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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
  21 KB (3,500 words) - 18:41, 23 April 2024
• 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
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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>.
  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
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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.
  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
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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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• 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.
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• 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;
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• 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
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• 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),
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• 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
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• 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
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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 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})
  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)
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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^{
  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
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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 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}
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• 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
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• 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 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
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• 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
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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);
  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
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• 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 (498 words) - 00:46, 4 August 2023
• 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
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• 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
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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
• 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
• 2006 AMC 12A Problems/Problem 12
  ...<math>1</math> cm less than that of the ring above it. The bottom ring has an outside diameter of <math>3</math> cm. What is the distance, in cm, from th <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 19
  <math>\textbf{(A) } 0\qquad\textbf{(B) } 1\qquad\textbf{(C) } 59\qquad\textbf{(D) } 89\qquad ...a triangle is <math>180</math> degrees. For an arithmetic progression with an odd number of terms, the middle term is equal to the average of the sum of
  2 KB (259 words) - 03:10, 22 June 2023
• 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
• 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>. Prove that there exist integers <math>m</m 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
• 2022 USAJMO Problems/Problem 1
  ...n infinite arithmetic sequence of integers <math>a_1,a_2,\cdots</math> and an infinite geometric sequence of integers <math>g_1,g_2,\cdots</math> satisfy ...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>.
  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
• 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 (263 words) - 12:29, 30 December 2023
• 2005 AIME II Problems/Problem 1
  ...uses a deck of <math> n </math> different cards, where <math> n </math> is an integer and <math> n \geq 6. </math> The number of possible sets of 6 cards 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 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
  ...hat the only such factor is 1, so we must add 1 to our previous sum to get an answer of <math>121 + 64 + 276 - 8 - 8 - 11 + 1 = \boxed{435}</math>. Assume that <math>n</math> (a positive integer) is a divisor of one of the numbers. Therefore, <math>n</math> can be expressed a
  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 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
  ...uses a deck of <math> n </math> different cards, where <math> n </math> is an integer and <math> n \geq 6. </math> The number of possible sets of 6 cards ...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 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 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
• Problem solving
  ...but perhaps they were missing something. By the end of MOP, I had learned a somewhat unsettling truth. The others knew fewer tricks than I did, not mor ...have learned also that the formula sheets were fool’s gold – he became a Rhodes scholar.)
  6 KB (1,039 words) - 17:43, 30 July 2018
• Line
  ...lope]] of the line and ''b'' is the [[y-intercept]]. Any two points define a line, and given specific <math>(x_1,y_1)</math> <math>(x_2,y_2)</math> on
  674 bytes (106 words) - 18:40, 9 May 2024
• Research Science Institute
  The '''Research Science Institute''' (RSI) is a 6-week residential program held each summer at the [[Massachusetts Institut ...h Science Institute in 1987, following Admiral Rickover's death, and after a nomadic existence in the Washington, DC area, RSI moved permanently to MIT
  2 KB (284 words) - 16:37, 6 July 2023
• 2005 AIME I Problems
  ...ngent to two circles adjacent to it. All circles are internally tangent to a circle <math> C </math> with radius 30. Let <math> K </math> be the area of ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 3
  Suppose <math>n</math> is such an [[integer]]. Because <math>n</math> has <math>3</math> proper divisors, it ...h>47</math>) so there are <math> {15 \choose 2} =105</math> ways to choose a pair of primes from the list and thus <math>105</math> numbers of the first
  2 KB (249 words) - 09:37, 23 January 2024
• 2005 AIME I Problems/Problem 4
  ...members left over. The director realizes that if he arranges the group in a formation with 7 more rows than columns, there are no members left over. Fi ...the number of students is <math>n(n + 7)</math> which must be 5 more than a perfect square, so <math>n \leq 14</math>. In fact, when <math>n = 14</mat
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  ...e, but not on the other. He wants to stack the eight coins on a table into a single stack so that no two adjacent coins are face to face. Find the numbe ...since after the first H no more tails can appear. The first H can occur in a maximum of eight times different positions, and then there is also the poss
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 7
  ...rilateral]] <math> ABCD,\ BC=8,\ CD=12,\ AD=10, </math> and <math> m\angle A= m\angle B = 60^\circ. </math> Given that <math> AB = p + \sqrt{q}, </math> label("$A$",(0,0),SW);
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 9
  ...> a,b, </math> and <math> c </math> are [[positive integer]]s, find <math> a+b+c+p+q+r. </math> ...}{6} = \frac{2}{3}</math> of all orientations, so from these cubes we gain a factor of <math>\left(\frac{2}{3}\right)^6</math>.
