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• 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

Page text matches

• 2016 AMC 10A Problems/Problem 10
  ...t colors as shown. The areas of the three differently colored regions form an arithmetic progression. The inner rectangle is one foot wide, and each of t <math>\textbf{(A) } 1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 4 \qquad \textbf{(D) } 6 \
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• 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 13
  ...d moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada been sitting before she got up? <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 * [[Math Olympiads for Elementary and Middle Schools]] (MOEMS) -- An extremely popular competition amongst elementary and middle school students
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• 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://www.imc-impea.org IMC-IMPEA] is an offline/online math contest for all grades level. The contest offers indivi ...f math contests open to individuals (homeschooled and otherwise), and also an up-to-date state-by-state list of regional meets.
  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 There's also some exams with the part of mathematics being as hard as an olympiad, but in the last years some of the questions in these exams are ev
  4 KB (406 words) - 11:20, 26 February 2021
• 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
• 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
• 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 The ICTM Contest Committee offers an Algebra contest for junior high schools. This individual competition consis
  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
  ...theme, team, and guts. Competitors form teams of up to four people. It is an annual contest that takes place in the spring. [https://winmac.org// websit ...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
  *[[Math-Con]] is an online national math competition [http://www.mathcon.org/ website] *[[Rapid Rabbit Relays]] (a middle school event at Detroit Country Day School)
  2 KB (305 words) - 01:06, 30 November 2023
• Oklahoma mathematics competitions
  ....com/wiki/index.php/OSSM_Math_and_Engineering_Day OSSM Awesome Contest] is an Oklahoma middle school math competition run by [https://www.ossm.edu/outrea ...ed) students from that year's MathCounts state competition will compete as a mixed team. Oklahoma does not participate in the high school contest.
  2 KB (279 words) - 19:41, 23 March 2017
• 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 ...h Contest[https://www.math.utah.edu/~bestvina/intermountain/index.html] is an college proof based math competition. It's format is similar to the Putnam,
  2 KB (231 words) - 13:07, 14 December 2019
• 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
  '''Art of Problem Solving''' ('''AoPS''') is an educational resources company founded by [[Richard Rusczyk]] in 2003. For a list of all the current staff at AoPS and some previous staff go to the [[A
  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
  ...d is recommended for high school students who are already studying math at an undergraduate level. ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato
  24 KB (3,177 words) - 12:53, 20 February 2024
• 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
  7 KB (901 words) - 14:11, 6 January 2022
• 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}}
  4 KB (623 words) - 13:11, 20 February 2024
• Physics competitions
  This is a resources page for students interested in '''Physics competitions'''. Plea ...emsolving.com/Forum/index.php?f=405 Introductory Physics Forum] as well as an [http://www.artofproblemsolving.com/Forum/index.php?f=332 Advanced Physics
  4 KB (506 words) - 11:46, 6 September 2023
• Computer science books
  * [http://www.amazon.com/exec/obidos/ASIN/0486240614/artofproblems-20 An Introduction to Information Theory] by J.R. Pierce. ...p://olympiads.win.tue.nl/ioi/study/books.html Reading List] (someone have a linky?)
  2 KB (251 words) - 00:45, 17 November 2023
• 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:
  9 KB (1,355 words) - 07:29, 29 September 2021
• 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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• 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>:
  12 KB (1,798 words) - 16:20, 14 March 2023
• 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.
