems/Problem 5
  44 bytes (5 words) - 18:41, 26 June 2009
• 2006 AIME A Problems/Problem 6
  44 bytes (5 words) - 12:05, 28 June 2009
• 2006 AIME A Problems/Problem 7
  44 bytes (5 words) - 13:37, 2 July 2009
• 2006 AIME A Problems/Problem 8
  44 bytes (5 words) - 13:38, 2 July 2009
• 2006 AIME A Problems/Problem 10
  45 bytes (5 words) - 11:33, 22 August 2009
• 2006 AIME A Problems/Problem 11
  45 bytes (5 words) - 15:15, 23 August 2009
• 2006 AIME A Problems/Problem 12
  45 bytes (5 words) - 11:15, 28 June 2009
• 2006 AIME A Problems/Problem 13
  45 bytes (5 words) - 12:14, 28 June 2009
• Graph of a function
  A '''graph''' is a visual representation of a [[function]]. If <math>f(x) = y</math> then the [[point]] <math>(x,y)</mat ...is the simplest thing to graph. The graph of <math>(2,5)</math> would be a dot 2 units to the right of <math>y</math>-axis and 5 units above the <math
  3 KB (497 words) - 05:11, 8 June 2023
• A:NOT
  64 bytes (10 words) - 17:30, 29 July 2020
• A:T
  52 bytes (7 words) - 15:40, 8 December 2020
• A:SAND
  51 bytes (7 words) - 07:44, 15 July 2020
• A:CP
  59 bytes (7 words) - 23:22, 11 July 2011
• A:PORT
  59 bytes (7 words) - 23:17, 11 July 2011
• A:DB
  39 bytes (5 words) - 18:26, 8 June 2020
• A:WL
  #REDIRECT [[A:AL]]
  18 bytes (3 words) - 18:07, 16 January 2024
• A:AL
  56 bytes (7 words) - 18:07, 16 January 2024
• Cycle of a permutation
  32 bytes (3 words) - 16:50, 31 October 2010
• A:FAQ
  26 bytes (3 words) - 23:58, 17 June 2011
• A:CR
  66 bytes (6 words) - 12:56, 19 August 2011
• If I have four boxes arranged in a $2 \times 2$ grid, in how many distinct ways can I place the digits $1$, $2$, and $3$ in the boxes, using each digit exactly once, such that each box contains at most one digit? (I only have one of each digit, so one box
  We can think of placing a <math>0</math> in the fourth box that will necessarily be empty. Now the pr
  328 bytes (60 words) - 17:19, 12 June 2022
• The perfect squares from $1$ through $2500,$ inclusive, are printed in a sequence of digits $1491625\ldots2500.$ How many digits are in the sequence?
  ...ts is <math>4^{2},</math> and the largest is <math>9^{2},</math> so that's a total of <math>6</math> perfect squares with <math>2</math> digits. ...s is <math>10^{2},</math> and the largest is <math>31^{2},</math> yielding a total of <math>22.</math>
  939 bytes (140 words) - 17:27, 12 June 2022
• A-Star Summer Math Camp
  #REDIRECT [[A* Summer Math Camp]]
  33 bytes (5 words) - 16:05, 28 November 2019
• A-Star Winter Math Camp
  A* Winter Math Camp has programs for 4th - 12th graders in Bay Area. The prog You can find more information about the A* Winter/Summer Math Camps here:
  629 bytes (95 words) - 00:49, 5 January 2018
• Creating a forum
  #REDIRECT [[AoPSWiki:FAQ#How do I create a forum%3F]]
  53 bytes (10 words) - 20:35, 17 September 2016
• A* Summer Math Camp
  ...an expert faculty via utilizing national and international competitions in a fun and challenging environment. You can find more information about the A* Summer Math Camp here:
  1 KB (177 words) - 19:31, 15 December 2017
• A* Winter Math Camp
  A* Winter Math Camp has programs for 4th - 12th graders in Bay Area. The prog You can find more information about the A* Winter/Summer Math Camps here:
  629 bytes (95 words) - 00:58, 5 January 2018
• Ali-A
  ...econd channel, More Ali-A, has a YouTube subscriber count of 5 million and a total of over 880 million video views. Ali-A plays only first-person shooter games on his main channel, but he plays a wide variety of games on his second channel, including Roblox, Pokemon GO,
  912 bytes (148 words) - 21:41, 17 July 2021
• 2008 AMC 12 A
  37 bytes (7 words) - 10:39, 5 May 2019
• Power of a Point
  #REDIRECT [[Power of a Point Theorem]]
  38 bytes (6 words) - 16:53, 15 April 2019
• Points $A$, $B$, $C$, and $D$ lie on a line, in that order. If $AB=2$ units, $BC=5$ units and $AD=14$ units, what is the ratio of $AC$ to $BD$? Express your answer as a common fraction.
  119 bytes (19 words) - 11:01, 20 March 2019
• A stock loses $10\%$ of its value on Monday. On Tuesday it loses $20\%$ of the value it had at the end of the day on Monday. What is the overall percent loss in value from the beginning of Monday to the end of Tuesday? Enter the answer as a percent.
  189 bytes (27 words) - 15:42, 15 April 2021
• Adopt a Page
  Adopt a page is an AoPS Wiki Charity devoted to adopting orphans.<br/> To adopt a page just go to the link below and pick out your favorite.<br/>
  354 bytes (57 words) - 15:52, 21 May 2019
• A choose b
  ...om{a}{b}=\frac{a!}{b!(a-b)!}</math>. This is assuming that of course <math>a \ge b </math>. a choose b counts the number of ways you can pick b things from a set of a things. For example <math>\binom{8}{2}=\frac{8!}{2!(8-2)!}=\frac{8(7)}{2}=\
  3 KB (407 words) - 20:03, 25 July 2021
• Let a and b be nonzero real numbers such that (2 - 7i)(a + bi)is pure imaginary. Find a/b.
  37 bytes (3 words) - 22:18, 7 September 2019
• In the diagram, the equilateral triangle has a base of $8$ m. What is the perimeter of the triangle?
  In the diagram, the equilateral triangle has a base of <math>8</math> m. What is the perimeter of the triangle?
  111 bytes (20 words) - 04:34, 11 May 2021
• The figure shows a square of side $y$ units divided into a square of side $x$ units and four congruent rectangles. What is the perimeter, in units, of one of the four congruent rectangles? Express your answer in terms of $y$.
  0 bytes (0 words) - 22:13, 4 March 2020
• Median of a Triangle in LaTeX
  The [[median]] of a [[triangle]] can be depicted using <math>\LaTeX</math>. We begin by drawing a basic triangle:
  865 bytes (142 words) - 14:06, 25 March 2020
• AoPS Online School/Introduction to Algebra A
  Please look [https://artofproblemsolving.com/school/course/algebra-a?gtmlist=Schedule_Side here]
  96 bytes (12 words) - 13:45, 25 September 2020
• Four fair coins are to be flipped. What is the probability that all four will be heads or all four will be tails? Express your answer as a common fraction.
  ...he deck is randomly arranged. What is the probability that the top card is a 5?
  527 bytes (81 words) - 21:02, 23 December 2020
• A:B
  28 bytes (4 words) - 12:49, 31 March 2021
• The local theater has one ticket window. In how many ways can six people line up to buy a ticket?
  ...ater has one ticket window. In how many ways can six people line up to buy a ticket?
  97 bytes (20 words) - 16:49, 28 February 2021
• Suppose that a and b are positive integers for which a has 3 factors and b has a factors. If b is divisible by a, then what is the least possible value of b?
  .../math> has <math>a</math> factors. If <math>b</math> is divisible by <math>a</math>, then what is the least possible value of $b?
  262 bytes (49 words) - 13:14, 3 August 2021
• How many four-digit, positive integers are there where each digit is a prime number?
  223 bytes (36 words) - 17:15, 12 June 2022
• The positive three-digit integer N has a ones digit of 0. What is the probability that N is divisible by 4? Express your answer as a common fraction.
  ...0). Since there are the same number of even digits as odd digits, there is a <math>\boxed{1/2}</math> probability that <math>N</math> is divisible by 4.
  427 bytes (71 words) - 17:35, 12 June 2022
• Suppose we flip four coins simultaneously: a penny, a nickel, a dime, and a quarter. What is the probability that the penny and dime both come up the same?
  460 bytes (67 words) - 17:43, 12 June 2022
• The equation $y = -16t^2 + 28t + 144$ describes the height (in feet) of a ball tossed up in the air at $28$ feet per second from a height of $144$ feet from the ground. In how many seconds will the ball hit the ground?
  10 bytes (1 word) - 02:18, 15 June 2022
• Smarty101:A
  0 bytes (0 words) - 22:11, 25 July 2022
• The table shows the percent of families in Mathville that have $0, 1, 2, 3$ and $4$ or more children. If there are a total of $10,250$ families, how many are there with no children?
  ...<math>0, 1, 2, 3</math> and <math>4</math> or more children. If there are a total of <math>10,250</math> families, how many are there with no children?
  214 bytes (35 words) - 09:55, 1 August 2022
• How many ways can $1995$ be factored as a product of two two-digit numbers? (Two factorizations of the form $a\cdot b$ and $b\cdot a$ are considered the same).
  1 bytes (0 words) - 21:31, 3 October 2022
• Area of a Triangle
  #REDIRECT [[Area#Area of a Triangle]]
  37 bytes (6 words) - 22:14, 17 November 2022
• A MatheMagician
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• Garmin launches a new FDA-cleared ECG app for the Venu 2 Plus
  ...ability to take an ECG reading from their wrist. The company is launching a new ECG app that allows users to record their heart rhythm and check for si ...ond ECG and view their heart rhythm results immediately on the watch or at a later time in the Garmin Connect app. When users take an ECG, the new app u
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• "* in a unit circle"
  0 bytes (0 words) - 12:14, 16 July 2023
• Insiders bet more on Fizz a social network that has now bubbled up at 80 college campuses
  Fizz, a burgeoning social network established by two Stanford dropouts, captured th Fizz operates on a distinct model that invites individuals with a Stanford email address to join its network, allowing them to contribute and
  10 KB (1,649 words) - 05:53, 11 August 2023
• Fat Man on a Seesaw
  The '''Fat Man on a Seesaw''' method is a physics approach to solving [[mean]] problems. It was developed by Ali Gure
  263 bytes (36 words) - 15:27, 12 November 2023
• Mean of a set
  18 bytes (2 words) - 15:30, 12 November 2023
• Mode of a set
  18 bytes (2 words) - 15:30, 12 November 2023
• For what real values of $c$ is $4x^2 + 5x^2 + 14x + x + c$ the square of a binomial?
  0 bytes (0 words) - 22:42, 26 January 2024
• A game board is constructed by shading two of the regions formed by the altitudes of an equilateral triangle as shown. What is the probability that the tip of the spinner will come to rest in a shaded region? Express your answer as a common fraction.
  3 bytes (0 words) - 23:23, 28 January 2024
• In triangle ABC, D be a point in BC, ∠BAD = 30 , ∠CAD = 90 , BD=1=AC, DC =
  0 bytes (0 words) - 13:26, 6 February 2024
• A
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• Power of a point theorem
  ...e point. Can be useful with [[cyclic quadrilaterals]] as well however with a slightly different application. ...</math> and <math> CD </math> intersect at a point <math> P </math> within a circle, then <math> AP\cdot BP=CP\cdot DP </math>
  6 KB (1,010 words) - 02:38, 7 May 2024

Page text matches

• Pythagorean Theorem
  ...n [[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 right tria In these proofs, we will let <math>ABC </math> be any right triangle with a right angle at <math>{} C </math>.
  5 KB (886 words) - 13:51, 15 May 2024
• 2005 AMC 12A Problems/Problem 16
  ...and the third is tangent to the first circle and the <math>y</math>-axis. A circle of radius <math>r > s</math> is tangent to both axes and to the seco <math> (\mathrm {A}) \ 5 \qquad (\mathrm {B}) \ 6 \qquad (\mathrm {C})\ 8 \qquad (\mathrm {D})
  2 KB (307 words) - 15:30, 30 March 2024
• Main Page
  If you don't know much about how to edit a wiki, you might want to read the [[AoPS_Wiki:Tutorial|tutorial]] and the [[
  2 KB (258 words) - 14:05, 15 November 2023
• 2016 AMC 10A Problems/Problem 2
  <math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2\qquad\textbf{(C)}\ 3\qquad\textbf{(D)}\ 4\qquad\ {{AMC10 box|year=2016|ab=A|num-b=1|num-a=3}}
  1 KB (190 words) - 10:58, 16 June 2023
• 2016 AMC 10A Problems/Problem 3
  <math>\textbf{(A)}\ \$37.50 \qquad\textbf{(B)}\ \$50.00\qquad\textbf{(C)}\ \$87.50\qquad\tex {{AMC10 box|year=2016|ab=A|num-b=2|num-a=4}}
  1 KB (176 words) - 10:58, 16 June 2023
• 2016 AMC 10A Problems/Problem 4
  <math>\textbf{(A) } -\frac{3}{8} \qquad \textbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \ Do note that the denominator of the answer will be a multiple of 5 and 8 (40) and that the answer will also be negative. The onl
  2 KB (257 words) - 10:57, 16 June 2023
• 2016 AMC 10A Problems/Problem 5
  A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. <math>\textbf{(A)}\ 48\qquad\textbf{(B)}\ 56\qquad\textbf{(C)}\ 64\qquad\textbf{(D)}\ 96\qqu
  1 KB (184 words) - 13:58, 22 August 2023
• 2016 AMC 10A Problems/Problem 6
  <math>\textbf{(A)}\ 13\qquad\textbf{(B)}\ 26\qquad\textbf{(C)}\ 102\qquad\textbf{(D)}\ 103\q ...act 1. Because 2 appears 10 times as a tens digit and 2 appears 3 times as a units digit, the answer is <math>10\cdot 10+1\cdot 3=\boxed{\textbf{(D) }10
  967 bytes (143 words) - 03:18, 27 June 2023
• 2016 AMC 10A Problems/Problem 7
  <math>\textbf{(A)}\ 50 \qquad\textbf{(B)}\ 60 \qquad\textbf{(C)}\ 75 \qquad\textbf{(D)}\ 90 Note: if the mean of a set is in the set, it can be discarded and the mean of the remaining number
  2 KB (268 words) - 18:19, 27 September 2023
• 2016 AMC 10A Problems/Problem 8
  <math>\textbf{(A)}\ 20 \qquad\textbf{(B)}\ 30\qquad\textbf{(C)}\ 35\qquad\textbf{(D)}\ 40\qq {{AMC10 box|year=2016|ab=A|num-b=7|num-a=9}}
  1 KB (169 words) - 14:59, 8 August 2021
• 2016 AMC 10A Problems/Problem 9
  A triangular array of <math>2016</math> coins has <math>1</math> coin in the <math>\textbf{(A)}\ 6\qquad\textbf{(B)}\ 7\qquad\textbf{(C)}\ 8\qquad\textbf{(D)}\ 9\qquad\t
  2 KB (315 words) - 15:34, 18 June 2022
• 2016 AMC 10A Problems/Problem 10
  A rug is made with three different colors as shown. The areas of the three di <math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \
  2 KB (337 words) - 14:56, 25 June 2023
• 2016 AMC 10A Problems/Problem 11
  <math>\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \ ...{2}</math>, and slide the left and right pieces next to each other to form a parallelogram with base <math>1</math> and height <math>\frac{8}{2}</math>.