  4 KB (600 words) - 21:44, 20 November 2023
• 2005 AIME I Problems/Problem 10
  ...</math> are <math> (p,q). </math> The [[line]] containing the [[median of a triangle | median]] to side <math> BC </math> has [[slope]] <math> -5. </ma pair A=(15,32), B=(12,19), C=(23,20), M=B/2+C/2, P=(17,22);
  5 KB (852 words) - 21:23, 4 October 2023
• 2005 AIME I Problems/Problem 11
  A [[semicircle]] with [[diameter]] <math> d </math> is contained in a [[square (geometry) | square]] whose sides have length 8. Given the maximum We note that aligning the base of the semicircle with a side of the square is certainly non-optimal. If the semicircle is tangent
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME I Problems/Problem 12
  ...005 </math> with <math> S(n) </math> [[even integer | even]]. Find <math> |a-b|. </math> ...<math>n</math>.) Thus, <math>S(n)</math> is odd if and only if there are an odd number of perfect squares less than <math>n</math>. So <math>S(1), S(2
  4 KB (647 words) - 02:29, 4 May 2021
• 2005 AIME I Problems/Problem 15
  ...math> is not [[divisor | divisible]] by the [[perfect square | square]] of a prime, find <math> m+n. </math> pair A=(0,0),B=(26,0),C=IP(circle(A,10),circle(B,20)),D=(B+C)/2,I=incenter(A,B,C);
  5 KB (906 words) - 23:15, 6 January 2024
• 2005 AIME II Problems/Problem 15
  ...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma ...points that can be the center of the circle with the desired properties is an [[ellipse]].
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2005 AIME II Problems/Problem 14
  pair A = origin; pair C = rotate(15,A)*(A+dir(-50));
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 15
  and define a [[sequence]] as follows: <math>x_1=x</math> and <math>x_{n+1}=f(x_n)</math> ...>9</math> operations; however, not all <math>9</math> of them may be <math>A: (-1)</math> otherwise we return back to <math>x_{10} = 1</math>, contradic
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  ...are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> triple Oxy = (0,0,0), A=(4*5^.5,-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h);
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 2
  ...The absolute value of the difference between the greatest element of <math>A</math> and the greatest element of <math>B</math> is <math>99</math>. Find ...e median) must be <math>2</math>. Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME I Problems/Problem 3
  ...gular and <math>12</math> of which are quadrilaterals. A space diagonal is a line segment connecting two non-adjacent vertices that do not belong to the ...f the [[polyhedron]] determines either an [[edge]], a face [[diagonal]] or a space diagonal. We have <math>{26 \choose 2} = \frac{26\cdot25}2 = 325</ma
  1 KB (156 words) - 17:56, 1 January 2016
• 2004 AIME I Problems/Problem 4
  ...re. The [[midpoint]]s of the line segments in set <math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k</math>. Find <ma ...midpoints lying on the sides determined by vertex <math>(0,0)</math> form a quarter-[[circle]] with [[radius]] 1.
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 6
  An integer is called snakelike if its decimal representation <math> a_1a_2a_3\ ...h>x_2x_40x_3</math>, <math>x_3x_40x_2</math>. Because we know that zero is a digit, there are <math>3\cdot{9\choose 3}=252</math> snakelike numbers whic
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME I Problems/Problem 7
  ...from <math>N</math> and the other is from <math>P</math>. Each summand is a term in the expansion of ...ed{588}</math>. This is a good solution because it guarantees we never add a product twice, and the pattern is simple to add by hand.
  5 KB (833 words) - 19:43, 1 October 2023
• 2004 AIME I Problems/Problem 8
  Define a regular <math> n </math>-pointed star to be the union of <math> n </math> l ...s intersects at least one of the other line segments at a point other than an endpoint,
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  The digits of a positive integer <math> n </math> are four consecutive integers in decreasi ...he absolute value of the difference between the greatest element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m.
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 15
  ...eated <math>8</math> more times. After the last fold, the strip has become a stack of <math>1024</math> unit squares. How many of these squares lie belo ...number of squares below the <math>n</math> square after the final fold in a strip of length <math>2^{k}</math>.
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  ...cdots77, </math> into which <math> + </math> signs are inserted to produce an arithmetic [[expression]]. For example, <math> 7+77+777+7+7=875 </math> cou ...</math>. Then the question is asking for the number of values of <math>n = a + 2b + 3c</math>.