  3 KB (337 words) - 03:35, 7 September 2020
• 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
  4 KB (520 words) - 12:11, 13 March 2024
• American Invitational Mathematics Examination
  ...ical Association of America]] (MAA). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC! ...tions. Qualification through USAMTS only is rare, however. Each answer is an integer from 000 to 999, inclusive, making guessing almost futile. Wrong a
  8 KB (1,057 words) - 12:02, 25 February 2024
• American Mathematics Competitions
  ...h as [[MIT]] now ask students for their AMC scores. "AMC" is also used as an abbreviation for American Math Contest, used to refer to the AMC 8, AMC 10, ...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
  5 KB (696 words) - 03:47, 24 December 2019
• 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);
  3 KB (415 words) - 18:01, 24 May 2020
• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (692 words) - 22:33, 15 February 2021
• United States of America Mathematical Olympiad
  ...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
  6 KB (869 words) - 12:52, 20 February 2024
• Informatics competitions
  '''Informatics competitions''' test a student's ability to understand, organize, and work with information on com * The [[United States of America Computing Olympiad]] [[USACO]] is an extremely challenging competition that helps select U.S. team members to co
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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.
  712 bytes (106 words) - 13:28, 15 January 2022
• 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.
  2 KB (267 words) - 17:06, 7 March 2020
• 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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• 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. * [[EPGY]] hosts an assortment of gifted programs. [http://epgy.stanford.edu/ Homepage]
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• Engineer
  An '''Engineer''' is a person that creates, designs, or updates machines or structures.
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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 means None Of The Above. Test topics usuall
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• AoPS Online School
  The '''AoPS Online School''' is an online school that hosts classes for outstanding middle and high school stu ...thematics competitions]]. Anyone may join these classes provided they have an AoPS account.
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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>. ...ry one that rests close to basic principles of [[modular arithmetic]], and an elegant method that relies on more powerful [[algebra]]ic tools.
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  The '''trivial inequality''' is an [[inequality]] that states that the square of any real number is nonnegativ ...application is maximizing and minimizing [[quadratic]] functions. It gives an easy proof of the two-variable case of the [[AMGM | Arithmetic Mean-Geometr
  3 KB (560 words) - 22:51, 13 January 2024
• 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
  2 KB (422 words) - 16:20, 5 March 2023
• 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:
  4 KB (675 words) - 00:05, 22 January 2024
• 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
  ...variables with each of those variables in a linear term on the other side. An example would be: <cmath>xy+66x-88y=23333</cmath>where <math>23333</math> i ...Factoring Trick, this equation can be transformed into: <cmath>(x+k)(y+j)=a+jk</cmath>
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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 ...for every correct response, 1 point for a blank response, and 0 points for an incorrect response.
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• Factoring
  ...oduct of different terms. This often allows one to find information about an expression that was not otherwise obvious. Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for differe
  3 KB (532 words) - 22:00, 13 January 2024
• 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
  2 KB (374 words) - 14:01, 21 August 2022
• 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
  11 KB (1,985 words) - 21:03, 5 August 2023
• 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.
  2 KB (264 words) - 12:04, 15 July 2021
• 1965 AHSME Problems/Problem 40
  ...> such that <math>P = x^4 + 6x^3 + 11x^2 + 3x + 31</math> is the square of an integer. Then <math>n</math> is: <math>\textbf{(A)}\ 4 \qquad
  3 KB (571 words) - 00:42, 22 October 2021
• 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
  578 bytes (81 words) - 19:22, 23 November 2015
• 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.
  9 KB (1,703 words) - 07:25, 24 March 2024
• Mill's Constant
  ...>\theta</math> such that <math>\lfloor\theta^{3^n}\rfloor</math> is always a prime number for all natural n. ...math>n</math> can be any number and <math>\theta</math> is an element from an set of numbers (that may be rational or irrational, and we are not sure) an
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• Combination
  ...cerned with finding the number of combinations of size <math>r</math> from an original set of size <math>n</math> ...ons are, their various types, and how to calculate each type! It serves as a great introductory video to combinations, permutations, and counting proble
  4 KB (615 words) - 11:43, 21 May 2021
• Cauchy-Schwarz Inequality
  ...yakovsky–Schwarz Inequality''' or informally as '''Cauchy-Schwarz''', is an [[inequality]] with many ubiquitous formulations in abstract algebra, calcu ..._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
  13 KB (2,048 words) - 15:28, 22 February 2024
• Factorial
  The '''factorial''' is an important function in [[combinatorics]] and [[analysis]], used to determine ...!=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
  10 KB (809 words) - 16:40, 17 March 2024
• ARML historical results
  This is a list of historical results from the [[American Regions Mathematics League]] == Division A Top Five Teams ==
  19 KB (2,632 words) - 14:31, 12 June 2022
• 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>.