  8 KB (1,016 words) - 00:17, 31 December 2023
• 2016 AMC 10A Problems/Problem 16
  .../math>. Which of the following transformations will return <math>\triangle A''B''C''</math> to <math>\triangle ABC</math>? <math>\textbf{(A)}</math> counterclockwise rotation about the origin by <math>90^{\circ}</ma
  1 KB (235 words) - 14:52, 25 June 2023
• 2016 AMC 10A Problems/Problem 12
  ...<math>1</math> and <math>2016</math>, inclusive. Which of the following is a correct statement about the probability <math>p</math> that the product of <math>\textbf{(A)}\ p<\dfrac{1}{8}\qquad\textbf{(B)}\ p=\dfrac{1}{8}\qquad\textbf{(C)}\ \dfr
  2 KB (297 words) - 14:54, 25 June 2023
• 2016 AMC 10A Problems/Problem 13
  Five friends sat in a movie theater in a row containing <math>5</math> seats, numbered <math>1</math> to <math>5</ma <math>\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qq
  2 KB (402 words) - 14:54, 25 June 2023
• List of United States elementary school mathematics competitions
  * [http://www.mindresearch.org/gameathon/ Game-a-thon] challenges students to design and build math games in this national c * [http://www.mathnasium.com/trimathlon National Trimathlon] is a fun filled Math Competition for kids in grades 3 through 6.
  4 KB (473 words) - 16:14, 1 May 2024
• List of United States middle school mathematics competitions
  ...ath/puzzle competitions for middle schoolers with a focus towards creating a fun and collaborative environment. * [[AMC 8]] hosted by the [[American Mathematics Competitions]] is a very large middle school math contest taken in-school. ([http://www.maa.org
  7 KB (792 words) - 10:14, 23 April 2024
• List of United States high school mathematics competitions
  * [http://homeschoolmathcontests.com Homeschool Math Contests] has a chart of math contests open to individuals (homeschooled and otherwise), an * [[Mu Alpha Theta]] -- hosts both local and a national competition.
  5 KB (583 words) - 17:26, 21 February 2024
• List of mathematics competitions
  This is a directory of internal links to more helpful pages about mathematics competi
  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 number of quality contests around the state for both middle and high school * [[Mobile Math Olympiad]] -- Winners are invited to a national competition. [http://www.southalabama.edu/mathstat/info/schedules/
  3 KB (411 words) - 21:32, 8 December 2014
• Mathematics summer program
  .... These programs mostly take place during the summer, but some last during a significant portion of the academic year. Selection is important in creatin ...programs listed below offer [[mathematics scholarships]] or financial aid. A few of them are free. When you discover more that do, please add that infor
  14 KB (1,869 words) - 20:06, 18 May 2024
• MathILy (serious mathematics infused with levity)
  '''{MathILy, MathILy-Er}''' are a pair of intensive 5-week-long residential summer math programs for high sch ...ts for about 7 hours a day total, in a morning and evening shift, six days a week. The lead instructors are mathematicians with Ph.D.s and the apprenti
  5 KB (706 words) - 23:49, 29 January 2024
• Mathematics scholarships
  ...generous stipends for up to $90,000 for students training to teach math at a New York City high school. [http://www.mathforamerica.org/home website]
  3 KB (409 words) - 15:02, 28 January 2022
• Mathematics, science, and technology scholarships
  ...t important such competition in the world. The 3 ISEF winners each receive a <dollar/>50,000 scholarship. [http://www.sciserv.org/isef website] ...in exchange for summer internships and a year of work after graduation in a DoD lab. [http://smart.asee.org/ website]
  7 KB (851 words) - 10:54, 29 January 2022
• Proton Energy Scholarship
  * Current high school senior enrolled in a U.S. high school * Declare/Intend to declare a science or technology-related major (pure mathematics does not count)
  894 bytes (123 words) - 15:48, 13 June 2022
• Mathematics competitions
  Math competitions differ widely in difficulty and content. Most test a broader range of mathematics curriculum and some focus on methods of [[math
  698 bytes (88 words) - 14:36, 24 January 2024
• MATHCOUNTS
  ...nals Countdown Round participants in the past decade. '''MATHCOUNTS''' is a large national [[mathematics competition]] and [[mathematics coaching]] pro 8 problems are given 2 at a time. Students have 6 minutes to complete each set of two problems. Studen
  10 KB (1,506 words) - 21:31, 14 May 2024
• California mathematics competitions
  *[http://www.mathnasium.com/trimathlon National Trimathlon] is a fun filled Math Competition for kids in grades 3 through 6. * [[Southern California Mathematics Competition]], a new contest hosted by USC [https://engage.usc.edu/scmathcompetition/home/ w
  4 KB (514 words) - 04:02, 21 September 2023
• Georgia mathematics competitions
  *[[GACS - Dwight Love Math Tournament]]—a large middle and high school tournament [http://www.gacs.pvt.k12.ga.us/facs *[[GACS - Dwight Love Math Tournament]]&mdash;a large middle and high school tournament. [http://www.gacs.pvt.k12.ga.us/fac
  4 KB (565 words) - 13:24, 13 September 2019
• Hawaii mathematics competitions
  * [[Math Matters]] is a local mathematics contest for the county of Maui. *[[Hawaii Mathematics Bowl]] is a statewide mathematics contest for the state of Hawaii.
  989 bytes (123 words) - 15:34, 29 August 2017
• ICTM Math Contest
  ...consists of five members who work together on 20 questions for 25 minutes. A school may field as many five member teams as it wishes. The individual com A school team will qualify by taking first place in a region with four or more schools competing.
  8 KB (1,182 words) - 14:26, 3 April 2024
• Kansas mathematics competitions
  ...vidual competition, where participants take tests on various subjects, and a team competition, Mathletics. The 2006 competition had over 1000 participan ...12). It is divided into three divisions, grades 5-6, 7-9 and 10-12, and is a 40 minute, 5 question test. First, second and third in each division win <m
  3 KB (473 words) - 16:11, 16 June 2020
• Massachusetts mathematics competitions
  *[[Winchester Math Competition]] (WinMaC) is a competition for fourth and fifth (until 2024 where they switched it up to 4 ...yle round, and team is a team round. This test is taken over the course of a day
  5 KB (733 words) - 20:58, 27 April 2024
• Michigan mathematics competitions
  *[[Rapid Rabbit Relays]] (a middle school event at Detroit Country Day School) *[[Michigan Math Meet]], a high school math competition organized by University of Michigan undergradu
  2 KB (305 words) - 01:06, 30 November 2023
• Nebraska mathematics competitions
  ...Nebraska-Omaha Math Contest]]. Consists of two problem solving rounds and a speed round - for grades 9-12.
  2 KB (200 words) - 10:11, 29 December 2020
• Oklahoma mathematics competitions
  ...ed) students from that year's MathCounts state competition will compete as a mixed team. Oklahoma does not participate in the high school contest. *[http://rocketcitymath.org/ Rocket City Math League] is a contest for middle school and high schools students. Jenks and Norman North
  2 KB (279 words) - 19:41, 23 March 2017
• Texas mathematics competitions
  *[[Texas A&M University math Contest]] [http://www.math.tamu.edu/teaching/undergrad/hi *[http://www.math.tamu.edu/outreach/highschoolcontest/ Texas A&M University Math Contest] [http://www.math.tamu.edu/outreach/highschoolcon
  2 KB (255 words) - 07:55, 2 March 2023
• Utah mathematics competitions
  *Mission Math Utah [http://missionmathutah.com/ (website)] is a contest for elementary and middle school students in Utah, run by Utah high
  2 KB (231 words) - 13:07, 14 December 2019
• Washington mathematics competitions
  * The [[Mustang Math Tournament]] a national team-based competition hosted by mathematically inclined high scho
  2 KB (263 words) - 01:25, 11 June 2022
• West Virginia mathematics competitions
  ...ion, a mental math competition (where only exact answers are correct), and a physical estimation competition. Grades 4-9 compete in this format. Grades
  2 KB (216 words) - 16:59, 17 December 2016
• Art of Problem Solving
  For a list of all the current staff at AoPS and some previous staff go to the [[A * [[AoPSWiki:Table of Contents]] -- a basic guide to AoPSWiki content
  5 KB (667 words) - 17:09, 3 July 2023
• Resources for mathematics competitions
  ...itions]] leads to links for many of these competition homepages. Here are a few examples: * [https://brilliant.org/ Brilliant] is a website where one can solve problems to gain points and go to higher levels
  16 KB (2,152 words) - 21:46, 6 May 2024
• Mathematics forums
  ...actice teaching math, and to socialize with other math students. Here are a few popular online math forums. ...x.php AoPS Forums] with thousands of brilliant, active members. Over half a million posts ranging from middle school math all the way up to graduate le
  869 bytes (123 words) - 16:55, 11 January 2020
• Mathematics websites
  ...reat '''Mathematics websites''' around the [[internet]]. Here we organize a list of those sites we feel are best for students with high interest in [[m * [https://www.pims.math.ca/resources/publications/pi-sky Pi in the Sky] is a mathematics magazine for high school students.
  4 KB (516 words) - 03:01, 13 April 2023
• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato ...3265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]
  24 KB (3,177 words) - 12:53, 20 February 2024
• Richard Rusczyk
  ...os Angeles. He graduated from Princeton University in 1993, and worked as a bond trader for D.E. Shaw & Company for four years.
  2 KB (254 words) - 09:04, 25 January 2019
• Math Olympiads for Elementary and Middle Schools
  '''Math Olympiads for Elementary and Middle Schools''' ('''MOEMS''') is a large and popular [[mathematics competition]] for students in grades 4 thro ...oblemsolving.com/Forum/index.php?f=343 MOEMS Forum] at [[AoPS]] as well as a private [http://www.artofproblemsolving.com/Forum/viewtopic.php?t=38770 MOE
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• Math textbooks
  This '''Math textbooks''' page is for compiling a list of [[textbook]]s for mathematics -- not problem books, contest books, ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato
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• Math circles
  ...ca. Inspired by Eastern European teaching models, math circles often take a problem-discussion approach to teaching and learning [[mathematics]]. * The [[Art of Problem Solving Foundation]] runs a local programs initiative to support the establishment of programs includin
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• William Lowell Putnam Mathematical Competition
  The '''William Lowell Putnam Mathematical Competition''' is a highly challenging, proof-oriented [[mathematics competition]] for undergra ...own=<u>Problem A/B, 1/2</u>: 7<br><u>Problem A/B, 3/4</u>: 8<br><u>Problem A/B, 5/6</u>: 9}}
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• Chemistry competitions
  This is a resources page for students interested in '''Chemistry competitions'''. Pl The U.S. National Chemistry Olympiad national exam (USNCO) is a 3-part, 4 hour and 45 minute exam administered in mid or late April by ACS
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• Physics competitions
  This is a resources page for students interested in '''Physics competitions'''. Plea
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• Computer science books
  ...p://olympiads.win.tue.nl/ioi/study/books.html Reading List] (someone have a linky?)
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• Physics books
  .../www.amazon.com/exec/obidos/ASIN/0756613647/artofproblems-20 Chaos: Making a New Science] by James Gleick ...le.amazon.com/Dreams-Final-Theory-Steven-Weinberg/dp/0679419233/ Dreams of a Final Theory] by Weinberg
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• Physics
  <strong>Physics</strong> is a branch of [[science]] that studies the properties of matter, energy, and ma Modern Physics is also a group of different subjects in physics:
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• 2005 Alabama ARML TST Problems/Problem 7
  {{ARML box|year=2005|state=Alabama|num-b=6|num-a=8}}
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• Newton
  A '''Newton''' (abbreviated N) is the [[Système international|metric]] measu
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• Mathematics
  ..., and the formation of turbulence around an aircraft. Mathematics provides a kind of "quality control" for the development of trustworthy theories and e Modern mathematics is built around a system of [[axiom|axioms]], which is a name given to "the rules of the game." Mathematicians then use various meth
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• AMC historical results
  ...at a consistent number of students qualify for AIME each year, rather than a fixed percentage. Usually, 6000-7000 competitors from the AMC 10 and 12 qua
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• Inequality
  Inequalities are arguably a branch of [[elementary algebra]], and relate slightly to [[number theory]]. For two [[number]]s <math>a</math> and <math>b</math>:
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• Science scholarships
  ...t important such competition in the world. The 3 ISEF winners each receive a <dollar/>50,000 scholarship. [http://www.sciserv.org/isef]
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• United States of America Mathematical Talent Search
  ...''United States of America Mathematical Talent Search''' ('''USAMTS''') is a [[mathematics competition]] in which students are challenged to write full Each problems are graded on a scale from 0 to 5. The full 5 points are awarded for correct, concise solut
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• Academic scholarships
  ...ries or are best listed under a category rather than a list, please create a new page for that category and list it here. ...deline for scholarships: if you're paying to apply then it is probably not a legitimate organization of scholarship.
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• AMC 8
  The '''AMC 8''' is a contest for students in grades 8 and below, hosted annually by the [[Americ Usually, high scoring students will be given a chance by their school to take the more challenging [[AMC 10]] exam. Howeve
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• 1999 JBMO Problems/Problem 1
  ...ath> b^3 + bx + y = 0</math> and <math> c^3 + cx + y = 0</math>. If <math> a,b,c</math> are all distinct numbers prove that their sum is zero. -y &= a^3 + ax \\
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• AMC 10
  ...an Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC. The AMC 10 is a 25 question, 75 minute multiple choice test. Problems generally increase in
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• AMC 12
  ...an Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC! The AMC 12 is a 25 question, 75 minute multiple choice test. Problems generally increase in
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• 1955 AHSME Problems/Problem 38
  <math>\textbf{(A)}\ 19 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ Define numbers <math>a, b, c,</math> and <math>d</math> to be the four numbers. In order to satisf
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• Knowledge bowl competitions
  * [[Texas A&M Regional Science Bowl]] [http://outreach.science.tamu.edu/sciencebowl.asp
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• American Invitational Mathematics Examination
  ...ical Association of America]] (MAA). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC! The AIME is a 15 question, 3 hour exam<math>^1</math> taken by high scorers on the [[AMC
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• Math Jam
  A '''Math Jam''' is a free online class or information session hosted by [[Art of Problem Solving
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• American Mathematics Competitions
  The '''American Mathematics Competitions''' (AMC) consist of a series of increasingly difficult tests for students in middle school and hi ...ilable nationwide. The amount of questions also decreased from 50 to 35 in a span of 17 years. In 1974, the Annual High School Contest was re-named the
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• 1954 AHSME Problems/Problem 46
  In the diagram, if points <math>A, B</math> and <math>C</math> are points of tangency, then <math>x</math> eq pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
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• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
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• United States of America Mathematical Olympiad
  ...tes of America Mathematical Olympiad''' ('''USAMO''') is the third test in a series of exams used to challenge bright students on the path toward choosi ...an Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC and of the recent expansion of USAMO participants
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• Academic competitions
  ...emonstrate their proficiency in academic subjects at a level above that of a typical classroom setting or [[standardized test | standardized testing]].
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• Math competition books
  ...ific individual [[mathematics competitions]]. Compilations of problems at a single level from multiple competitions can be found in the [[math books]] ...mazon.com/exec/obidos/ASIN/0387242996/artofproblems-20 The IMO Compendium: A Collection of Problems Suggested for The International Mathematical Olympia
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• Informatics competitions
  '''Informatics competitions''' test a student's ability to understand, organize, and work with information on com
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• Valentin Vornicu
  '''Valentin Vornicu''' is a former [[Art of Problem Solving]] administrator and webmaster. In 2002, he Vornicu is a multiple time participant in the [[IMO]] for Romania.
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• American Regions Mathematics League
  The '''American Regions Math League''' (ARML) is a [[mathematical problem solving]] competition primarily for U.S. high school ...am already. Contact ARML for details. If your area does not already have a team, ask ARML how to start one.