  11 KB (1,857 words) - 21:55, 19 June 2023
• 2004 AIME II Problems/Problem 12
  ...and circles of radius 2 centered at <math> C </math> and <math> D. </math> A circle contained within the trapezoid is [[tangent]] to all four of these c pair A=(0,0), B=(6,0), D=(1, 24^.5), C=(5,D.y), O = (3,(r^2 + 6*r)^.5);
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 10
  ...{40} </math> whose binary expansions have exactly two <math>1</math>'s. If a number is chosen at random from <math> S, </math> the [[probability]] that ...again, so if we raise <math>2^{3}</math>, which we know already works, to an odd power, we will also satisfy the congruence. Thus, <math>2^{3}, 2^{9}, 2
  8 KB (1,283 words) - 19:19, 8 May 2024
• 2004 AIME II Problems/Problem 7
  <math> ABCD </math> is a rectangular sheet of paper that has been folded so that corner <math> B </m pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect
  9 KB (1,501 words) - 05:34, 30 October 2023
• 2004 AIME II Problems/Problem 6
  ...es the rest equally between the other two. Given that each monkey receives a [[whole number]] of bananas whenever the bananas are divided, and the numbe ...> is divisible by <math>72</math> (however, since the denominator contains a <math>27</math>, the factors of <math>3</math> cancel, and it only really n
  6 KB (950 words) - 14:18, 15 January 2024
• 2004 AIME II Problems/Problem 5
  ...off, so the second quarter of the work was completed behind schedule. Then an additional <math>100</math> workers were laid off, so the third quarter of A train is traveling at <math>1000</math> miles per hour and has one hour to
  4 KB (592 words) - 19:02, 26 September 2020
• 2004 AIME II Problems/Problem 4
  First, let's count numbers with only a single digit. We have nine of these for each length, and four lengths, so ...2^n - 2</math> ways to arrange them in an <math>n</math>-digit number, for a total of <math>(2^1 - 2) + (2^2 - 2) + (2^3 -2) + (2^4 - 2) = 22</math> suc
  3 KB (508 words) - 01:16, 19 January 2024
• 2004 AIME II Problems/Problem 3
  A solid rectangular block is formed by gluing together <math> N </math> [[con ...e [[prime factorization]] of <math>231 = 3\cdot7\cdot11</math>, so we have a variety of possibilities; for instance, <math>l - 1 = 1</math> and <math>m
  2 KB (377 words) - 11:53, 10 March 2014
• 2004 AIME II Problems
  ...a, b, c, d, e, </math> and <math> f </math> are positive integers, <math> a </math> and <math> e </math> are relatively prime, and neither <math> c </m A jar has 10 red candies and 10 blue candies. Terry picks two candies at rand
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1984 AIME Problems
  ...8}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ldo A point <math>P</math> is chosen in the interior of <math>\triangle ABC</math
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  ...re added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of <math>1986_{}^{}</math>. What was the number of the page t ...ath> parallel to the sides of the triangle. If these three segments are of an equal length <math>d</math>, find <math>d</math>.
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  An ordered pair <math>(m,n)</math> of non-negative integers is called "simple" ...er we mean a positive integral divisor other than 1 and the number itself. A natural number greater than 1 will be called "nice" if it is equal to the p
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...ability that a randomly chosen positive divisor of <math>10^{99}</math> is an integer multiple of <math>10^{88}</math>. Find <math>m + n</math>. ...C</math>, <math>\tan \angle CAB = 22/7</math>, and the altitude from <math>A</math> divides <math>BC</math> into segments of length <math>3</math> and <
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  Ten points are marked on a circle. How many distinct convex polygons of three or more sides can be dra ...ose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1990 AIME Problems
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. ...}</math> be a regular <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea
  6 KB (870 words) - 10:14, 19 June 2021
• 1991 AIME Problems
  Given a rational number, write it as a fraction in lowest terms and calculate the product of the resulting numerat Suppose <math>r^{}_{}</math> is a real number for which
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  A positive integer is called ascending if, in its decimal representation, the ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, s
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  ...n for office, a candidate made a tour of a country which we assume lies in a plane. On the first day of the tour he went east, on the second day he went :(a) the winner caught <math>15</math> fish;
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  ...\,</math> consists of those positive multiples of 3 that are one less than a perfect square. What is the remainder when the 1994th term of the sequence ...circle of radius 20. Square <math>ABCD\,</math> is constructed with <math>A\,</math> and <math>B\,</math> on the larger circle, <math>\overline{CD}\,</
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1995 AIME Problems
  Starting at <math>(0,0),</math> an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or do ...us <math>9</math>. The circle of radius <math>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the le
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1996 AIME Problems
  ...mn, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...at <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
  6 KB (931 words) - 17:49, 21 December 2018
• 1997 AIME Problems
  The nine horizontal and nine vertical lines on an <math>8\times8</math> checkerboard form <math>r</math> rectangles, of which ...he two-digit number to the left of the three-digit number, thereby forming a five-digit number. This number is exactly nine times the product Sarah sho
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1998 AIME Problems
  ...ums those three values. The [[probability]] that all three players obtain an [[odd]] sum is <math>m/n,</math> where <math>m</math> and <math>n</math> ar ...allelogram]]. Extend <math>\overline{DA}</math> through <math>A</math> to a point <math>P,</math> and let <math>\overline{PC}</math> meet <math>\overli
  7 KB (1,084 words) - 02:01, 28 November 2023
• 1999 AIME Problems
  Find the smallest prime that is the fifth term of an increasing arithmetic sequence, all four preceding terms also being prime. ...> <math>(10,114),</math> <math>(28,153),</math> and <math>(28,84).</math> A line through the origin cuts this figure into two congruent polygons. The
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2001 AIME I Problems
  A finite set <math>\mathcal{S}</math> of distinct real numbers has the follow ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>.