  16 KB (2,658 words) - 16:02, 8 May 2024
• 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
  3 KB (551 words) - 16:22, 13 September 2023
• 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>.
  3 KB (542 words) - 17:45, 21 March 2023
• Brute forcing
  ...an 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 ([[angle]]s, [[length]]s, [[area] ...level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]].
  7 KB (1,296 words) - 14:22, 22 October 2023
• 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
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's theorem
  ...se]] of [[Ptolemy's Inequality]]. Ptolemy's theorem frequently shows up as an intermediate step in problems involving inscribed figures. Given a [[cyclic quadrilateral]] <math>ABCD</math> with side lengths <math>{a},{b},{c},{d}</math> and [[diagonal]]s <math>{e},{f}</math>:
  7 KB (1,198 words) - 20:39, 9 March 2024
• Elementary symmetric sum
  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:
  2 KB (275 words) - 12:51, 26 July 2023
• 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.
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...larly well-suited to studying properties of individual prime numbers. From an algebraic perspective, number theory can perhaps best be described as the s An elliptic curve is the set of points <math>(x,y)</math> satisfying some two
  5 KB (849 words) - 16:14, 18 May 2021
• 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.
  2 KB (198 words) - 17:47, 3 November 2021
• 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
  ...birthday problem]]. Probability can be loosely defined as the chance that an event will happen. Before reading about the following topics, a student learning about probability should learn about introductory [[counti
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• Jeff Boyd
  ...://www.artofproblemsolving.com/Resources/AoPS_R_A_MATHCOUNTS.php/ Building a Successful MathCounts Program].
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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
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• 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>.
  2 KB (277 words) - 16:21, 29 April 2023
• 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 ...This is a simple application of the rearrangement inequality. It is also an application of the [[greedy algorithm]], so one possible interpretation of
  5 KB (804 words) - 13:54, 26 January 2023
• 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
  8 KB (1,298 words) - 15:07, 23 May 2024
• 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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• 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 ...his, take an acute triangle and swap its orthocenter and any vertex to get an obtuse triangle. It is easy to verify that this placement of the orthocente
  8 KB (1,408 words) - 11:54, 8 December 2021
• 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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• 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
  '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jensen in 1906. Let <math>{F}</math> be a [[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb
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• Michigan Mathematics Prize Competition
  ...er 5-question proof test. The top 100 finishers of Part II are honored at an awards banquet every year, with the top 50 receiving scholarships. Part I is normally on a Wednesday morning in the first or second week of October.
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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. ...ms cannot be completely solved through casework. However, it is crucial as an intermediate step across all of mathematics, not just in competitions.
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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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• 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
  ...function has the rule that it takes its input value, and squares it to get an output value. One can call this function <math>f</math>. ...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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• 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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• 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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• 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
  ...mber bases''' are 2, 8, 10 and 16. The base doesn't necessarily have to be an [[integer]]. There are [[complex base | complex]], [[irrational base | irra [[Binary]] is base 2. It's a favorite among computer programmers. It has just two digits: <math>0</math>
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• Partition of an interval
  A '''partition of an interval''' is a division of an [[interval]] into several disjoint sub-intervals. Partitions of intervals 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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• 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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• Multiple
  ...m</math> and <math>n</math> are integers. (In this case, <math>k</math> is a multiple of <math>n</math>, as well). Every nonzero integer has an [[infinite]] number of multiples. As an example, some of the multiples of 15 are 15, 30, 45, 60, and 75.