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• Joining an ARML team
  The SD team is entering just its third year of participation, so a permanent process of selecting team members has yet to be decided upon. The A student who wishes to attend practice should try to take the SDMO. If a student wishes to speak with one of the coaches for the team, they may do s
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• Engineering competitions
  * [[Toy Challenge]] is a national toy design contest for grades 5-8. [http://www.toychallenge.com/ * [[TEAMS]] -- A [[JETS]] competition. [http://www.jets.org/programs/teams.cfm Website]
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• Science summer programs
  * [[MehtA+ AI in Visual Arts Camp]] is a mini-bootcamp for grades 5-12 [https://mehtaplustutoring.com/ai-in-visual-a * [[MehtA+ AI in Visual Arts Camp]] is a mini-bootcamp for grades 5-12 [https://mehtaplustutoring.com/ai-in-visual-a
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• Computer summer programs
  * [[MehtA+ AI in Visual Arts Camp]] is a mini-bootcamp for grades 5-12 [https://mehtaplustutoring.com/ai-in-visual-a * [[MehtA+ AI in Visual Arts Camp]] is a mini-bootcamp for grades 5-12 [https://mehtaplustutoring.com/ai-in-visual-a
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• Technology summer programs
  * [[MehtA+ AI in Visual Arts Camp]] is a mini-bootcamp for grades 5-12 [https://mehtaplustutoring.com/ai-in-visual-a * [[MehtA+ Competitive Programming Bootcamp]] is a mini-bootcamp for grades 7-12 [https://mehtaplus.com/courses/competitive-pr
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• Engineering summer programs
  * [[MehtA+ AI in Visual Arts Camp]] is a mini-bootcamp for grades 5-12 [https://mehtaplustutoring.com/ai-in-visual-a * [[MehtA+ AI in Visual Arts Camp]] is a mini-bootcamp for grades 5-12 [https://mehtaplustutoring.com/ai-in-visual-a
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• Mathematics Olympiads
  ...onal Mathematics Olympiad]] (IMO) is the most famous math Olympiad. In USA a Math Olympiad would be [[United States of America Mathematical Olympiad]] t
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• Proof writing
  ...ance, when a mathematician uses the word "trivial" in a proof, they intend a different meaning to how the word is understood by the wider population. St ...n, accuracy, and clarity. A single word can change the intended meaning of a proof, so it is best to be as precise as possible.
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• Gifted programs
  The following is a list of '''Gifted programs''' and organizations that run gifted programs.
  1,017 bytes (131 words) - 17:51, 18 July 2020
• Biology scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using
  1 KB (184 words) - 21:01, 12 December 2018
• Chemistry scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using
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• Physics scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using ...ntzen Award] for female sophomores and juniors in college who are pursuing a bachelors degree in physics or geosciences.
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• Technology scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using
  3 KB (345 words) - 17:54, 16 June 2023
• Computer science and technology scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using ...rship.aspx Computer science scholarships] with full or partial tuition and a salaried summer internship.
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• Math team
  ...r Elementary and Middle Schools]] (MOEMS). A math team can be as simple as a normal school math team, to the most advanced international math olympiad t
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• Engineering scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using ...eering (NACME) Scholarships] for minority engineering students enrolled at a partner university.
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• Community Service Scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using ...programs/scholarship.php YouthLaunch Scholarship for Outstanding Service]. A $3,000 scholarship for Texas students who demonstrate commitment to their c
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• Engineer
  An '''Engineer''' is a person that creates, designs, or updates machines or structures.
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• Princeton University Mathematics Competition
  The '''Princeton University Mathematics Competition (PUMaC)''' is a high school math competition held by [[Princeton University]] students sinc ...are two divisions in the competition: Division A and Division B. Division A is for experienced solvers and has harder problems, while Division B is for
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• Rocket City Math League
  The '''Rocket City Math League''' is a national [[mathematics competition]] for middle and high school students ru
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• Mandelbrot Competition
  ...competition]] for primarily high school students. It was often regarded as a predecessor to the Olympiad-level American mathematics competitions. ...he has compiled two books consisting of past contests. Lehoczky now enjoys a successful career on Wall Street and Rusczyk runs the [[Art of Problem Solv
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• Mu Alpha Theta
  ...national Mu Alpha Theta convention is held during the summer that includes a large competition. The test is a multiple choice test with 4 choices and an extra choice called NOTA, which
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• San Diego Math Olympiad
  ...Math League]] (SDML). There is a division for middle school students and a division for high school students.
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• AoPS Online School
  ...ebra 1) — [https://artofproblemsolving.com/school/course/catalog/algebra-a Details] ([https://artofproblemsolving.com/store/item/intro-algebra Compani ...ine School/Special AMC 8 Problem Seminar A | Special AMC 8 Problem Seminar A]] — [https://artofproblemsolving.com/school/course/catalog/maa-amc8-speci
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• Power Mean Inequality
  The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] I ...th> is flipped, but as <math>k_2<0</math>, <math>x^\frac{1}{k_2}</math> is a decreasing function, the inequality sign is flipped again after applying <m
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• Wilson's Theorem
  Suppose first that <math>p</math> is composite. Then <math>p</math> has a factor <math>d > 1</math> that is less than or equal to <math>p-1</math>. ...ique, and each number is the inverse of its inverse. If one integer <math>a</math> is its own inverse, then
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• 1953 AHSME Problems/Problem 13
  A triangle and a trapezoid are equal in area. They also have the same altitude. If the base <math>\textbf{(A)}\ 36\text{ inches} \qquad
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• Stanford University Mathematics Camp
  After a daily lecture, students are then tutored by counselors on problem sets from
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• Trivial Inequality
  <cmath>ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath> .../math> is positive and <math>ax^2+bx+c\le c-\frac{b^2}{4a}</math> if <math>a</math> is negative.
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• Arithmetic mean
  The '''arithmetic mean''' of a [[set]] of numbers (or variables) is the sum of all the numbers, divided by
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• Completing the square
  ...n equation to make that equation a [[perfect square]]. This makes solving a lot of equations easy. In fact, all [[quadratic equation]]s can be solved ...2-6x</math> part. If <math>9</math> was added to this, then we would have a [[perfect square]], <math>(x-3)^2=x^2-6x+9</math>. To do this, add <math>7
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• Heron's Formula
  ...rmula) is a [[mathematical formula | formula]] for finding the [[area]] of a [[triangle]] given only the three side lengths. ...any triangle with side lengths <math>{a}, {b}, {c}</math>, the area <math>{A}</math> can be found using the following formula:
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• Polynomial
  ...algebra]] often an arbitrary [[field]]). Note that a [[constant]] is also a polynomial. ===A More Precise Definition===
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• Simon's Favorite Factoring Trick
  ...of the equation and a product of variables with each of those variables in a linear term on the other side. An example would be: <cmath>xy+66x-88y=23333 ...Factoring Trick, this equation can be transformed into: <cmath>(x+k)(y+j)=a+jk</cmath>
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• USAMO historical results
  ...ct score on the [[USAMO]] has been 42 points for 6 problems. Prior to 1996 a perfect score was 100 points across 5 problems.
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• Quadratic formula
  ...''' is a general [[expression]] for the [[root (polynomial)|solutions]] to a [[quadratic equation]]. It is used when other methods, such as [[completing ===General Solution For A Quadratic by Completing the Square===
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• University of South Carolina High School Math Contest
  The individual test is a 30 question, multiple choice test. The time limit is 90 minutes. Calculat The test is scored as 5 points for every correct response, 1 point for a blank response, and 0 points for an incorrect response.
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• Factoring
  ...rs, is to write the expression (often an [[integer]] or [[polynomial]]) as a product of different terms. This often allows one to find information abou Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for differe
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• Calculus
  ...t line]] to a curve at a [[point]] and how to find the [[area]] bounded by a curve. What is surprising is that these two problems are fundamentally conn [[Limit]]s are heavily used in calculus. The formal notion of a limit is what differentiates calculus from precalculus mathematics.
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• Proof by contradiction
  ...fs look better --> This technique usually works well on problems where not a lot of information is known, and thus we can create some using proof by con ...d as a rational fraction of the form <math>\frac{b}{a}</math>, where <math>a</math> and <math>b</math> are two [[relatively prime]] integers. Now, sinc
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• Mathematical problem solving
  ...ng]] involves using all the tools at one's disposal to attack a problem in a new way. == A Historical Example ==
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• Pigeonhole Principle
  ...ts, the principle may be referred to as the '''Dirichlet box principle'''. A common phrasing of the principle uses balls and boxes and is that if <math> ...nhole principle is as follows: suppose for contradiction that there exists a way to place <math>n</math> balls into <math>k</math> boxes where <math>n>k
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• Quadratic equation
  ...[[equation]] of the form <math> {a}{x}^2+{b}{x}+{c}=0</math>, where <math>a</math>, <math>b</math> and <math>c</math> are [[constant]]s (that is, they The purpose of factoring is to turn a general quadratic into a product of [[binomial]]s. This is easier to illustrate than to describe.
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• Relatively prime
  Consecutive positive integers are always relatively prime, since, if a prime <math>p</math> divides both <math>n</math> and <math>n+1</math>, then ...exist some <math>x,y\in \mathbb{Z}</math> such that <math>ax+by=1</math> (a special case of [[Bezout's Lemma]]). The [[Euclidean Algorithm]] can be use
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• 1965 AHSME Problems/Problem 40
  <math>\textbf{(A)}\ 4 \qquad First, we wish to factor <math>P</math> into a more manageable form.
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• Trigonometric substitution
  ...n]]s such as <math>\sqrt{a^2+x^2},\sqrt{a^2-x^2}</math> or <math>\sqrt{x^2-a^2}</math> === <math>\sqrt{a^2+x^2}</math> ===
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• Manhattan Mathematical Olympiad
  ...ril. It is divided into three divisions, grades 5-6, 7-8, and 9-12, and is a three hour, 4 question test, modeled after other math olympiads such as the
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• Combinatorics
  ...] in a given [[graph]] is a combinatorial problem, as is the derivation of a non-[[recursive]] formula for the [[Fibonacci numbers]], and so too methods ...ion of problems found in mathematics competitions and can be approached by a variety of techniques, such as [[generating functions]] or the [[Principle
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• Geometric mean
  .../math> and <math>c</math> is the number <math>a</math> such that <math>a + a = b + c</math>, while the geometric mean of the numbers <math>b</math> and ...length of [[line segment]] <math>AM</math> is the geometric mean of <math>a</math> and <math>b</math>.
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• Principle of Inclusion-Exclusion
  ...ized method/formula to find the number of [[element]]s in the [[union]] of a given group of [[set]]s, the size of each set, and the size of all possible ...counted once and only once. In particular, memorizing a formula for PIE is a bad idea for problem solving.
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• Googol
  Googol is a huge number. It has a 1 followed by 100 zeroes or <math>10^{100}.</math>
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• Mill's Constant
  ...>\theta</math> such that <math>\lfloor\theta^{3^n}\rfloor</math> is always a prime number for all natural n.
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• Combination
  ...'binomial coefficient''', is a way of choosing <math>r</math> objects from a set of <math>n</math> where the order in which the objects are chosen is ir ...ons are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting proble
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• AoPSWiki
  :''This is a neutral self-referential article. For a non-encyclopedic version, see [[AoPSWiki:About]].'' The '''AoPSWiki''' is a wiki, or a publicly editable resource, that is dedicated to students of problem solvin
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• 2020 CIME II Problems/Problem 10
  ...n <math>a+b+1=ab\Rightarrow (a-1)(b-1)=2</math> from whence we have <math>(a,b,c)\in\{(2,3,1),(3,2,1)\}</math>. ...math> we have <math>6c\leq 2c+6\Rightarrow c\leq3/2\Rightarrow c=1</math>; a contradiction since <math>c\geq 2</math>.
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• Cauchy-Schwarz Inequality
  ..._2 + \cdots + a_nb_n)^2,</cmath> with equality if and only if there exists a constant <math>t</math> such that <math>a_i = t b_i</math> for all <math>1 ...cdot \overrightarrow{w}|</cmath> with equality if and only if there exists a scalar <math>t</math> such that <math>\overrightarrow{v} = t \overrightarro
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• Factorial
  ...!=n \cdot (n-1) \cdots 2 \cdot 1 = \prod_{i=1}^n i</math>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n ...</math> (remember! this is 1, not 0! (the '!' was an exclamation mark, not a factorial sign))
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• Discriminant
  ...has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. The discriminant can tell us something about the roots of a given [[polynomial]] <math>p(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0</math> of deg
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• Menelaus' theorem
  ...earity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R;
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• Number theory
  Number theory is a broad topic, and may cover many diverse subtopics, such as: Some branches of number theory may only deal with a certain subset of the real numbers, such as [[integer]]s, [[positive]] numb
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• San Diego ARML
  ...]] was a coach for 2007-2009, and [[user:MCrawford | Mathew Crawford]] was a coach for 2006-2007. ...n division A. San Diego Sun (B) finished 33rd (nationally) in division B: a very strong showing considering that this was the first time for most of th
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• ARML historical results
  This is a list of historical results from the [[American Regions Mathematics League]] == Division A Top Five Teams ==
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• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
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• Parabola
  ...[[locus]] of points that are equidistant from a point (the [[focus]]) and a line (the [[directrix]]). ...la. The first is [[polynomial]] form: <math>y = a{x}^2+b{x}+c</math> where a, b, and c are [[constant]]s. This is useful for manipulating the polynomial
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• Chebyshev's Inequality
  Chebyshev's inequality is a consequence of the [[Rearrangement inequality]], which gives us that the su
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• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...is a positive integer [[relatively prime]] to <math>a</math>, then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>.
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• Brute forcing
  ...e, through bashing calculations, and can actually sometimes be faster than a more creative approach, and is thus an important tool to have. ...te force would be to list all 91 possibilities (although this would not be a smart time to use brute force).
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• Geometric inequality
  A '''geometric inequality''' is an [[inequality]] involving various measures ...level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]].
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• Brahmagupta's Formula
  ...ath>c</math>, <math>d</math> are the four side lengths and <math>s = \frac{a+b+c+d}{2}</math>. ...^2+b^2-2ab\cos B=c^2+d^2-2cd\cos D</math>. <math>\cos B=-\cos D</math>, so a little rearranging gives
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• Ptolemy's theorem
  ...orem''' gives a relationship between the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] Given a [[cyclic quadrilateral]] <math>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
  An '''elementary symmetric sum''' is a type of [[summation]]. ...>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
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• Cyclic quadrilateral
  ...[triangles]] are cyclic, the same is not true of quadrilaterals. They have a number of interesting properties. In a quadrilateral <math>ABCD</math>:
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• Algebra
  ...the [[variable]], an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic ...o add to the confusion, "[[algebra (structure) |algebra]]" is the name for a certain kind of structure in modern algebra.
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• Number theory/Advanced
  ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta
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• 1959 IMO Problems/Problem 2
  <math> \sqrt{x+\sqrt{2x-1}} + \sqrt{x-\sqrt{2x-1}} = A,</math> given (a) <math>A=\sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admitted for square root
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• 1970 IMO Problems/Problem 6
  In a plane there are <math>100</math> points, no three of which are collinear.
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• Algebra/Intermediate
  ...algebra involves familiarity with introductory topics to a high level and a multitude of new topics. ...Olympiad Training (WOOT) Program] -- Intermediate to Olympiad training in a variety of subjects, including algebra.
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• Harmonic mean
  ...n (like [[arithmetic mean]] and [[geometric mean]]). The harmonic mean of a [[set]] of <math>n</math> [[positive]] [[real number]]s <math> x_1, x_2... ...-Geometric mean-Harmonic mean inequality]]. The Inequality states that for a set of positive numbers <math>x_1, x_2,\ldots,x_n</math>: <math>\frac{x_1+x
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• Euler's totient function
  '''Euler's totient function''' <math>\phi(n)</math> applied to a [[positive integer]] <math>n</math> is defined to be the number of positive ...ere the <math>p_i </math> are distinct [[prime number]]s. Now, we can use a [[PIE]] argument to count the number of numbers less than or equal to <mat
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• Alabama ARML
  * In 2006, Alabama came in 15th in the A division. * In 2007, Alabama placed 9th in the A division and had several top scoring students including Seva Tchernov who p
  2 KB (258 words) - 00:50, 28 December 2021
• Probability
  Before reading about the following topics, a student learning about probability should learn about introductory [[counti ...dergraduate level. Once measure theory is covered, probability does become a lot easier to use and understand.
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• Jeff Boyd
  ...://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MathCounts Program].
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• LaTeX
  ..., you will learn much of what you'll need to express math and science like a professional, much, much better than writing papers with Word! * Click [https://youtu.be/HfRsDKBrSaA here] for a Quick Start <math>\text{\LaTeX}</math> video.