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  ...plate will contain at least one palindrome (a three-letter arrangement or a three-digit arrangement that reads the same left-to-right as it does right- ...gram shows twenty congruent circles arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  ...centric circles with radii <math> 1, 2, 3, \dots, 100 </math> are drawn in a plane. The interior of the circle of radius 1 is colored red, and each regi ...et <math> \mathcal{S} = \{8, 5, 1, 13, 34, 3, 21, 2\}. </math> Susan makes a list as follows: for each two-element subset of <math> \mathcal{S}, </math>
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math>x^2 - y^ ...h>m/n</math> be the probability that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are relatively prime positive
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...rom left to right, each pair of consecutive digits of <math>N</math> forms a perfect square. What are the leftmost three digits of <math>N</math>? Each of the 2001 students at a high school studies either Spanish or French, and some study both. The numb
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R= ...th>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c</math>.
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...<math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many seven-letter good words are there? In a regular tetrahedron, the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 2
  ...ven interval <math>p \leq x \leq 15</math>) when <math>x=15</math>, giving a minimum of <math>\boxed{015}</math>. ...equal to <math>15 - \varepsilon</math>, where <math>\varepsilon</math> is an almost neglectable value. Because of the small value <math>\varepsilon</mat
  1 KB (184 words) - 20:16, 14 January 2023
• 1983 AIME Problems/Problem 5
  Because we want the largest possible <math>w</math>, let's find an expression for <math>z</math> in terms of <math>w</math>. An alternate way to solve this is to let <math>x=a+bi</math> and <math>y=c+di</math>.
  4 KB (672 words) - 10:17, 17 March 2023
• 1983 AIME Problems/Problem 7
  ...three had been sitting next to each other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? We will place <math>A</math>, <math>B</math>, and <math>C</math> with and without the restriction
  9 KB (1,392 words) - 20:37, 19 January 2024
• 1983 AIME Problems/Problem 9
  We can rewrite the numerator to be a perfect square by adding <math>-\dfrac{12x \sin x}{x \sin x}</math>. Thus, ...\frac{y\pm\sqrt{y^2-144}}{18}</math>. We know that <math>y</math> must be an integer and as small as it can be, so <math>y</math> = 12. We plug this bac
  4 KB (722 words) - 20:25, 14 January 2023
• 1983 AIME Problems/Problem 11
  The solid shown has a square base of side length <math>s</math>. The upper edge is parallel to th triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
  5 KB (865 words) - 21:11, 6 February 2023
• 1983 AIME Problems/Problem 13
  ...>For <math>\{1, 2, 3, \ldots, n\}</math> and each of its non-empty subsets a unique '''alternating sum''' is defined as follows. Arrange the numbers in Let <math>S</math> be a non-[[empty set | empty]] [[subset]] of <math>\{1,2,3,4,5,6\}</math>.
  5 KB (894 words) - 22:02, 5 April 2024
• 1983 AIME Problems/Problem 14
  ...<math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> have equal length. ...ath>C</math>, and with radius <math>6</math>) that intersects circle <math>A</math> at <math>Q</math>. The rest is just finding lengths, as follows.