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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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• 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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• Pi
  '''Pi''' is an [[irrational number]] (in fact, [[transcendental number]], as proved by Lin Pi is the [[ratio]] of the [[circumference]] ([[perimeter]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.14159
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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 ...ample of a [[linear recursion]] with constant coefficients. There is also an explicit formula [[#Binet's formula|below]].
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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>: The inequality is easier to understand given an example. Since the sequence <math>(5,1)</math> majorizes <math>(4,2)</math
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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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• 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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• 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 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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• 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
  ...e given two sides and the included angle. All triangles with two sides and an include angle are [[congruent]] by the Side-Angle-Side congruence postulate ...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''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai S Schur's inequality states that for all non-negative <math>a,b,c \in \mathbb{R}</math> and <math>r>0</math>:
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• Trigonometric identities
  ...functions that are true for all input values. Trigonometric functions have an abundance of identities, of which only the most widely used are included in ...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
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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
  '''Iff''' is an abbreviation for the phrase "if and only if." It is also known as a [[conditional|biconditional]] statement.
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• Irrational number
  ...t cannot be expressed as the [[ratio]] of two [[integer]]s. Equivalently, an irrational number, when expressed in [[decimal notation]], never terminates ...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 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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• 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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• Roots of unity
  The '''Roots of unity''' are a topic closely related to [[trigonometry]]. Roots of unity come up when we e First, we note that since we have an '''n'''th degree polynomial, there will be '''n''' complex roots.
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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> An [[angle]] <math>\angle BAC</math> also has an external angle bisector, which bisects external angle <math>BAC</math>:
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• Sequence
  A '''sequence''' is an ordered list of terms. Sequences may be either [[finite]] or [[infinite]]. A sequence of real numbers is simply a function <math>f : \mathbb{N} \rightarrow \mathbb{R}</math>. For instance,
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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 ...e with common difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with common difference <math>-8</math>; however, <math>
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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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• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or sometimes [[natural number]] or [[who ...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
  ...ng an [[integer]] for the base in our [[positional number system]], we use an [[improper fraction]] for the base. ...te 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 of the unknowns to be solved for are [[function]]s. ===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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• 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 ...<math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\frac 1{2q}</math>.
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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 In contest math, trigonometry is an integral subfield of both [[geometry]] and [[algebra]]. Many essential resu
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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 An approximation for Euler's number is <math>e\approx 2.7182818284590452...</m
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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 === Using an infinite series ===
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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 ...', and let <math>\mathrm{Li}(x)=\int_2^x \frac{1}{\ln t}\; dt</math>. Then an equivalent statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm
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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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• 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 ...ty, suppose that <math>P</math> is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It is clearly bou
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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. ...ents of a non-trivial cycle, and also the only possible lowest elements of an infinitely increasing sequence.
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• Incenter
  ...r is the center of the [[incircle]]. Every [[nondegenerate]] triangle has a unique incenter. ...th>BC</math> and <math>BC</math> and <math>CA</math> and therefore lies on an angle bisector of <math>ACB</math>. Since it lies within the triangle <mat
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• Majorization
  We say a [[nonincreasing]] [[sequence]] of [[real number]]s <math> a_1, \ldots ,a_n< ...^n a_i \le \sum_{i=k}^n b_i</math>, with equality when <math>k=1 </math>. An interesting consequence of this is that the finite sequence <math>\{a_i\} <
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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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• Algebraic geometry
  ...p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine algebraic variety'''. ...eties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying
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• Prime Number Theorem
  is easier, though still non-trivial!) It gives an constants <math>A</math> and <math>B</math>,
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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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• Well-Ordering theorem
  ...]] <math>(S,\prec)</math> for which each set <math>A\subseteq S</math> has a [[minimum]] element.
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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]]
  577 bytes (81 words) - 10:33, 18 April 2019
• 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 ==
  51 KB (6,175 words) - 20:58, 6 December 2023
• 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
  2 KB (301 words) - 17:46, 16 March 2012
• 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.