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• Contest strategies
  ...ying video games because it's easy to get drowsy during the test. Getting a good night sleep can help refresh the brain, so it can be ready for the tes In many geometry problems, a visual aid can help solvers visualize and look for clues. Even in non-geom
  3 KB (538 words) - 13:13, 16 January 2021
• Divisibility
  In [[number theory]], '''divisibility''' is the ability of a number to evenly divide another number. The study of divisibility resides a ...that <math>a</math> is a '''multiple''' of <math>b</math>, and that <math>a</math> is '''divisible''' or '''evenly divisible''' by <math>b</math>.
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• Rearrangement Inequality
  ...ss than or equal to exactly <math>{i}</math> of the other members of <math>A</math>, then <math>{b_k}</math> is also greater than or equal to exactly <m ...re more pennies than nickels, more nickels than dimes, and so on. This is a simple application of the rearrangement inequality. It is also an applicat
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• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam The set of real numbers, denoted by <math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used subse
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• Divisibility rules
  ...its of the number are divisible by <math>2^n</math>. Thus, in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, A number is divisible by 3 or 9 if and only if the sum of its digits is divis
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• Induction
  ...lue (the Base Case); it is then shown that if the desired result holds for a certain value, it then holds for another, closely related value. Typically Other, odder inductions are possible. If a problem asks you to prove something for all integers greater than 3, you ca
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• Incenter/excenter lemma
  ...ircumcenter. Then, the theorem states that <math>L</math> is the center of a circle through <math>I</math>, <math>B</math>, <math>I_A</math>, and <math> ...try. Along with the larger lemma, two smaller results follow: first, <math>A</math>, <math>I</math>, <math>L</math>, and <math>I_A</math> are collinear,
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• Orthic triangle
  ...h>E</math>, and <math>F</math> denote the feet of the altitudes from <math>A</math>, <math>B</math>, and <math>C</math>, respectively. Then, <math>\tria ...e EFH = \theta = 90^\circ - \angle C</math>. The same argument (with <math>A</math> instead of <math>B</math>) shows that <math>\angle DFH = 90^\circ -
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• Triangle
  A '''triangle''' is a type of [[polygon]]. {{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}
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• Greatest common divisor
  ...mon factor''')) of two or more [[integer]]s is the largest integer that is a [[divisor]] of all the given numbers. A very useful property of the GCD is that it can be represented as a sum of the given numbers with integer coefficients. From here it immediatel
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• Vertex
  ==Vertex of a parabola== ...value of the graph is minimal (if <math>a > 0</math>) or maximal (if <math>a < 0</math>).
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• Overcounting
  ...ea of strategic overcounting is fundamental to [[combinatorics]] and plays a role in incredibly important counting tools such as [[combinations]] and th An example of a classic problem is as follows:
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• Constructive counting
  ...s a [[counting]] technique that involves constructing an item belonging to a set. Along with the construction, one counts the total possibilities of eac '''Solution''': We can construct a four-digit by picking the first digit, then the second, and so on until the
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• Jensen's Inequality
  Let <math>{F}</math> be a [[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb If <math>{F}</math> is a concave function, we have:
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• Michigan Mathematics Prize Competition
  ...he top 1000 finishers of this round are selected to take Part II, which is a harder 5-question proof test. The top 100 finishers of Part II are honored Part I is normally on a Wednesday morning in the first or second week of October.
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• Geometric probability
  When dealing with a [[probability]] problem involving [[discrete]] quantities, we often times j ...etric probability'''. We can usually translate a probability problem into a [[geometry]] problem. We can use [[length]] for one dimension, [[area]] f
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• Casework
  ...ts individually, then adding together the totals of each part. Casework is a very general problem-solving approach, and as such has wide applicability. ...problems where casework produces the most elegant solution, in those where a shorter answer exists, casework may be considered [[brute force]]. This is
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• Euclidean algorithm
  ...orithm that finds the [[greatest common divisor]] (GCD) of two elements of a [[Euclidean domain]], the most common of which is the [[nonnegative]] [[int ...c idea is to repeatedly use the fact that <math>\gcd({a,b}) \equiv \gcd({b,a - b})</math>
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• Conic section
  ..., and every conic section can be described in [[Cartesian coordinates]] by a [[polynomial]] [[equation]] of degree two or less. ...cone or equivalently [[perpendicular]] to the axis. (This is really just a special case of the ellipse -- see the next bullet point.)
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• Generating function
  ...se [[coefficient]]s, <math>c_0, c_1, c_2, \ldots</math>, give the terms of a [[sequence]] which is of interest. Therefore the power series (i.e. the gen If we let <math>A(k)={n \choose k}</math>, then we have <math>{n \choose 0}+{n \choose 1}x +
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• Binomial Theorem
  The '''Binomial Theorem''' states that for [[real]] or [[complex]] <math>a</math>, <math>b</math>, and [[non-negative]] [[integer]] <math>n</math>, <center><math>(a+b)^n = \sum_{k=0}^{n}\binom{n}{k}a^{n-k}b^k</math></center>
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• Prime number
  A '''prime number''' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | divis ...</math> must [[#Importance of Primes|have]] a prime factor, which leads to a direct contradiction.
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• Recursion
  ...tion]]) in terms of previously defined values. The most famous example of a recursive definition is that of the [[Fibonacci sequence]]. If we let <mat Often, it is convenient to convert a recursive definition into a closed-form definition. For instance, the sequence defined recursively by
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• Function
  A '''function''' is a rule that maps one set of values to another set of values, assigning to eac ...is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if
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• Complementary counting
  ...ting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ...is the [[complement]] of <math>B</math>. In most instances, though, <math>A</math> is obvious from context and is committed from mention.
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• Bijection
  ...[Function#The_Inverse_of_a_Function|two-sided inverse]] exactly when it is a bijection between its [[domain]] and [[range]]. ...are frequently used in [[combinatorics]] in order to count the elements of a set whose size is unknown. Bijections are also very important in [[set the
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• Combinatorial identity
  ...binatorially by noting that any combination of <math>r</math> objects from a group of <math>m+n</math> objects must have some <math>0\le k\le r</math> o ...he addends represented in the summation and the sum itself is highlighted, a hockey-stick shape is revealed. We can also flip the hockey stick because p
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• Integer
  ...th>a\div b</math> may or may not be an integer, depending on whether <math>a</math> can be divided by <math>b</math> with no remainder. ...ward definition, an integer is a number that is '''not''' a [[decimal]] or a [[fraction]].
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• Prime factorization
  ...ental Theorem of Arithmetic]] tells us that every [[positive integer]] has a unique prime factorization, up to changing the order of the terms. The form of a prime factorization is
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• Composite number
  A comprehensive video explaining composite numbers: https://youtu.be/SMOGYNYD A '''composite number''' is a [[positive integer]] with at least one [[divisor]] different from 1 and its
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• Natural number
  ...th>. The set of '''natural numbers''', denoted <math>\mathbb{N}</math>, is a subset of the set of [[integer]]s, <math>\mathbb{Z}</math>. Some texts use
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• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. {{asy image|<asy>unitsize(2cm);draw(unitcircle,blue);</asy>|right|A basic circle.}}
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• Ellipse
  An '''ellipse''' is a type of [[conic section]]. An ellipse is formed by cutting through a [[cone]] at an [[angle]].
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• Divisor
  ...hat <math>n=kd</math> or, in other words, if <math>\frac nd</math> is also a natural number (i.e <math>d</math> divides <math>n</math>). See [[Divisibil A common notation to indicate a number is a divisor of another is <math>n|k</math>. This means that <math>n</math> divi
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• Base numbers
  ...each number in 2746 is actually just a placeholder which shows how many of a certain power of 10 there are. The first digit to the left of the decimal ...digits 0-9. Usually, the base, or '''radix''', of a number is denoted as a subscript written at the right end of the number (e.g. in our example above
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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</math> ...t everyone knows and loves. Most numbers in the world are written without a specified radix and usually it can just be assumed that they are in base 1
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• Partition of an interval
  A '''partition of an interval''' is a division of an [[interval]] into several disjoint sub-intervals. Partition Let <math>[a,b]</math> be an interval of [[real number]]s.
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• Henstock-Kurzweil integral
  ...ble generalizations of the [[Integral|Riemann integral]], but it also uses a strikingly simple and elegant idea. It was developed independently by [[Ral Let <math>f:[a,b]\rightarrow\mathbb{R}</math>
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• 2001 AMC 12 Problems/Problem 1
  <math>\textbf{(A)}\ 2S + 3\qquad \textbf{(B)}\ 3S + 2\qquad \textbf{(C)}\ 3S + 6 \qquad\text 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
  ...ath>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a <math>\text{(A)}\ 2\qquad \text{(B)}\ 3\qquad \text{(C)}\ 6\qquad \text{(D)}\ 8\qquad \tex
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• 2001 AMC 12 Problems/Problem 3
  <math>\textbf{(A)}\,\textdollar 28000 \qquad \textbf{(B)}\,\textdollar 32000 \qquad \textbf{ Let <math>A</math>, <math>T</math> be Kristin's annual income and the income tax total,
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• Square (geometry)
  A '''square''' is a [[quadrilateral]] in which all [[edge|sides]] have equal length and all [[a pair A, B, C, D;
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• Standardized test
  ...ifferent is that the test can't test you on multivariable calculus because a lot of students don't get to that in highschool.
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• Scholastic Aptitude Test
  The '''Scholastic Aptitude Test''', or '''SATI ''', or just simply "SAT," is a [[standardized test]] that tests the critical reasoning skills that are nee ...of points obtained in each section, and thus on the entire test, is always a multiple of 10.
  908 bytes (146 words) - 14:44, 12 December 2023
• Mathematical Kangaroo
  ...in the US that are ranked high within their state will also be awarded as a state winner. ...ve to be in their home state or country, so competitors often register for a testing site that is the closest or most convenient for them despite being
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• American Scholastic Mathematics Association
  ...wards the team's total. The test is 35 minutes long and assumes the use of a calculator. ...hools. The student with the highest score in each school will also receive a certificate
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• Online Math League
  ...It is now part of a series known as [[Academic Leagues]] which encompasses a variety of subject areas.
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• Multiple
  ...m</math> and <math>n</math> are integers. (In this case, <math>k</math> is a multiple of <math>n</math>, as well). An equivalent phrasing is that <math>k</math> is a multiple of <math>m</math> exactly when <math>k</math> is [[divisibility |
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• Least common multiple
  ...[[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\}</math> is conventionally repres ...s of each until you find a multiple that is common to all of them. This is a tedious method, so it is usually only used when the numbers are small. For
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• Economics scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[user:MCrawford | Mathew
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• Common factorizations
  ...e be solved by [[Newton sums]] or problems that give a polynomial, and ask a question about the roots. Combined with [[Vieta's formulas]], these are ex \text{\textbullet}&&a^2+b^2+c^2+2(ab+bc+ac)&=(a+b+c)^2\\
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• Math Bee
  '''Math Bee''' is a [[mathematics competition]] for students in grades K through 8 of Indian or .... Participants are given two phases of math problems to be solved without a calculator. The first phase has 25 free-response questions to be solved in
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• Math Day at the Beach
  '''Math Day at the Beach''' is a [[mathematical problem solving]] festival for Southern California high scho ...esent high schools and have 6 members each. The competition takes place on a Saturday in March.
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• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician Ptolemy. The inequality states that in for four points <math>A, B, C, D </math> in the plane,
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• Cevian
  ...(or the extension of that side). In the below diagram, <math>AD</math> is a cevian. label("$A$",(10,50),N);
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• Median of a triangle
  A '''median''' of a [[triangle]] is a [[cevian]] of the triangle that joins one [[vertex]] to the [[midpoint]] of In the following figure, <math>AM</math> is a median of triangle <math>ABC</math>.
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• Altitude
  <i>A line segment that passes through a vertex of a triangle and is perpendicular to the line opposite to the vertex.</i>
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• Pi
  Pi is the [[ratio]] of the [[circumference]] ([[perimeter]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.14159 ...i</math> is to inscribe a unit circle in a square of side length 2. Using a computer, random points are placed inside the square. Because the area of
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• Phi
  <math>\phi</math> appears in a variety of different mathematical contexts: it is the limit of the ratio of ...ngle]] is a rectangle with side lengths of 1 and <math>\phi</math>; it has a number of interesting properties.
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• Fibonacci sequence
  The '''Fibonacci sequence''' is a [[sequence]] of [[integer]]s in which the first and second terms are both e ...th> for <math>n \geq 3</math>. This is the simplest nontrivial example of a [[linear recursion]] with constant coefficients. There is also an explicit
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• Chain Rule
  ...<math>h'(x_0)</math>,<math>f'(g(x_0))</math>, and <math>g'(x_0)</math> is a matrix.) ...as <math>\Delta x</math> approaches <math>0</math>. This can be made into a rigorous proof. (But we do have to worry about the possibility that <math>
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• Muirhead's Inequality
  ...>p</math> [[Majorization|majorizes]] a sequence <math>q</math>, then given a set of positive reals <math>x_1,x_2,\cdots,x_n</math>: A common [[Brute forcing|bruteforce]] technique with inequalities is to clear
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• Cardinality
  '''Cardinality''' is a property of [[set]]s. For [[finite]] sets, the cardinality of is the numbe ...4\}| = 2</math>. Sometimes, the notations <math>n(A)</math> and <math>\# (A)</math> are used.
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• Injection
  ...t takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the [[range]] is the image of exactly one ...: A function <math>f:A\to B</math> is an injection if for all <math>x,y\in A</math>, if <math>f(x)=f(y)</math> then <math>x=y</math>.
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• 1988 AJHSME Problems/Problem 5
  If <math>\angle \text{CBD}</math> is a [[right angle]], then this protractor indicates that the measure of <math>\ pair A,B,C,D;
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• Fundamental Theorem of Calculus
  The '''Fundamental Theorem of Calculus''' establishes a link between the two central operations of [[calculus]]: [[derivative|diffe ...imes <math>t=1</math> and <math>t=2</math> geometrically, as an area under a curve.
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• Polygon
  A '''polygon''' is a closed [[planar figure]] consisting of straight [[line segment]]s. There ar A polygon can be [[regular polygon| regular]] or irregular. A polygon is regular if all sides are the same length and all angles are [[co
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• Isoperimetric Inequalities
  ...qualities''' are [[inequalities]] concerning the [[area]] of a figure with a given [[perimeter]]. They were worked on extensively by [[Lagrange]]. ...cle has the largest area. Conversely, of all plane figures with area <math>A</math>, the circle has the least perimeter.
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• Writing scholarships
  ...f '''writing scholarships''' — scholarships granted based primarily upon a submitted essay — is primarily for American students. This list can be r ...be in a creative fundraiser, an inventive new app, series of blog posts on a particular theme, or another idea). The application deadline is 11:59 pm PS
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• National Scholarships
  The following is a list of large scholarships open to US students, regardless of interests. * [[AXA Achievement Scholarship]] State winners win a \$10,000 and national winners with \$25,000 for their achievements. [http:/
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• 2011 AMC 10B Problems/Problem 22
  ...ength <math>1</math> and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the py <math> \textbf{(A)}\ 5\sqrt{2} - 7 \qquad\textbf{(B)}\ 7 - 4\sqrt{3} \qquad\textbf{(C)}\ \fra
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• 2011 AMC 10B Problems/Problem 4
  ...A</math> dollars and Bernardo had paid <math>B</math> dollars, where <math>A < B</math>. How many dollars must LeRoy give to Bernardo so that they share ...textbf{(C)}\ \dfrac{B - A}{2}\qquad\textbf{(D)}\ B - A \qquad\textbf{(E)}\ A + B </math>
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• Computer science scholarships
  ...they can finance their educations! If you are unfamiliar with how to edit a Wiki and don't have time to learn, please contact [[Mathew Crawford]] using
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• Semiperimeter
  ...r <math>\frac{P}{2}</math>, where <math>P</math> is the total perimeter of a figure. It is typically denoted <math>s</math>. ...the simplest is <math>A=rs</math>, where <math>A</math> is the [[area]] of a [[triangle]] and <math>r</math> is the triangle's [[inradius]] (that is, th
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• Bretschneider's formula
  Suppose we have a [[quadrilateral]] with [[edge]]s of length <math>a,b,c,d</math> (in that order) and [[diagonal]]s of length <math>p, q</math>. <math>[ABCD]=\frac{1}{4} \cdot \sqrt{4p^2q^2-(b^2+d^2-a^2-c^2)^2}</math>.