  13 KB (2,149 words) - 18:44, 5 February 2024
• 1983 AIME Problems/Problem 15
  ...r arc <math>AB</math> is a rational number. If this number is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <mat pair A=(-0.91,-0.41);
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 1
  ...8}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an [[arithmetic progression]] with common difference 1, and <math>a_1+a_2+a_3+ One approach to this problem is to apply the formula for the sum of an [[arithmetic series]] in order to find the value of <math>a_1</math>, then
  4 KB (576 words) - 21:03, 23 December 2023
• 1984 AIME Problems/Problem 6
  ...t <math>(14, 92)</math>, <math>(17, 76)</math>, and <math>(19, 84)</math>. A [[line]] passing through <math>(17,76)</math> is such that the total area o ...t <math>M</math>. If we draw the feet of the [[perpendicular]]s from <math>A,C</math> to the line (call <math>E,F</math>), we see that <math>\triangle A
  6 KB (1,022 words) - 19:29, 22 January 2024
• 1984 AIME Problems/Problem 9
  ...math> is <math>12 \mbox { cm}^2</math>. These two faces meet each other at a <math>30^\circ</math> angle. Find the [[volume]] of the tetrahedron in <mat path3 rightanglemark(triple A, triple B, triple C, real s=8)
  6 KB (947 words) - 20:44, 26 November 2021
• 1984 AIME Problems/Problem 12
  ...<math>f(7+x)=f(7-x)</math> for all <math>x</math>. If <math>x=0</math> is a root for <math>f(x)=0</math>, what is the least number of roots <math>f(x)= Since <math>0</math> is a root, all multiples of <math>10</math> are roots, and anything congruent to
  3 KB (588 words) - 14:37, 22 July 2020
• 1984 AIME Problems/Problem 14
  Take an even positive integer <math>x</math>. <math>x</math> is either <math>0 \bmo Clearly, if <math>x \ge 44</math>, it can be expressed as a sum of 2 odd composites. However, if <math>x = 42</math>, it can also be ex
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1984 AIME Problems/Problem 15
  ...4,16,36,64</math>, we can express the difference of the two polynomials by a quartic polynomial that has roots at <math>t=4,16,36,64</math>, so ...hly recommend never following this unless you have no idea what to do with an hour of your time. Even so, learning the above solutions will be more benef
  6 KB (1,051 words) - 04:52, 8 May 2024
• 1985 AIME Problems/Problem 15
  ...| regular]] [[hexagon]], as shown in the second figure, so as to fold into a [[polyhedron]]. What is the [[volume]] (in <math>\mathrm{cm}^3</math>) of t ...ng a hexagonal face (rotated <math>60^\circ</math> from each other) yields a [[cube (geometry) | cube]], so the volume is <math>\frac12 \cdot 12^3 = 864
  2 KB (245 words) - 22:44, 4 March 2024
• 1985 AIME Problems/Problem 14
  ...h of the two players earned <math>\frac{1}{2}</math> point if the game was a tie. After the completion of the tournament, it was found that exactly half ...verage of <math>n - 1</math>, while the bottom 10 got 90 points total, for an average of 9. Thus we must have <math>n > 10</math>, so <math>n = 15</math
  5 KB (772 words) - 22:14, 18 June 2020
• 1985 AIME Problems/Problem 12
  ...= \frac{n}{729}</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of ...> let <math>P(k)</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>k</math> meters. We wish to find <
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 11
  An [[ellipse]] has [[focus | foci]] at <math>(9,20)</math> and <math>(49,55)</ An ellipse is defined to be the [[locus]] of points <math>P</math> such that t
  5 KB (932 words) - 17:00, 1 September 2020
• 1985 AIME Problems/Problem 10
  where <math>x</math> is a [[real number]], and <math>\lfloor z \rfloor</math> denotes the greatest [[ ...oor</math>, then <math>k=20a+b</math> for some non-negative integers <math>a</math>, <math>b</math>, where there are <math>12</math> values for <math>b<
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 9
  ...alpha</math>, which is a [[positive]] [[rational number]], is expressed as a [[fraction]] in lowest terms, what is the sum of its numerator and denomina real r = 8/15^0.5, a = 57.91, b = 93.135;
  5 KB (763 words) - 16:20, 28 September 2019
• 1985 AIME Problems/Problem 8
  ...<math>a_7 = 2.86</math>. It is desired to replace each <math>a_i</math> by an [[integer]] approximation <math>A_i</math>, <math>1\le i \le 7</math>, so t ...6,A_7)</math> with the smallest <math>M</math>, and the rest would just be a piece of cake.
  2 KB (377 words) - 02:17, 16 February 2021
• 1986 AIME Problems/Problem 15
  ...<math>AB</math> is <math>60</math>, and that the [[median]]s through <math>A</math> and <math>B</math> lie along the lines <math>y=x+3</math> and <math> Let <math>\theta_1</math> be the angle that the median through <math>A</math> makes with the positive <math>x</math>-axis, and let <math>\theta_2<
  11 KB (1,722 words) - 09:49, 13 September 2023
• 1986 AIME Problems/Problem 14
  The shortest distances between an interior [[diagonal]] of a rectangular [[parallelepiped]], <math>P</math>, and the edges it does not m We invert the above equations to get a system of linear equations in <math>\frac {1}{h^2}</math>, <math>\frac {1}{
  2 KB (346 words) - 13:13, 22 July 2020
• 1986 AIME Problems/Problem 13
  ...tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by <tt>TH</tt>, <tt>HH</tt>, and etc. For ex ...tches to <tt>H</tt> four times; hence it follows that our string will have a structure of <tt>THTHTHTH</tt>.