  3 KB (422 words) - 11:01, 25 December 2020
• 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
• 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
• 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
  ...[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect sq 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
  ...s either [[finite]] or [[countably infinite]]. The most common example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. We give an [[indirect proof]] here. This is one of the most famous indirect proofs and
  2 KB (403 words) - 20:53, 13 October 2019
• 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
• 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 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
• LaTeX:Symbols
  This article provides a short list of commonly used LaTeX symbols. [http://detexify.kirelabs.org/classify.html Detexify] is an online application which allows you to draw the symbol you'd like and shows
  16 KB (2,324 words) - 16:50, 19 February 2024
• 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 ...ether, the sum is <math>0.\overline{999}</math>, or, more precisely, 1. As an example, <math>.\overline{123}+.\overline{876}=.\overline{999} \Longrightar
  2 KB (237 words) - 19:14, 20 November 2023
• 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 ...irst digit deleted. Now, we know that <math>N<1000</math> (because this is an AIME problem). Thus, <math>N</math> has <math>1,</math> <math>2</math> or <
  4 KB (622 words) - 03:53, 10 December 2022
• 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
  12 KB (1,784 words) - 16:49, 1 April 2021
• 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)
  13 KB (2,058 words) - 12:36, 4 July 2023
• 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)
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
  13 KB (1,971 words) - 13:03, 19 February 2020
• 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)
  13 KB (1,953 words) - 00:31, 26 January 2023
• 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) \
  13 KB (1,955 words) - 21:06, 19 August 2023
• 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
  12 KB (1,792 words) - 13:06, 19 February 2020
• 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^{
  13 KB (1,948 words) - 12:26, 1 April 2022
• 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
  10 KB (1,547 words) - 04:20, 9 October 2022
• 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?
  13 KB (1,987 words) - 18:53, 10 December 2022
• 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
  13 KB (2,049 words) - 13:03, 19 February 2020
• 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
  12 KB (1,781 words) - 12:38, 14 July 2022
• 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 \
  1 KB (152 words) - 16:11, 8 December 2013
• 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 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
  1 KB (227 words) - 17:21, 8 December 2013
• 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
  4 KB (696 words) - 09:47, 10 August 2015
• 2006 AMC 12B Problems/Problem 20
  \mathrm{(A)}\ \frac 18 Let <math>k</math> be an arbitrary integer. For which <math>x</math> do we have <math>\lfloor\log_{1
  3 KB (485 words) - 14:09, 21 May 2021
• 2006 AMC 12B Problems/Problem 21
  ...and <math>D</math>. What is the perimeter of the rectangle? (The area of an ellipse is <math>ab\pi</math> where <math>2a</math> and <math>2b</math> are \mathrm{(A)}\ \frac {16\sqrt {2006}}{\pi}
  2 KB (339 words) - 13:15, 12 July 2015
• 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 12A Problems/Problem 10
  ...how many real values of <math>x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer? <math> \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qqu
  1 KB (167 words) - 23:23, 16 December 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 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);
  6 KB (958 words) - 23:29, 28 September 2023
• 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
  5 KB (908 words) - 19:23, 22 September 2022
• 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 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 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 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 11
  An envelope contains eight bills: <math>2</math> ones, <math>2</math> fives, < <math>\textbf{(A) }\ {{{\frac{1}{4}}}} \qquad \textbf{(B) }\ {{{\frac{2}{5}}}} \qquad \textb
  4 KB (607 words) - 21:01, 20 May 2023
• 2005 AMC 12B Problems/Problem 18
  ...ing of those points <math>C</math> such that <math>\triangle ABC</math> is an acute triangle. What is the closest integer to the area of the region <math \mathrm{(A)}\ 25 \qquad
  2 KB (262 words) - 21:20, 21 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
  4 KB (761 words) - 09:10, 1 August 2023
• 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
• 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
• Algebraically closed
  ...lex 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
• 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

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