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• Vector
  ...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
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• Logarithm
  ...would be a pain to have to calculate any time you wanted to use it (say in a comparison of large numbers). Its natural logarithm though (partly due to ...ly 7 digits before the decimal point. Comparing the logs of the numbers to a given precision can allow easier comparison than computing and comparing th
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• Law of Cosines
  ...sines''' 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 common ...h>, <math>b</math> and <math>c</math> opposite [[angle]]s of measure <math>A</math>, <math>B</math> and <math>C</math>, respectively, the Law of Cosines
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• Schur's Inequality
  Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>: <cmath>a^r(a-b)(a-c)+b^r(b-a)(b-c)+c^r(c-a)(c-b) \geq 0</cmath>
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• Trigonometric identities
  ...ath>(\cos (x), \sin (x))</math> is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, <math>\sin ...ful in [[integration]] and in deriving the half-angle identities. They are a simple rearrangement of the two above.
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• Chicago ARML
  ...teams each year to compete at the [[American Regions Mathematics League]], a primarily high school [[mathematics competition]]. * Chicago A
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• Iff
  It is also known as a [[conditional|biconditional]] statement. In order to prove a statement of the form "<math>p</math> iff <math>q</math>," it is necessary
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• Massachusetts Institute of Technology
  The '''Massachusetts Institute of Technology''' (MIT) is a private university well known for its excellence in math and science educat
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• Vocabulary and spelling competitions
  ...grades. Over 3 million students participate each year. 2 AoPSers have won a total of $40,000 at this competition. [http://nwpc.rd.com Sign up today!]
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• Irrational number
  An '''irrational number''' is a [[real number]] that cannot be expressed as the [[ratio]] of two [[integer] ...erhood- stumbled upon <math>\sqrt{2}</math> as the length of a diagonal of a square with side lengths 1 in the sixth century <math>B.C</math>. The Pytha
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• Complex plane
  ...e system) corresponds to the complex number [[zero (constant) | 0]], while a point two units to the right and one unit down from the origin corresponds
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• Complex number
  ...n has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we ad ...s the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.
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• Rational number
  A '''rational number''' is a [[number]] that can be represented as the [[ratio]] of two [[integer]]s. ...l because every integer <math>a</math> can be represented as <math>a=\frac{a}{1}</math>
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• Cis
  '''Cis''' notation is a [[polar form | polar]] notation for [[complex number]]s. For all complex nu
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• Polar form
  ...ows us to graph complex numbers given an [[angle]] <math>\theta</math> and a [[radius]] or [[magnitude]] <math>r</math>. ...th>. (See [[cis]] if you do not understand this notation.) This represents a complex number <math>z</math> that is <math>r</math> units away from the or
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• Common divisor
  ...of all those integers. For any [[set]] of [[positive integer]]s, there is a [[greatest common divisor]].
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• Vocabulary competitions
  ...grades. Over 3 million students participate each year. 2 AoPSers have won a total of $40,000 at this competition. [http://nwpc.rd.com Sign up today!]
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• Georgia ARML
  ...cally, in 1992 Georgia won. In 2012, Georgia A1 came in sixth place in the A division.
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• Roots of unity
  The '''Roots of unity''' are a topic closely related to [[trigonometry]]. Roots of unity come up when we e ...making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik
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• Angle bisector
  ...internal) angle bisector of <math>\angle BAC</math> is the line from <math>A</math> such that the angle between this line and <math>\overline{AB}</math> pair A,B,C,D,E,F;
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• Mathew Crawford
  ...l Computing. After spending several on Wall Street and eventually running a finance operation from the basement of his apartment, Crawford founded his * Perfect score on the [[AIME]] as a freshman.
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• Without loss of generality
  ...math. The term is used to indicate that the following proof emphasizes on a particular case, but doesn’t affect the validity of the proof in general. Be careful when using WLOG in a proof. By using it, you must be certain that your statement actually DOES w
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• Law of Sines
  The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines]] and the [[law of tan ...h>\triangle ABC</math>, where <math>a</math> is the side opposite to <math>A</math>, <math>b</math> opposite to <math>B</math>, <math>c</math> opposite
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• The Art and Craft of Problem Solving
  ...Zeitz]], former coach of the [[U.S. IMO]] team, is generally thought of as a good mathematical book for when one is finished with the Art of Problem Sol
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• Sequence
  A '''sequence''' is an ordered list of terms. Sequences may be either [[fini A sequence of real numbers is simply a function <math>f : \mathbb{N} \rightarrow \mathbb{R}</math>. For instance,
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• Science
  [[Mathematics]] has a special place in science. It is not possible to perform experimental tests
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• Geometric sequence
  ...'geometric sequence''', sometimes called a '''geometric progression''', is a [[sequence]] of numbers such that the ratio between any two consecutive ter ...e with common ratio <math>2</math> and <math>100, -50, 25, -25/2</math> is a geometric sequence with common ratio <math>-1/2</math>; however, <math>1, 3
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• Arithmetic sequence
  ...ithmetic sequence''', sometimes called an '''arithmetic progression''', is a [[sequence]] of numbers such that the difference between any two consecutiv ...and <math>c</math> are in arithmetic progression if and only if <math>b - a = c - b</math>.
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• Fermat's Last Theorem
  ...> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...ater than the second to be the sum of two like powers. ''I have discovered a truly marvelous demonstration of this proposition that this margin is too n
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• Homothety
  ...ath> and factor <math>k</math> sends point <math>A</math> to a point <math>A' \ni HA'=k\cdot HA</math> This is denoted by <math>\mathcal{H}(H, k)</math> ...is <math>\frac{OD'}{OD}</math>. Additionally, <math>ABCD</math> and <math>A'B'C'D'</math> are homothetic with respect to <math>O</math>.
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• 2005 USAMO Problems/Problem 4
  ...math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
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• Geometry
  '''Geometry''' is the field of [[mathematics]] dealing with figures in a given [[space]]. It is one of the two oldest branches of mathematics, along :''“Through any line and a point not on the line, there is exactly one line passing through that point
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• 2000 AMC 12 Problems/Problem 1
  <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ ...re small. If we want to make sure that this is correct, we could test with a smaller number, like <math>30</math>. It becomes much more clear that this
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• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or some ...closely tied to [[modular arithmetic]] and [[number theory]]. Often, when a Diophantine equation has infinitely many solutions, [[parametric form]] is
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• Area
  ...size of the region that a two-[[dimension]]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elemen
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• Improper fractional base
  An '''improper fractional base''' is a type of [[number base]]. Instead of using an [[integer]] for the base in o ...ntation in an improper fractional base. (Note that this means there is not a unique representation for each number in an improper fractional base.)
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• Fraction
  A '''fraction''' is the [[ratio]] of two [[number]]s. Most commonly, we cons ...erator is greater than the denominator is called an [[improper fraction]]. A fraction where the numerator is the same as the denominator such as <math>\
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• Functional equation
  A '''functional equation''', roughly speaking, is an equation in which some o ===The Inverse of a Function===
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• Exponential form
  ...> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. pair A,B,C,D,E;
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• Floor function
  ...]</math>. The action of this function is the same as "rounding down." On a [[positive]] argument, this function is the same as "dropping everything af ...+b\rfloor\ge \lfloor a\rfloor+\lfloor b \rfloor</math> for all real <math>(a,b)</math>.
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• Pascal's triangle
  ...ns the values from the [[binomial expansion]]; its various properties play a large role in [[combinatorics]]. ...2 is <math>{4 \choose 2} = 6</math>. Pascal's Triangle thus can serve as a "look-up table" for binomial expansion values. Also, many of the character
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• Newton's Sums
  ...ficient way of finding the sums of [[root]]s of a [[polynomial]] raised to a power. They can also be used to derive several [[factoring]] [[identity|id Consider a polynomial <math>P(x)</math> of degree <math>n</math>,
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• 1985 IMO Problems/Problem 5
  A circle with center <math>O</math> passes through the vertices <math>A</math> and <math>C</math> of the triangle <math>ABC</math> and intersects t .../math> is the Miquel Point of quadrilateral <math>ACNK</math>, so there is a spiral similarity centered at <math>M</math> that takes <math>KN</math> to
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• Exponentiation
  ...math>a^b</math>, where <math>b</math> is the exponent (or power) and <math>a</math> is the [[base]]. In the [[order of operations]], it is the second operation performed if a equation has [[parentheses]] or the first one performed when there is no pa
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• Rational approximation
  The main theme of this article is the question how well a given [[real number]] <math>x</math> can be approximated by [[rational numb ...th> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\fr
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• Missouri ARML
  ...t the [[American Regions Mathematics League]]. The team was started about a decade ago by then [[Washington University in St. Louis]] student [[Tim San
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• Physics summer programs
  ...MehtA%2B_Machine_Learning_Bootcamp MehtA+ AI/Machine Learning Bootcamp] is a virtual research program for grades 8-12 [https://mehtaplustutoring.com/ai- ...MehtA%2B_Machine_Learning_Bootcamp MehtA+ AI/Machine Learning Bootcamp] is a virtual research program for grades 8-12 [https://mehtaplustutoring.com/ai-
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• Trigonometry
  ...s and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way towards ...definition of trigonometry is preferred in order to extend trigonometry to a complex domain.
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• Euler's number
  '''Euler's number''' is a [[constant]] that appears in a variety of mathematical contexts. It is defined as the positive real number The <math>\ln</math> (natural logarithm) function is equivalent to a [[logarithm]] with base <math>e</math>. In addition, the function <math>\ex
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• Exponential function
  ...unction]] <math>f(x) = e^x</math>, [[exponentiation]] by ''[[e]]''. It is a very important function in [[analysis]], both [[real]] and [[complex]]. Exponential functions are functions that grows or decays at a constant percent rate.
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• Mathematical Olympiad Summer Program
  The '''Mathematical Olympiad Program''' (abbreviated '''MOP''') is a 3-week intensive problem solving camp held at the Carnegie Mellon Universit ...ternational Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results of the USAMO and the TST are we
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• Complex analysis
  ...ctions of a real variable that cannot occur in complex variables. Here are a few spectacular results in complex analysis. ...ly connected]] [[domain]] ''D'', and let <math>\Gamma\subseteq D</math> be a [[simple closed Jordan curve]]. Then for any <math>z_0</math> in the interi
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• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says ..._1}+\cdots+\frac{1}{x_n}} \ge \sqrt[n_4]{\frac{x_1^{n_4}+\cdots+x_a^{n_4}}{a}}</cmath> where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math
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• Harmonic series
  Generally, a '''harmonic series''' is a [[series]] whose terms involve the [[reciprocal]]s of the [[positive intege The [[zeta-function]] is a harmonic series when the input is one.
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• Goldbach Conjecture
  The '''Goldbach Conjecture''' is a yet unproven [[conjecture]] stating that every [[even integer]] greater tha In 1742, the Prussian mathematician Christian Goldbach wrote a letter to [[Leonhard Euler]] in which he proposed the following conjecture:
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• Twin Prime Conjecture
  The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) that states that there are [[infinite]]ly many pairs of [[twin ...twin primes. Unfortunately, it has been shown that this sum converges to a constant <math>B</math>, known as [[Brun's constant]]. This could mean eit
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• Fallacy
  ...tempted [[proof]] that is [[logic]]ally flawed in some way. The fact that a proof is fallacious says nothing about the validity of the original proposi Let <math>a=b</math>.
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• Riemann Hypothesis
  The '''Riemann Hypothesis''' is a famous [[conjecture]] in [[analytic number theory]] that states that all no * <math>|M(x)|\le cx^{1/2+\epsilon}</math> for a constant c and where <math>M(x)=\sum_{n\le x}\mu(n)</math>
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• Quadratic residues
  ...] <math>m</math> if there is some integer <math>n</math> so that <math>n^2-a</math> is [[divisibility | divisible]] by <math>m</math>. ...lo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</math>
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• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =
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• AMC 12 Problems and Solutions
  ! Year || Test A || Test B
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• AMC 10 Problems and Solutions
  ! Year || Test A || Test B
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• Fundamental Theorem of Algebra
  ...tant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some Suppose that <math>P(z)</math> is a complex polynomial of degree <math>n</math> with no complex roots; without
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• Collatz Problem
  ...ath> as it was for <math>b</math>; any time the lead coefficient still has a factor of 2. ...<math>n=4c+3</math>; namely they are the only possible lowest elements of a non-trivial cycle, and also the only possible lowest elements of an infinit
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• Algebraic number
  ...\sqrt[3]{4}+\sqrt[7]{97}}</math>. A number that is not algebraic is called a [[transcendental number]], such as <math>e</math> or <math>\pi</math>. ...umbers are rational. For example, consider <math>\sqrt{2}</math>, which is a root of <math>x^{2}-2</math>.
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• Nonnegative number
  ...''' if it is greater than or equal to [[zero (constant)|zero]]. Basically, a nonnegative number is not negative.
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• Incenter
  ...r is the center of the [[incircle]]. Every [[nondegenerate]] triangle has a unique incenter. Consider a triangle <math>ABC</math>. Let <math>I</math> be the intersection of the r
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• Triangle Inequality
  The '''Triangle Inequality''' says that in a [[nondegenerate]] [[triangle]] <math>ABC</math>: .... The lengths <math>a_1, a_2, \ldots, a_n</math> can only be the sides of a nondegenerate <math>n</math>-gon if <math>a_i < a_1 + \ldots + a_{i -1} + a
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• Majorization
  We say a [[nonincreasing]] [[sequence]] of [[real number]]s <math> a_1, \ldots ,a_n<
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• Surjection
  ..., the range is a subset of the codomain.) In adjectival form, we say that a function is ''surjective'' or ''onto''. ...defined by <math>f(x) = x+1</math> is not surjective because there exists a [[natural number]] which is not one more than any other natural number.
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• International Mathematical Olympiad
  ...r, countries from around the world send a team of 6 students to compete in a grueling competition. ...rm of a [[proof writing|mathematical proof]]. Since there are 6 problems, a perfect score is 42 points.
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• Sam Vandervelde
  ...ational [[MATHCOUNTS]]. He is now the head and a teacher of Proof School, a school with the aim of supporting kids who love math.
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• American High School Mathematics Examination
  The AHSME was a 30 question, 90 minute, multiple choice exam (except in 1973, in which it h
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• Naoki Sato
  ...aining]] program and [[Art of Problem Solving]] instructor. His career as a [[mathematics]] educator and [[mathematics competition]] champion include t
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• Range
  ...th>\{b\in B\mid \mathrm{there\ is\ some\ } a\in A\mathrm{\ such\ that\ } f(a)=b\}</math> is called the '''range''' or '''image''' of <math>f</math>. A function is a [[surjection]] exactly when the range is equal to the codomain.
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• Algebraic geometry
  ...ath> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine a ...s are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying
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• Prime Number Theorem
  constants <math>A</math> and <math>B</math>, <cmath> \pi(x) \sim \frac{x}{A \log x - B} . </cmath>
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• Note that with two sequences <math>\mathbf{a}</math> and <math>\mathbf{b}</math>, and <math>\lambda_a = \lambda_b = 1/2< ...nces of nonnegative reals, and let <math>\{ \lambda_i \}_{i=1}^n</math> be a sequence of nonnegative reals such that <math>\sum \lambda = 1</math>. The
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• Infinite
  A [[set]] <math>S</math> is said to be '''infinite''' if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <mat In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element
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• Euler's Polyhedral Formula
  ...yhedron lie in the interior of the polyhedron rather than along an edge or a face? (1988 AIME #10)
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• Physics forums
  ...teaching physics, and to socialize with other physics students. Here are a few popular online physics forums.
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• Southern California ARML
  ...ely, Kent Merryfield, who used to coach this team, passed away in 2018. As a result, for more information, please contact Dr. Olga Korosteleva at olga.k
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• Well-Ordering theorem
  ...]] <math>(S,\prec)</math> for which each set <math>A\subseteq S</math> has a [[minimum]] element.