  4 KB (772 words) - 21:09, 7 May 2024
• Analytic continuation
  ...</math>. We can now define it as the analytic continuation and treat it as an extension of the original function, so in this example, we might find that
  1 KB (180 words) - 20:12, 19 August 2015
• 1986 AIME Problems/Problem 11
  ...h>\delta_{i, j}</math> by using combinatorial identities, which is left as an exercise for the reader. Thus, since the two matrices multiply to form <mat {{AIME box|year=1986|num-b=10|num-a=12}}
  6 KB (872 words) - 16:51, 9 June 2023
• 1986 AIME Problems/Problem 9
  ...arallel]] to the sides of the [[triangle]]. If these three segments are of an equal length <math>d</math>, find <math>d</math>. pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425));
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1986 AIME Problems/Problem 6
  ...re added, one of the page numbers was mistakenly added twice, resulting in an incorrect sum of <math>1986_{}^{}</math>. What was the number of the page t ...}</math>, plus the additional page number <math>k</math>. We can establish an upper and lower bound for the number of pages contained in the book, where
  2 KB (336 words) - 14:13, 6 September 2020
• 1987 AIME Problems/Problem 12
  ...ath>, where <math>n</math> is a [[positive integer]] and <math>r</math> is a [[positive]] [[real number]] less than <math>1/1000</math>. Find <math>n</m ...ess than <math>\frac{1}{1000}</math>. However, we still have to make sure a sufficiently small <math>r</math> exists.
  4 KB (673 words) - 19:48, 28 December 2023
• 1987 AIME Problems/Problem 10
  ...times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.) ...math>x - 75</math> steps. Since Bob and the escalator were both moving at a constant speed over the time it took Bob to climb, the [[ratio]] of their d
  7 KB (1,187 words) - 16:21, 27 January 2024
• 1987 AIME Problems/Problem 4
  ...60,15)</math>, <math>(48,0)</math>, <math>(80,0)</math>. This triangle has an area of <math>(80-48)*15*0.5=240</math>. {{AIME box|year=1987|num-b=3|num-a=5}}
  2 KB (371 words) - 17:25, 13 February 2024
• 1987 AIME Problems/Problem 1
  An [[ordered pair]] <math>(m,n)</math> of [[non-negative]] [[integer]]s is cal ...an integer <math>a</math> is <math>(a + 1)</math> because there are <math>(a - 1)</math> positive integers that are less than it, in addition to <math>0
  1 KB (191 words) - 14:42, 17 September 2016
• 1988 AIME Problems/Problem 15
  In an office at various times during the day, the boss gives the secretary a letter to type, each time putting the letter on top of the pile in the secr While leaving for lunch, the secretary tells a colleague that letter <math>8</math> has already been typed but says nothin
  7 KB (1,186 words) - 10:16, 4 June 2023
• 1988 AIME Problems/Problem 13
  ...ath> and <math>b</math> are integers such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>. 0 = ax^{17} + bx^{16} + 1 = a(F_{17}\cdot x + F_{16}) + b(F_{16}\cdot x + F_{15}) + 1 &\Longrightarrow (a
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1988 AIME Problems/Problem 12
  ...ments indicated in the figure. Find the product <math>abc</math> if <math>a + b + c = 43</math> and <math>d = 3</math>. <math>\frac {d}{a + d} = \frac {[PBC]}{[ABC]}</math>
  4 KB (727 words) - 23:37, 7 March 2024
• 1988 AIME Problems/Problem 10
  ...ron lie in the interior of the polyhedron rather than along an [[edge]] or a [[face]]? The polyhedron described looks like this, a truncated cuboctahedron.
  6 KB (902 words) - 17:40, 19 May 2024
• 1988 AIME Problems/Problem 9
  A little bit of checking tells us that the units digit must be 2. Now our cub ...value for the hundreds digit is <math>4</math>, and so <math>442</math> is a valid solution.
  6 KB (893 words) - 08:15, 2 February 2023
• 1988 AIME Problems/Problem 8
  ...for the first one, we know that <math>f(x,x) = x</math> in order to obtain an integer answer. So, we have to transform <math>f(14,52)</math> to this form Hence, <math>f(x,y) = \mathrm{lcm}(x,y)</math> is a solution to the functional equation.