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• Rectangle
  A '''rectangle''' is a [[quadrilateral]] in which all [[angle]]s are [[right angle]]s. pair A, B, C, D;
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• Rhombus
  ...eral]] all of whose sides are [[congruent (geometry) | congruent]]. It is a special type of [[parallelogram]], and its properties (aside from those pro * If all of a rhombus' [[angle]]s are [[right angle]]s, then the rhombus is a [[square (geometry) | square]].
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• Kite
  ...etric figure that lies in a plane. [[Quadrilateral]] <math>ABCD</math> is a kite if and only if <math>AB=BC</math> and <math>CD=DA</math>. Thus, there
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• Trapezoid
  ...'' is a cool and pretty geometric figure that lies in a plane. It is also a type of [[quadrilateral]]. ...[[parallelogram]] is also a trapezoid. (Careful: some authors insist that a trapezoid must have exactly one pair of parallel sides.)
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• Isosceles trapezoid
  ...sceles trapezoid''' is a geometric figure that lies in a [[plane]]. It is a specific type of [[trapezoid]] in which the legs have the same length. Its * properties of a [[trapezoid]]
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• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
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• Mock AIME
  A '''Mock AIME''' is a contest that is intended to mimic the [[AIME]] competition. (In more recent ...msolving.com/community/c594864h3000448_released_tlmsg_presents_amongsusime A(mongsus)IME]
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• Commutative property
  An [[operation]] (especially a [[binary operation]]) is said to have the '''commutative property''' or to ...eal number]]s, [[integer]]s, etc.) because <math>\displaystyle a + b = b + a</math>. However, the operation of [[division]] is not commutative over the
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• Permutation
  ...are <math>r!</math> (the [[factorial]] of <math>r</math>) permutations of a set with <math>r</math> distinct objects. ...ite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.
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• Algebraic topology
  ...of [[topology]] using methods from [[abstract algebra]]. In general, given a [[topological space]], we can associate various algebraic objects, such as ...a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate a
  3 KB (479 words) - 15:35, 1 December 2015
• Bertrand's Postulate
  ...t is, in fact, a theorem. A more widely known version states that there is a prime between <math>n</math> and <math>2n</math>. ...in the [[Prime Number Theorem|prime number theorem]] article but requires a closer look at the [[combinations|binomial coefficient]] <math>\binom{2n}{n
  2 KB (309 words) - 21:43, 11 January 2010
• Andrew Wiles
  '''Andrew Wiles''' was born on April 11, 1953. Wiles is a professor and Chair of the [[mathematics]] department at [[Princeton Univer
  312 bytes (42 words) - 21:59, 2 May 2018
• Riemann zeta function
  The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture
  9 KB (1,547 words) - 03:04, 13 January 2021
• Mock USAMO
  ...mitate the [[USAMO]]. When a mock USAMO is run on [[AoPS]]/[[MathLinks]], a very wide time window is often allowed to take the mock USAMO.
  2 KB (205 words) - 19:56, 4 March 2020
• Modular arithmetic/Introduction
  ...is article is to explain the basics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily ...ple, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
  15 KB (2,396 words) - 20:24, 21 February 2024
• Perfect square
  An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^ An integer <math>n</math> is a perfect square [[iff]] it is a [[quadratic residue]] [[modulo]] all but finitely [[prime]]s.
  954 bytes (155 words) - 01:14, 29 November 2023
• Convex function
  A convex function is a continuous function whose value at the midpoint of every interval in its do A [[function]] <math>f: I \mapsto \mathbb{R}</math> for some interval <math>I
  2 KB (417 words) - 00:10, 20 February 2016
• Uncountable
  A [[set]] <math>S</math> is said to be '''uncountable''' if there is no [[inj ...the elements of <math>A</math> (in other words, take an injection <math>f: A \to \mathbb{N}</math>, and denote <math>\omega_i = f(i)</math>).
  2 KB (403 words) - 20:53, 13 October 2019
• AIME Problems and Solutions
  ...ot in the AoPSWiki, please consider adding them. Also, if you notice that a problem in the Wiki differs from the original wording, feel free to correct
  3 KB (391 words) - 16:00, 21 February 2024
• Gamma function
  The '''Gamma function''' is a generalization of the notion of a [[factorial]] to [[complex number|complex numbers]]. ...plex numbers. We can then use the identity to extend the Gamma function to a [[meromorphic]] function on the full [[complex plane]], with simple poles a
  840 bytes (137 words) - 22:26, 22 June 2009
• Modular arithmetic/Intermediate
  ...or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...Z}_n</math> for short). This structure gives us a useful tool for solving a wide range of number-theoretic problems, including finding solutions to [[D
  14 KB (2,317 words) - 19:01, 29 October 2021
• Right triangle
  ...ight triangle''' is any [[triangle]] with an angle of 90 degrees (that is, a [[right angle]]). pair A, B, C;
  3 KB (499 words) - 23:41, 11 June 2022
• Subset
  ...r than <math>B</math> itself. In the latter case, <math>A</math> is called a ''proper subset''. The following is a true statement:
  1 KB (217 words) - 09:32, 13 August 2011
• 2006 AMC 10A
  ...005 AMC 10B|B]]|after=[[2007 AMC 10A|2007 AMC 10A]], [[2007 AMC 10B|B]]|ab=A}}
  2 KB (180 words) - 18:06, 6 October 2014
• Limit
  ...of mathematics. It took several centuries to articulate the definition of a limit and to make it rigorous. ...ue to which a function grows close when its argument is near (but not at!) a particular value. For example,
  7 KB (1,325 words) - 13:51, 1 June 2015
• 2006 AIME I Problems
  In quadrilateral <math> ABCD , \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <mat ...th> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 15
  Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math ...the numbers in this list up in the following way: Whenever a positive and a negative number are adjacent in this progression, pair them up and remove t
  6 KB (910 words) - 19:31, 24 October 2023
• 2006 AIME I Problems/Problem 14
  A tripod has three legs each of length <math>5</math> feet. When the tripod i triple O=(0,0,0),T=(0,0,5),C=(0,3,0),A=(-3*3^.5/2,-3/2,0),B=(3*3^.5/2,-3/2,0);
  6 KB (980 words) - 21:45, 31 March 2020
• 2006 AIME I Problems/Problem 13
  ...st integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...> is even, hence <math>k+n-1</math> is odd, and <math>S_n</math> cannot be a perfect square. Hence <math>k</math> must be even. In particular, as <math>
  10 KB (1,702 words) - 00:45, 16 November 2023
• 2006 AIME I Problems/Problem 12
  ...e have <math>a^3 + b^3 = (a+b)^3 \rightarrow ab(a+b) = 0</math>. But <math>a+b = 2\cos 4x\cos x</math>, so we require <math>\cos x = 0</math>, <math>\co ...A = \{150, 126, 162, 198, 112.5, 157.5\}</math> and thus <math>\sum_{x \in A} x = \boxed{906}</math>.
  911 bytes (147 words) - 18:54, 13 December 2017
• LaTeX:Symbols
  This article provides a short list of commonly used LaTeX symbols. .../math> on the web, (technically an AJAX library simulating it.)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list o
  16 KB (2,324 words) - 16:50, 19 February 2024
• 2006 AIME I Problems/Problem 11
  ...h]] <math> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules: * The cube immediately on top of a cube with edge-length <math> k </math> must have edge-length at most <math>
  3 KB (436 words) - 05:40, 4 November 2022
• 2006 AIME I Problems/Problem 10
  ...are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> ...h>, which can be easily solved to be <math>6x = 2y + 5</math>. Thus, <math>a^2 + b^2 + c^2 = \boxed{065}</math>.
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 9
  ...8 a_{12} = 2006, </math> find the number of possible ordered pairs <math> (a,r). </math> <cmath>\log_8 a_1+\log_8 a_2+\ldots+\log_8 a_{12}= \log_8 a+\log_8 (ar)+\ldots+\log_8 (ar^{11}) \\
  4 KB (651 words) - 18:27, 22 May 2021
• 2006 AIME I Problems/Problem 8
  ...area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</math pair A=(0,0), B=(4.2,0), C=(5.85,-1.6), D=(4.2,-3.2), EE=(0,-3.2), F=(-1.65,-1.6),
  5 KB (730 words) - 15:05, 15 January 2024
• 2006 AIME I Problems/Problem 7
  ...atio of shaded region <math> D </math> to the area of shaded region <math> A. </math> pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10);
  4 KB (709 words) - 01:50, 10 January 2022
• 2006 AIME I Problems/Problem 6
  ...cimal| decimals]] of the form <math> 0.\overline{abc} </math> where <math> a, b, c </math> are distinct [[digit]]s. Find the sum of the elements of <mat Another method, albeit a little risky, that can be used is to note that the numbers between 1 and 99
  2 KB (237 words) - 19:14, 20 November 2023
• 2006 AIME I Problems/Problem 5
  ...n be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s. Find <math> <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2
  3 KB (439 words) - 18:24, 10 March 2015
• 2006 AIME I Problems/Problem 4
  A number in decimal notation ends in a zero for each power of ten which divides it. Thus, we need to count both t ...; every n! for <math>n\geq 50</math> has one more in addition to that, for a total of <math>51</math> extra; and similarly there are <math>26</math> ext
  2 KB (278 words) - 08:33, 4 November 2022
• 2006 AIME I Problems/Problem 3
  ...th>a_n</math> must be divisible by <math>7.</math> But <math>a_n</math> is a nonzero digit, so the only possibility is <math>a_n = 7.</math> This gives ...uation to get <cmath>35=14b+7</cmath> <cmath>b=2.</cmath> Now, since <math>a=7</math>, <math>b=2</math>, and <math>c=5</math>, our number <math>N=100a+1
  4 KB (622 words) - 03:53, 10 December 2022
• 2006 AIME I Problems/Problem 2
  ...th> and let <math> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math> Alternatively, for ease of calculation, let set <math>\mathcal{B}</math> be a 10-element subset of <math>\{1,2,3,\ldots,100\}</math>, and let <math>T</ma
  1 KB (189 words) - 20:05, 4 July 2013
• 2006 AIME I Problems/Problem 1
  In [[quadrilateral]] <math> ABCD</math>, <math>\angle B </math> is a [[right angle]], [[diagonal]] <math>\overline{AC}</math> is [[perpendicular pair C=(0,0), D=(0,-14),A=(-(961-196)^.5,0),B=IP(circle(C,21),circle(A,18));
  2 KB (217 words) - 21:43, 2 February 2014
• MediaWiki
  ...iginally created to run [//en.wikipedia.org Wikipedia], but is now used in a variety of wiki projects.
  505 bytes (73 words) - 01:18, 10 November 2023
• Math and CS Research
  Math and CS Research is a highly-selective mathematics and computer science research publication writ ...conversation and interest by providing editors, writers, and readers with a platform to express and exchange ideas in mathematics and computer science.
  2 KB (222 words) - 15:22, 31 March 2021
• 2022 AIME I Problems/Problem 1
  .../math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial. for some constants <math>a,b,c</math> and <math>d.</math>
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• 2021 JMC 10
  <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qqu <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2
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• 2006 AMC 12B Problems
  \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } <math>\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D)
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• 2006 AMC 12A
  ...C 12B|B]]|after=[[2006 AMC 12B]] , [[2007 AMC 12A]], [[2007 AMC 12B|B]]|ab=A}}
  1 KB (168 words) - 21:51, 6 October 2014
• 2004 AMC 12A
  ...C 12B|B]]|after=[[2004 AMC 12B]] , [[2005 AMC 12A]], [[2005 AMC 12B|B]]|ab=A}} ...ofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=25 2004 AMC A Math Jam Transcript]
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• 2005 AMC 12A
  {{AMC12 box|year=2005|ab=A|before=[[2004 AMC 12A]], [[2004 AMC 12B|B]]|after=[[2005 AMC 12B]], [[2006 ...ofproblemsolving.com/Community/AoPS_Y_MJ_Transcripts.php?mj_id=48 2005 AMC A Math Jam Transcript]
  2 KB (202 words) - 21:30, 6 October 2014
• 2002 AMC 12A
  {{AMC12 box|year=2002|ab=A|before=[[2001 AMC 12]]|after=[[2002 AMC 12B]], [[2003 AMC 12A]], [[2003 AMC
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• 2003 AMC 12A
  {{AMC12 box|year=2003|ab=A|before=[[2002 AMC 12A]], [[2002 AMC 12B|B]]|after=[[2003 AMC 12B]] , [[2004
  1 KB (162 words) - 21:52, 6 October 2014
• 2006 AMC 12A Problems
  {{AMC12 Problems|year=2006|ab=A}} <math> \mathrm{(A) \ } 31\qquad \mathrm{(B) \ } 32\qquad \mathrm{(C) \ } 33\qquad \mathrm{(D)
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• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
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• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} <math>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
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• 2003 AMC 12A Problems
  {{AMC12 Problems|year=2003|ab=A}} <math> \mathrm{(A) \ } 0\qquad \mathrm{(B) \ } 1\qquad \mathrm{(C) \ } 2\qquad \mathrm{(D) \
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• 2002 AMC 12A Problems
  {{AMC12 Problems|year=2002|ab=A}} <math> \mathrm{(A) \ } \frac{7}{2}\qquad \mathrm{(B) \ } 4\qquad \mathrm{(C) \ } 5\qquad \mat
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• 2000 AMC 12 Problems
  <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ <math>\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{
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• 2001 AMC 12 Problems
  <math>\text{(A)}\ 2S + 3\qquad \text{(B)}\ 3S + 2\qquad \text{(C)}\ 3S + 6 \qquad\text{(D) ...ath>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2002 AMC 12B Problems
  ...numbers in the set <math>\{9, 99, 999, 9999, \ldots, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are distinc <math>\mathrm{(A)}\ 0
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• 2003 AMC 12B Problems
  \text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. 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 green pill cost?
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• 2004 AMC 12B Problems
  <math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \q In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and <math>d</math> are 0, 1, 2, and
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• 2005 AMC 12B Problems
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \mathrm{(A)}\ 100 \qquad
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• 2006 AMC 12B Problems/Problem 1
  \text {(A) } - 2006 \qquad \text {(B) } - 1 \qquad \text {(C) } 0 \qquad \text {(D) } {{AMC12 box|year=2006|ab=B|before=First Question|num-a=2}}
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• 2006 AMC 12B Problems/Problem 2
  <math>\text {(A) } - 72 \qquad \text {(B) } - 27 \qquad \text {(C) } - 24 \qquad \text {(D) <math>3\spadesuit -9=-72 \Rightarrow \text{(A)}</math>
  473 bytes (71 words) - 10:44, 4 July 2013
• 2006 AMC 12B Problems/Problem 3
  ...nthers. The two teams scored a total of 34 points, and the Cougars won by a margin of 14 points. How many points did the Panthers score? \text {(A) } 10 \qquad \text {(B) } 14 \qquad \text {(C) } 17 \qquad \text {(D) } 20
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• 2006 AMC 12B Problems/Problem 4
  ...2.99</math>, <math>1.99</math>, and <math>0.99</math>. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of \text {(A) } 5 \qquad \text {(B) } 10 \qquad \text {(C) } 15 \qquad \text {(D) } 20 \
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• 2006 AMC 12B Problems/Problem 5
  ...ast at a speed of 3 miles per hour, while Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many min \text {(A) } 30 \qquad \text {(B) } 50 \qquad \text {(C) } 60 \qquad \text {(D) } 90
  654 bytes (115 words) - 21:47, 1 August 2020
• 2006 AMC 12B Problems/Problem 6
  \text {(A) } 129 \qquad \text {(B) } 137 \qquad \text {(C) } 174 \qquad \text {(D) } ...ies from the lemon juice and <math>386</math> calories from the sugar, for a total of <math>25+386=411</math> calories per <math>600</math> grams. We wa
  978 bytes (145 words) - 13:57, 4 December 2015
• 2006 AMC 12B Problems/Problem 7
  \text {(A) } 4 \qquad \text {(B) } 12 \qquad \text {(C) } 16 \qquad \text {(D) } 24 \ {{AMC12 box|year=2006|ab=B|num-b=6|num-a=8}}
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• 2006 AMC 12B Problems/Problem 8
  ...c 14x + b</math> intersect at the point <math>(1,2)</math>. What is <math>a + b</math>?<!-- don't remove the following tag, for PoTW on the Wiki front \text {(A) } 0 \qquad \text {(B) } \frac 34 \qquad \text {(C) } 1 \qquad \text {(D) }
  1 KB (235 words) - 00:46, 6 January 2022
• 2006 AMC 12B Problems/Problem 9
  \text {(A) } 21 \qquad \text {(B) } 34 \qquad \text {(C) } 51 \qquad \text {(D) } 72 ...>\binom{5}{2}=10</math> and <math>\binom{7}{2}=21</math> choices for <math>a</math> and <math>b</math>. Thus there are altogether <math>3+10+21=\boxed{3
  3 KB (409 words) - 17:10, 30 April 2024
• 2006 AMC 12B Problems/Problem 10
  In a triangle with integer side lengths, one side is three times as long as a second side, and the length of the third side is 15. What is the greatest p \text {(A) } 43 \qquad \text {(B) } 44 \qquad \text {(C) } 45 \qquad \text {(D) } 46
  977 bytes (156 words) - 13:57, 19 January 2021
• 2006 AMC 12B Problems/Problem 11
  Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of \text {(A) } \frac 67 \qquad \text {(B) } \frac {13}{14} \qquad \text {(C) } 1 \qquad
  927 bytes (137 words) - 10:45, 4 July 2013
• 2006 AMC 12B Problems/Problem 12
  <math>\text {(A) } -p \qquad \text {(B) } 0 \qquad \text {(C) } 2 \qquad \text {(D) } 4 \qq Substituting <math>(0,-p)</math>, we find that <math>y = -p = a(0)^2 + b(0) + c = c</math>, so our parabola is <math>y = ax^2 + bx - p</mat
  1 KB (239 words) - 22:20, 17 October 2020
• 2006 AMC 12B Problems/Problem 13
  pair A=origin, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/ draw(B--C--D--A--B--F--D--E--B);
  3 KB (447 words) - 03:49, 16 January 2021
• 2006 AMC 12B Problems/Problem 14
  Elmo makes <math>N</math> sandwiches for a fundraiser. For each sandwich he uses <math>B</math> globs of peanut butter \mathrm{(A)}\ 1.05
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• 2006 AMC 12B Problems/Problem 15
  ...ve radii 2 and 4, respectively, and are externally tangent. Points <math> A</math> and <math> B</math> are on the circle centered at <math> O</math>, a pair A, B, C, D;
  3 KB (458 words) - 16:40, 6 October 2019
• 2006 AMC 12B Problems/Problem 16
  Regular hexagon <math>ABCDEF</math> has vertices <math>A</math> and <math>C</math> at <math>(0,0)</math> and <math>(7,1)</math>, res \mathrm{(A)}\ 20\sqrt {3}
  1 KB (203 words) - 16:36, 18 September 2023
• 2006 AMC 12B Problems/Problem 17
  ...in the ratio <math>1:2:3:4:5:6</math>. What is the probability of rolling a total of <math>7</math> on the two dice? \mathrm{(A)}\ \frac 4{63}
  1 KB (188 words) - 22:10, 9 June 2016
• 2006 AMC 12B Problems/Problem 18
  ...ne unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? \mathrm{(A)}\ 120
  2 KB (354 words) - 16:57, 28 December 2020
• 2006 AMC 12B Problems/Problem 19
  ...s. On a family trip his oldest child, who is 9, spots a license plate with a 4-digit number in which each of two digits appears two times. "Look, daddy! \mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\m
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• 2006 AMC 12B Problems/Problem 20
  \mathrm{(A)}\ \frac 18 For a fixed <math>k</math> the length of the interval <math>\left[ 10^k, \frac{10
  3 KB (485 words) - 14:09, 21 May 2021
• 2006 AMC 12B Problems/Problem 21
  ...06</math>. An ellipse with area <math>2006\pi</math> passes through <math>A</math> and <math>C</math> and has foci at <math>B</math> and <math>D</math> \mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi}
  2 KB (339 words) - 13:15, 12 July 2015
• 2006 AMC 12B Problems/Problem 22
  ...ath>c</math> are positive integers with <math>a+b+c=2006</math>, and <math>a!b!c!=m\cdot 10^n</math>, where <math>m</math> and <math>n</math> are intege \mathrm{(A)}\ 489
  5 KB (881 words) - 15:52, 23 June 2021
• 2006 AMC 12B Problems/Problem 23
  ...re <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? pair A = (B.y,0);
  7 KB (1,169 words) - 14:04, 10 June 2022
• 2006 AMC 12B Problems/Problem 24
  \mathrm{(A)}\ \dfrac{\pi^2}{9} A &= \left(\frac{\pi}{2}\right)^2 - 2 \cdot \frac12 \cdot \left(\frac{\pi}{6}
  3 KB (563 words) - 22:45, 24 October 2021
• 2006 AMC 12B Problems/Problem 25
  A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by \mathrm{(A)}\ 165
  5 KB (924 words) - 12:02, 15 June 2022
• 2006 AMC 12A Problems/Problem 10
  <math> \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qqu ...t{120-\sqrt{x}}</math> to be an integer, <math>120-\sqrt{x}</math> must be a perfect square.
  1 KB (167 words) - 23:23, 16 December 2021
• 2006 AMC 12A Problems/Problem 11
  ...xt{one point}\qquad\mathrm{(C)}\ \text{two lines}\qquad\mathrm{(D)}\ \text{a circle}\qquad\mathrm{(E)}\ \text{the entire plane}</math> {{AMC12 box|year=2006|ab=A|num-b=10|num-a=12}}
  898 bytes (133 words) - 17:52, 3 July 2013
• 2006 AMC 12A Problems/Problem 13
  The [[vertex|vertices]] of a <math>3-4-5</math> [[right triangle]] are the centers of three mutually ext <math> \mathrm{(A) \ } 12\pi\qquad \mathrm{(B) \ } \frac{25\pi}{2}\qquad \mathrm{(C) \ } 13\p
  1 KB (184 words) - 13:57, 19 January 2021
• 2006 AMC 12A Problems/Problem 14
  ..."change" received in the form of goats or pigs as necessary. (For example, a <math>390</math> dollar debt could be paid with two pigs, with one goat rec <math> \textbf{(A) } 5\qquad \textbf{(B) } 10\qquad \textbf{(C) } 30\qquad \textbf{(D) } 90\q
  3 KB (442 words) - 03:13, 8 August 2022
• 2006 AMC 12A Problems/Problem 15
  <math> \mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ ...<math>\frac{5\pi}{3} - \frac{3\pi}{2} = \frac{\pi}{6} \Rightarrow \mathrm{(A)}</math>.
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• 2006 AMC 12A Problems/Problem 16
  ...nters <math>A</math> and <math>B</math> have radius 3 and 8, respectively. A [[common internal tangent line | common internal tangent]] intersects the c pair A=(0,0), Ep=(5,0), B=(5+40/3,0);
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• 2006 AMC 12A Problems/Problem 17
  ...ies on the circle, on the same side of <math>\overline{BE}</math> as <math>A</math>. Segment <math>AF</math> is tangent to the circle, and <math>AF=\sqr label("A", (-s,0), W);
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• 2006 AMC 12A Problems/Problem 18
  <math> \mathrm{(A) \ } \{x|x\ne 0\}\qquad \mathrm{(B) \ } \{x|x<0\}</math> {{AMC12 box|year=2006|ab=A|num-b=17|num-a=19}}
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• 2006 AMC 12A Problems/Problem 19
  ...| radii]] <math>4</math> and <math>9</math>, respectively. The equation of a common external [[tangent line|tangent]] to the circles can be written in t <math> \mathrm{(A) \ } \frac{908}{119}\qquad \mathrm{(B) \ } \frac{909}{119}\qquad \mathrm{(C
  2 KB (253 words) - 22:52, 29 December 2021
• 2006 AMC 12A Problems/Problem 20
  A bug starts at one vertex of a cube and moves along the edges of the cube according to the following rule. <math> \textbf{(A) } \frac{1}{2187}\qquad \textbf{(B) } \frac{1}{729}\qquad \textbf{(C) } \fr
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• 2006 AMC 12A Problems/Problem 21
  <math> \mathrm{(A) \ } 98\qquad \mathrm{(B) \ } 99\qquad \mathrm{(C) \ } 100\qquad \mathrm{(D <math>S_1</math> is a circle with a radius of <math>7</math>. So, the area of <math>S_1</math> is <math>49\pi <
  2 KB (280 words) - 17:35, 17 September 2023
• 2006 AMC 12A Problems/Problem 22
  ...>. The [[probability]] that three entire sides of hexagon are visible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</m <math> \mathrm{(A) \ } 2\sqrt{2}+2\sqrt{3}\qquad \mathrm{(B) \ } 3\sqrt{3}+\sqrt{2}\qquad \ma
  2 KB (343 words) - 15:39, 14 June 2023
• 2006 AMC 12A Problems/Problem 23
  ...h>S=(a_1,a_2,\ldots ,a_n)</math> of <math>n</math> real numbers, let <math>A(S)</math> be the sequence ...ath>x>0</math>, and let <math>S=(1,x,x^2,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>?
  3 KB (466 words) - 22:40, 29 September 2023
• 2006 AMC 12A Problems/Problem 24
  <math> \mathrm{(A) \ } 6018\qquad \mathrm{(B) \ } 671,676\qquad \mathrm{(C) \ } 1,007,514\qqu <cmath>\sum_{a+b+c=2006}{\frac{2006!}{a!b!c!}x^ay^bz^c}</cmath>
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  <math> \mathrm{(A) \ } 277\qquad \mathrm{(B) \ } 311\qquad \mathrm{(C) \ } 376\qquad \mathrm{ ...on can be solved fairly directly by casework and pattern-finding. We give a somewhat more general attack, based on the solution to the following proble
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 20
  (\text {A}) \ 0 \qquad (\text {B}) \ 2005 \qquad (\text {C})\ 4010 \qquad (\text {D}) ...different form(although they'll all satisfy the final equation), so we get a different starting value of <math>x</math>. Every time we have two choices
  3 KB (437 words) - 23:49, 28 September 2022
• 2005 AMC 12A Problems/Problem 25
  <math>(\mathrm {A}) \ 72\qquad (\mathrm {B}) \ 76 \qquad (\mathrm {C})\ 80 \qquad (\mathrm {D ...triangle. This triangle will have a side length of <math>\sqrt{2}</math>; a quick further examination of this cube will show us that this is the only p
  4 KB (495 words) - 01:36, 26 May 2024
• 2005 AMC 12B Problems/Problem 1
  A scout troop buys <math>1000</math> candy bars at a price of five for <math>2</math> dollars. They sell all the candy bars at t \textbf{(A) }\ 100 \qquad
  1 KB (179 words) - 13:53, 14 December 2021
• 2005 AMC 12B Problems/Problem 2
  A positive number <math>x</math> has the property that <math>x\%</math> of <m \textbf{(A) }\ 2 \qquad
  1 KB (145 words) - 13:56, 14 December 2021
• 2005 AMC 12B Problems/Problem 3
  <math>\textbf{(A) }\ \frac15 \qquad\textbf{(B) }\ \frac13 \qquad\textbf{(C) }\ \frac25 \qqua This was just a simple manipulation of the equation. No solving was needed!
  978 bytes (156 words) - 14:14, 14 December 2021
• 2005 AMC 12B Problems/Problem 4
  ...how many of the remaining quizzes can she earn a grade lower than an <math>A</math>? \textbf{(A) }\ 1 \qquad
  1 KB (197 words) - 14:16, 14 December 2021
• 2005 AMC 12B Problems/Problem 5
  ...e tiles of size <math>1</math> foot by <math>1</math> foot. Each tile has a pattern consisting of four white quarter circles of radius <math>1/2</math> \textbf{(A) }\ 80-20\pi \qquad
  2 KB (223 words) - 14:30, 15 December 2021
• 2005 AMC 12B Problems/Problem 6
  ...point on line <math>AB</math> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>? \textbf{(A) }\ 3 \qquad
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• 2005 AMC 12B Problems/Problem 7
  \mathrm{(A)}\ 6 \qquad ...lowing 4 equations (using the logic that if <math>|a|=b</math>, then <math>a</math> is either <math>b</math> or <math>-b</math>):
  2 KB (357 words) - 20:15, 27 December 2020
• 2005 AMC 12B Problems/Problem 8
  For how many values of <math>a</math> is it true that the line <math>y = x + a</math> passes through the vertex of the parabola <math>y = x^2 + a^2</math> ?
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• 2005 AMC 12B Problems/Problem 9
  On a certain math exam, <math>10\%</math> of the students got <math>70</math> po <math>\textbf{(A) }\ 0 \qquad \textbf{(B) }\ 1 \qquad \textbf{(C) }\ 2 \qquad \textbf{(D) }\
  2 KB (280 words) - 15:35, 16 December 2021
• 2005 AMC 12B Problems/Problem 10
  The first term of a sequence is <math>2005</math>. Each succeeding term is the sum of the cubes <math>\textbf{(A) } 29 \qquad \textbf{(B) } 55 \qquad \textbf{(C) } 85 \qquad \textbf{(D) }
  1 KB (204 words) - 14:37, 15 December 2021
• 2005 AMC 12B Problems/Problem 11
  <math>\textbf{(A) }\ {{{\frac{1}{4}}}} \qquad \textbf{(B) }\ {{{\frac{2}{5}}}} \qquad \textb ...>14</math> ways to attain a sum of <math>20</math> or greater, so there is a total probability of <math>\dfrac{14}{28}=\boxed{\textbf{(D) }\frac{1}{2}}<
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• 2005 AMC 12B Problems/Problem 12
  <math>\textbf{(A) }\ {{{1}}} \qquad \textbf{(B) }\ {{{2}}} \qquad \textbf{(C) }\ {{{4}}} \qq Let <math>x^2 + px + m = 0</math> have roots <math>a</math> and <math>b</math>. Then
  2 KB (317 words) - 12:27, 16 December 2021
• 2005 AMC 12B Problems/Problem 13
  <math>\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3 {{AMC12 box|year=2005|ab=B|num-b=12|num-a=14}}
  1 KB (203 words) - 19:57, 24 December 2020
• 2005 AMC 12B Problems/Problem 14
  A circle having center <math>(0,k)</math>, with <math>k>6</math>, is tangent \mathrm{(A)}\ 6\sqrt{2}-6 \qquad
  2 KB (278 words) - 21:12, 24 December 2020
• 2005 AMC 12B Problems/Problem 15
  \mathrm{(A)}\ 1 \qquad .../math>. The last digit of <math>221</math> is <math>1</math>, and <math>10(a+b+c+d)</math> won't affect the units digits, so <math>(e+f+g+h)</math> must
  2 KB (411 words) - 21:02, 21 December 2020
• 2005 AMC 12B Problems/Problem 16
  \mathrm {(A)}\ \sqrt{2} \qquad ...ce, the centers of the spheres are <math>(\pm 1, \pm 1, \pm 1)</math>. For a sphere centered at the origin to contain all eight spheres, its radius must
  2 KB (364 words) - 04:54, 16 January 2023
• 2005 AMC 12B Problems/Problem 17
  How many distinct four-tuples <math>(a,b,c,d)</math> of rational numbers are there with <cmath>a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?</cm
  1 KB (159 words) - 21:18, 21 December 2020
• 2005 AMC 12B Problems/Problem 18
  Let <math>A(2,2)</math> and <math>B(7,7)</math> be points in the plane. Define <math>R< \mathrm{(A)}\ 25 \qquad
  2 KB (262 words) - 21:20, 21 December 2020
• 2005 AMC 12B Problems/Problem 19
  \mathrm{(A)}\ 88 \qquad let <math>x=10a+b</math>, then <math>y=10b+a</math> where <math>a</math> and <math>b</math> are nonzero digits.