  4 KB (538 words) - 13:24, 12 October 2021
• 1988 AIME Problems/Problem 6
  ...od will be to use the given numbers to set up equations to solve for <math>a</math> and <math>b</math>, and then calculate <math>(*)</math>. \hline a & & 103 & & \\
  5 KB (878 words) - 23:06, 20 November 2023
• 1988 AIME Problems/Problem 5
  ...y]] that a randomly chosen positive [[divisor]] of <math>10^{99}</math> is an integer multiple of <math>10^{88}</math>. Find <math>m + n</math>. ...99}</math> which are divisible by <math>10^{88}</math>; it is easy to draw a [[bijection]] to the number of factors that <math>10^{11} = 2^{11}5^{11}</m
  822 bytes (108 words) - 22:21, 6 November 2016
• 1988 AIME Problems/Problem 4
  So <math>n \ge 20</math>. We now just need to find an example where <math>n = 20</math>: suppose <math>x_{2k-1} = \frac{19}{20}</ ...> grow by approximately <math>2</math> every time we choose a negative and a positive, so if we have <math>20</math> numbers, namely <math>10</math> pos
  2 KB (394 words) - 10:21, 27 January 2024
• 1989 AIME Problems/Problem 15
  ...th>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math ...ath>YPL</math> have their areas in ratio <math>XP:PL</math> (as they share a common height from <math>Y</math>), and the same is true of triangles <math
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 13
  Let <math>S</math> be a [[subset]] of <math>\{1,2,3,\ldots,1989\}</math> such that no two members o ...6,10]</math>. Now, <math>1989 = 180\cdot 11 + 9</math>. Because this isn't an exact multiple of <math>11</math>, we need to consider some numbers separat
  2 KB (274 words) - 04:07, 17 December 2023
• 1989 AIME Problems/Problem 11
  ...ach between 1 and 1000 inclusive, with repetitions allowed. The sample has a unique [[mode]] (most frequent value). Let <math>D</math> be the difference ...h>n-1 \approx 3</math>. Substituting this result and some arithmetic gives an answer of <math>\boxed{947}</math>.
  5 KB (851 words) - 18:01, 28 December 2022
• 1989 AIME Problems/Problem 10
  ...h>\beta</math>, <math>\gamma</math>, be the angles opposite them. If <math>a^2+b^2=1989c^2</math>, find pair A = (0,0), B = (3, 0), C = (1, 4);
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1989 AIME Problems/Problem 9
  ...d in the 1960s by three American mathematicians when they showed there was a positive integer such that <cmath>133^5+110^5+84^5+27^5=n^{5}.</cmath> Find which arrives at a contradiction. Therefore, we conclude that <math>n=\boxed{144}.</math>
  6 KB (874 words) - 15:50, 20 January 2024
• 1989 AIME Problems/Problem 7
  ...nd <math>596</math>, one obtains the squares of three consecutive terms of an arithmetic series. Find <math>k</math>. ...a + d,\ a + 2d</math>, making their squares <math>a^2,\ a^2 + 2ad + d^2,\ a^2 + 4ad + 4d^2</math>.
  2 KB (320 words) - 15:21, 14 May 2024
• 1989 AIME Problems/Problem 6
  ...the same time Allie leaves <math>A</math>, Billie leaves <math>B</math> at a speed of <math>7</math> meters per second and follows the [[straight]] path pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  6 KB (980 words) - 15:08, 14 May 2024
• 1989 AIME Problems/Problem 4
  ...t <math>b+c+d</math> is a [[perfect square]] and <math>a+b+c+d+e</math> is a [[perfect cube]], what is the smallest possible value of <math>c</math>? ...re is a cubed term in <math>5^2 \cdot y^3</math>, <math>3^3</math> must be a factor of <math>c</math>. <math>3^35^2 = \boxed{675}</math>, which works as
  3 KB (552 words) - 12:41, 3 March 2024
• 1989 AIME Problems/Problem 3
  Suppose <math>n</math> is a [[positive integer]] and <math>d</math> is a single [[digit]] in [[base 10]]. Find <math>n</math> if ...present [[rational number]]s. To figure out which rational number, we sum an [[infinite]] [[geometric series]], <math>0.d25d25d25\ldots = \sum_{n = 1}^\
  3 KB (499 words) - 22:17, 29 March 2024
• 1989 AIME Problems/Problem 1
  ...have that <math>a=868</math> as per the equation <math>(a+1)^2 = a \cdot (a+2) +1</math>. ...prove that one more than the product of four consecutive integers must be a perfect square:
  4 KB (523 words) - 00:12, 8 October 2021
• 1990 AIME Problems/Problem 15
  Find <math>ax^5 + by^5</math> if the real numbers <math>a,b,x,</math> and <math>y</math> satisfy the equations ...he values of the starting term that make the sequence geometric, and <math>a,b</math> are the appropriately chosen constants such that those special sta
  4 KB (644 words) - 16:24, 28 May 2023
• 1990 AIME Problems/Problem 14
  ...ments <math>\overline{CP}</math> and <math>\overline{DP}</math>, we obtain a [[triangular pyramid]], all four of whose faces are [[isosceles triangle]]s pair D=origin, A=(13,0), B=(13,12), C=(0,12), P=(6.5, 6);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1990 AIME Problems/Problem 13
  Let <math>T = \{9^k : k ~ \mbox{is an integer}, 0 \le k \le 4000\}</math>. Given that <math>9^{4000}_{}</math> ha ...tive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 12
  ...^{}_{}</math>, and <math>d^{}_{}</math> are positive integers. Find <math>a + b + c + d^{}_{}</math>. ...\cdot 12\sin 45 = 12\sqrt{2}</math> (or notice that the triangle formed is a <math>45 - 45 - 90</math> [[right triangle]]).