  2 KB (283 words) - 20:02, 24 December 2020
• 2005 AMC 12B Problems/Problem 20
  Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set <math What is the minimum possible value of <math>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</math>
  3 KB (463 words) - 19:28, 6 November 2022
• 2005 AMC 12B Problems/Problem 21
  A positive integer <math>n</math> has <math>60</math> divisors and <math>7n</ <math>\mathrm{(A)}\ {{{0}}} \qquad \mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad
  888 bytes (140 words) - 20:04, 24 December 2020
• 2005 AMC 12B Problems/Problem 22
  A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is define \textbf{(A)}\ 1 \qquad
  4 KB (660 words) - 17:40, 24 January 2021
• 2005 AMC 12B Problems/Problem 23
  ...y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math> \textbf{(A)}\ \frac {15}{2} \qquad
  5 KB (786 words) - 11:36, 19 May 2024
• 2005 AMC 12B Problems/Problem 24
  ...<math>2</math>. The <math>x</math>-coordinates of the three vertices have a sum of <math>m/n</math>, where <math>m</math> and <math>n</math> are relati <math> \mathrm{(A)}\ {{{14}}}\qquad\mathrm{(B)}\ {{{15}}}\qquad\mathrm{(C)}\ {{{16}}}\qquad\m
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• 2005 AMC 12B Problems/Problem 25
  ...the six [[vertex|vertices]] of a regular [[octahedron]], with each ant at a different vertex. Simultaneously and independently, each ant moves from its <math>\mathrm{(A)}\ \frac {5}{256}
  10 KB (1,840 words) - 21:35, 7 September 2023
• Mathematical convention
  A '''mathematical convention''' is a fact, name, [[notation]], or usage which is generally agreed upon by [[math ...longer a name or notation has been in use, the more likely it is to become a mathematical convention. Unfortunately, some notational questions stubborn
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• Power of a Point Theorem/Introductory Problem 1
  Applying the [[Power of a Point Theorem]], we get <math> 3\cdot(3+5) = x (x+10) \rightarrow x^2 + 10x ''Back to the [[Power of a Point Theorem]].''
  448 bytes (67 words) - 15:15, 23 March 2020
• Power of a Point Theorem/Introductory Problem 2
  Applying the Power of a Point Theorem gives <math> 6\cdot x = 4\cdot 1 </math>, so <math> x = \frac ''Back to the [[Power of a Point Theorem]].''
  289 bytes (45 words) - 13:14, 16 July 2017
• Power of a Point Theorem/Introductory Problem 4
  ...</math> of a given circle are perpendicular to each other and intersect at a right angle at <math>E</math> . Given that <math> BE = 16, DE = 4, </math> pair A = (3,0);
  1 KB (177 words) - 02:14, 26 November 2020
• Root (polynomial)
  ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...ath> with all <math>c_j \in \mathbb C</math> and <math>c_n \neq 0</math>, a general degree-<math>n</math> polynomial. The degree of <math>P(x)</math> i
  8 KB (1,427 words) - 21:37, 13 March 2022
• Vertical line test
  ...y of determining whether or not a plotted [[graph of a function|graph]] is a [[function]]. The vertical line test states that a [[relation]] is a [[function]] if no vertical [[line]] intersects the graph in more than one
  659 bytes (114 words) - 10:41, 27 April 2024
• Algebraically closed
  ...eld. The field of [[complex number]]s, denoted <math>\mathbb{C}</math>, is a well-known example of an algebraically closed field.
  343 bytes (52 words) - 17:17, 12 March 2014
• 2006 AMC 10A Problems
  {{AMC10 Problems|year=2006|ab=A}} <math>\mathrm{(A)}\ 31\qquad\mathrm{(B)}\ 32\qquad\mathrm{(C)}\ 33\qquad\mathrm{(D)}\ 34\qqu
  13 KB (2,028 words) - 16:32, 22 March 2022
• 1995 AHSME Problems/Problem 6
  The figure shown can be folded into the shape of a cube. In the resulting cube, which of the lettered faces is opposite the fa label("$A$", (1.3,0.5), E);
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• 2006 AMC 10A Problems/Problem 6
  <math> \textbf{(A) } \frac17\qquad \textbf{(B) } \frac27\qquad \textbf{(C) } 1\qquad \textbf{ {{AMC10 box|year=2006|ab=A|num-b=5|num-a=7}}
  590 bytes (84 words) - 14:28, 31 May 2023
• 2006 AMC 10A Problems/Problem 8
  A [[parabola]] with equation <math>y=x^2+bx+c</math> passes through the point <math> \textbf{(A) } 2\qquad \textbf{(B) } 5\qquad \textbf{(C) } 7\qquad \textbf{(D) } 10\qqu
  2 KB (348 words) - 23:10, 16 December 2021
• 2006 AMC 10A Problems/Problem 11
  ...oint}\qquad \textbf{(C) } \textrm{two\,lines} \qquad \textbf{(D) } \textrm{a\,circle} \qquad \textbf{(E) } \textrm{the\,entire\,plane} </math> {{AMC10 box|year=2006|ab=A|num-b=10|num-a=12}}
  820 bytes (123 words) - 08:05, 17 December 2021
• 2006 AMC 10A Problems/Problem 12
  Rolly wishes to secure his dog with an 8-foot rope to a [[square (geometry) | square]] shed that is 16 feet on each side. His prel <math> \textbf{(A) } I,\,\textrm{ by }\,8\pi\qquad \textbf{(B) } I,\,\textrm{ by }\,6\pi\qqua
  3 KB (424 words) - 10:14, 17 December 2021
• 2006 AMC 10A Problems/Problem 13
  ...d loses otherwise. How much should the player win if the game is fair? (In a fair game the probability of winning times the amount won is what the playe <math>\textbf{(A) } \textdollar12\qquad\textbf{(B) } \textdollar30\qquad\textbf{(C) } \textd
  1 KB (207 words) - 09:39, 25 July 2023
• 2006 AMC 12A Problems/Problem 12
  A number of linked rings, each <math>1</math> cm thick, are hanging on a peg. The top ring has an outside diameter of <math>20</math> cm. The outsid <math>\textbf{(A) } 171\qquad\textbf{(B) } 173\qquad\textbf{(C) } 182\qquad\textbf{(D) } 188
  2 KB (292 words) - 11:56, 17 December 2021
• 2006 AMC 10A Problems/Problem 15
  ...ns counterclockwise at <math>300 m/min</math> and uses the outer lane with a radius of <math>60</math> meters, starting on the same radial line as Odell <math>\textbf{(A) } 29\qquad\textbf{(B) } 42\qquad\textbf{(C) } 45\qquad\textbf{(D) } 47\qqu
  3 KB (532 words) - 17:49, 13 August 2023
• 2006 AMC 10A Problems/Problem 16
  A circle of radius <math>1</math> is [[tangent]] to a circle of radius <math>2</math>. The sides of <math>\triangle ABC</math> ar D('B', (0,0),SW); D('C',(4*t,0), SE); D('A', (2*t, 8), N);
  5 KB (732 words) - 23:19, 19 September 2023
• 2006 AMC 10A Problems/Problem 17
  pair A,B,C,D,E,F,G,H,W,X,Y,Z; A=(0,2); B=(1,2); C=(2,2); D=(3,2);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2006 AMC 10A Problems/Problem 18
  A license plate in a certain state consists of <math>4</math> digits, not necessarily distinct, <math>\textbf{(A) } 10^4\times 26^2\qquad\textbf{(B) } 10^3\times 26^3\qquad\textbf{(C) } 5\
  2 KB (254 words) - 14:39, 5 April 2024
• 2006 AMC 10A Problems/Problem 19
  <math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 59\qquad\textbf{(D) } 89\qquad The sum of the angles of a triangle is <math>180</math> degrees. For an arithmetic progression with an
  2 KB (259 words) - 03:10, 22 June 2023
• 2006 AMC 10A Problems/Problem 20
  ...the probability that some pair of these integers has a difference that is a multiple of <math>5</math>? <math>\textbf{(A) } \frac{1}{2}\qquad\textbf{(B) } \frac{3}{5}\qquad\textbf{(C) } \frac{2}{3
  1 KB (187 words) - 08:21, 17 March 2023
• 2006 AMC 10A Problems/Problem 21
  ...igit positive integers have at least one digit that is a <math>2</math> or a <math>3</math>? <math>\textbf{(A) } 2439\qquad\textbf{(B) } 4096\qquad\textbf{(C) } 4903\qquad\textbf{(D) }
  3 KB (525 words) - 20:25, 30 April 2024
• 2006 AMC 10A Problems/Problem 24
  Centers of adjacent faces of a unit cube are joined to form a regular [[octahedron]]. What is the volume of this octahedron? <math>\textbf{(A) } \frac{1}{8}\qquad\textbf{(B) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4
  2 KB (292 words) - 10:19, 19 December 2021
• Set
  ...ms, see [[Zermelo-Fraenkel Axioms]]. In this article we shall present just a brief discussion of the most common properties of sets and operations relat ...g: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset.
  11 KB (2,021 words) - 00:00, 17 July 2011
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] Let <math>f:(0,+\infty)\to\mathbb C</math> be a bounded function. Assume that
  6 KB (1,034 words) - 07:55, 12 August 2019
• Computer science
  ...nder certain constraints. For example, the [[P versus NP]] problem is such a problem.
  1 KB (157 words) - 23:11, 16 November 2023
• 2006 USAMO Problems
  Let <math>p</math> be a prime number and let <math>s</math> be an integer with <math>0<s<p</math>. if and only if <math>s</math> is not a divisor of <math>p-1</math>.
  3 KB (520 words) - 09:24, 14 May 2021
• 1991 AJHSME Problems
  <math>\text{(A)}\ 222,222,222,222 \qquad <math>\text{(A)}\ 4 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 12 \qquad \text{(D)}\ 16 \qqua
  17 KB (2,246 words) - 13:37, 19 February 2020
• 1990 AJHSME Problems/Problem 1
  <math>\text{(A)}\ 947 \qquad \text{(B)}\ 1037 \qquad \text{(C)}\ 1047 \qquad \text{(D)}\ 1 ...ath> and <math>\overline{def}</math>. Their [[sum]] is equal to <math>100(a+d)+10(b+e)+(c+f)</math>.
  1 KB (191 words) - 17:12, 29 October 2016
• 2022 USAJMO Problems/Problem 1
  ...exist integers <math>a,d,r</math> with <math>m \nmid d</math> and <math>m|a+(n-1)d-gr^{n-1}</math> for all integers <math>n>1</math>. <cmath>m | a+nd-gr^n \; (1),</cmath>
  4 KB (792 words) - 00:29, 13 April 2024
• Prime ideal
  In [[ring theory]] we say that an [[ideal]] <math>P</math> of a [[ring]] <math>R</math> is '''prime''' if <math>P\ne R</math> and for any i ...nd for any <math>a,b\in R</math> if <math>ab\in P</math> then either <math>a\in P</math> or <math>b\in P</math>.
  967 bytes (176 words) - 18:08, 7 April 2012
• 2006 AIME II Problems/Problem 2
  ...\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>. ...equality]] and applying the well-known logarithmic property <math>\log_{c} a + \log_{c} b = \log_{c} ab</math>, we have that
  1 KB (164 words) - 14:58, 14 April 2020
• 3D Geometry
  ...awing on a piece of paper is 2-dimensional since it has length and width. A baseball, on the other hand, is three-dimensional because it not only has l ...e can do this by looking at certain cross-section(s) of the diagram one at a time.
  2 KB (264 words) - 06:43, 24 May 2024
• 2005 AIME II Problems/Problem 1
  A game uses a deck of <math> n </math> different cards, where <math> n </math> is an inte We must find a [[factoring|factorization]] of the left-hand side of this equation into thr
  1 KB (239 words) - 11:54, 31 July 2023
• 2005 AIME II Problems/Problem 2
  ...e indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the [[probability]] each guest got o ...We need only calculate the probability the first and second person all get a roll of each type, since then the rolls for the third person are determined
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 3
  An [[infinite]] [[geometric series]] has sum 2005. A new series, obtained by squaring each term of the original series, has 10 t ...>10 = \frac a{1 + r}</math>. Then <math>a = 2005 - 2005r</math> and <math>a = 10 + 10r</math> so <math>2005 - 2005r = 10 + 10r</math>, <math>1995 = 201
  3 KB (581 words) - 07:54, 4 November 2022
• 2005 AIME II Problems/Problem 4
  Assume that <math>n</math> (a positive integer) is a divisor of one of the numbers. Therefore, <math>n</math> can be expressed a If <math>n</math> is the power of a single prime, then there are 11 possibilities (<math>2^1</math> to <math>2^
  3 KB (377 words) - 18:36, 1 January 2024
• 2005 AIME II Problems/Problem 13
  Let <math> P(x) </math> be a polynomial with integer coefficients that satisfies <math> P(17)=10 </math> ...it has roots at <math>17</math> and <math>24</math>. Hence <math>P(x)-x+7=A(x-17)(x-24)</math>. In particular, this means that
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems/Problem 5
  ...b) </math> of [[integer]]s such that <math> \log_a b + 6\log_b a=5, 2 \leq a \leq 2005, </math> and <math> 2 \leq b \leq 2005. </math> ...r <math>\log b=2\log a </math>, so either <math> b=a^3 </math> or <math> b=a^2 </math>.
  3 KB (547 words) - 19:15, 4 April 2024
• 2005 AIME II Problems/Problem 6
  ...original stack, the stack is named magical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original pos ...of it. There are <math>\frac{130}{2} = 65</math> cards from each of piles A, B in front of card 131. This suggests that <math>n = 131 + 65 = 196</math>
  2 KB (384 words) - 00:31, 26 July 2018
• 2005 AIME II Problems
  A game uses a deck of <math> n </math> different cards, where <math> n </math> is an inte ...e indistinguishable from one another. She then randomly put three rolls in a bag for each of the guests. Given that the probability each guest got one r
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 7
  {{AIME box|year=2005|n=II|num-b=6|num-a=8}}
  2 KB (279 words) - 12:33, 27 October 2019
• 2005 AIME II Problems/Problem 8
  ...e circles are all [[collinear]]. A [[chord]] of <math> C_3 </math> is also a common external tangent of <math> C_1 </math> and <math> C_2. </math> Given pair A = OP(cir3, t), B = IP(cir3, t), T1 = IP(cir1, t), T2 = IP(cir2, t);
  4 KB (693 words) - 13:03, 28 December 2021
• 2005 AIME II Problems/Problem 9
  ...math>n</math>. So, we'd like to somehow convert our given expression into a form from which we can apply De Moivre's Theorem. ...1 \Rightarrow n \equiv 1 \bmod{4}</math>. Thus since <math>1000</math> is a multiple of <math>4</math> exactly one quarter of the residues are congruen
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ...> AB </math> with <math> AE<BF </math> and <math> E </math> between <math> A </math> and <math> F, m\angle EOF =45^\circ, </math> and <math> EF=400. </m ...D--A);draw(E--O--F);draw(G--O); dot(A^^B^^C^^D^^E^^F^^G^^O); label("\(A\)",A,(-1,1));label("\(B\)",B,(1,1));label("\(C\)",C,(1,-1));label("\(D\)",D,(-1,
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  Given that <math> O </math> is a regular [[octahedron]], that <math> C </math> is the [[cube (geometry) | cu ...ertices]] of the octahedron be <math>A, B, C, D, E, F</math> so that <math>A</math> and <math>F</math> are opposite each other and <math>AF = s\sqrt2</m
  3 KB (436 words) - 03:10, 23 September 2020
• 2005 AIME II Problems/Problem 11
  ...math> be a positive integer, and let <math> a_0, a_1,\ldots,a_m </math> be a sequence of reals such that <math>a_0 = 37, a_1 = 72, a_m = 0, </math> and Thus the product <math>a_{k}a_{k+1}</math> is a [[monovariant]]: it decreases by 3 each time <math>k</math> increases by 1.
  3 KB (499 words) - 18:52, 21 November 2022
• Hyperbola
  A '''hyperbola''' ( pronounced: /hīˈpərbələ/ ) is defined as the [[locus ...lso, the graph of <math>xy=k</math> for some real number <math>k</math> is a hyperbola.
  666 bytes (112 words) - 10:06, 30 January 2019

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