  6 KB (906 words) - 13:23, 5 September 2021
• 1990 AIME Problems/Problem 7
  .../math> can be written in the form <math>ax+2y+c=0_{}^{}</math>. Find <math>a+c_{}^{}</math>. ...at it has lengths of side <math>15,\ 20,\ 25</math>, indicating that it is a <math>3-4-5</math> [[right triangle]]. At this point, we just need to find
  8 KB (1,319 words) - 11:34, 22 November 2023
• 1990 AIME Problems/Problem 3
  ...lar polygon|regular]] <math>r~\mbox{gon}</math> and <math>P_2^{}</math> be a regular <math>s~\mbox{gon}</math> <math>(r\geq s\geq 3)</math> such that ea The formula for the interior angle of a regular sided [[polygon]] is <math>\frac{(n-2)180}{n}</math>.
  3 KB (516 words) - 19:18, 16 April 2024
• 1990 AIME Problems/Problem 1
  ...neither the [[perfect square | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. We need <math>n - T = 500</math>, where <math>n</math> is an integer greater than 500 and <math>T</math> is the set of numbers which con
  2 KB (283 words) - 23:11, 25 June 2023
• 2006 AMC 10B Problems/Problem 1
  <math> \textbf{(A)} -2006\qquad \textbf{(B)} -1\qquad \textbf{(C) } 0\qquad \textbf{(D) } 1\q ...raised to an odd integer is <math>-1</math> and <math>-1</math> raised to an even integer exponent is <math>1</math>:
  606 bytes (80 words) - 10:32, 19 December 2021
• 1991 AIME Problems/Problem 15
  ...ositive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> is also an integer. Find this <math>n^{}_{}</math>. If this is an integer, we can write <math>17^2 + n^4 = m^2</math>, for an integer <math>m</math>. Thus, <math>(m - n^2)(m + n^2) = 289\cdot 1 = 17\cd
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 13
  ...s that, when two socks are selected randomly without replacement, there is a probability of exactly <math>\frac{1}{2}</math> that both are red or both a ...math>. Then by complementary counting, the number of ways to get a red and a blue sock must be equal to <math>1-\frac12=\frac12=\frac{2rb}{(r+b)(r+b-1)}
  7 KB (1,328 words) - 20:24, 5 February 2024
• 2006 AMC 10B Problems/Problem 15
  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 (445 words) - 22:01, 20 August 2022
• 1991 AIME Problems/Problem 12
  ...--B--C--D--cycle);draw(P--R..Q--S); draw(P--Q--R--S--cycle); label("\(A\)",A,NW);label("\(B\)",B,NE);label("\(C\)",C,SE);label("\(D\)",D,SW); label("\(P ...bus; <math>PQ = \sqrt{15^2 + 20^2} = 25</math>. Since the [[diagonal]]s of a rhombus are [[perpendicular bisector]]s, we have that <math>OP = 15, OQ = 2
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...ath>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. for (int a=1; a<24; a+=2)
  4 KB (740 words) - 19:33, 28 December 2022
• 1991 AIME Problems/Problem 10
  ...ath>S_b^{}</math> in alphabetical order. When <math>p</math> is written as a [[fraction]] in [[irreducible fraction|lowest terms]], what is its [[numera Let us make a chart of values in alphabetical order, where <math>P_a,\ P_b</math> are the
  5 KB (813 words) - 06:10, 25 February 2024
• 2006 AMC 10B Problems/Problem 17
  Bob and Alice each have a bag that contains one ball of each of the colors blue, green, orange, red, <math> \textbf{(A) } \frac{1}{10}\qquad \textbf{(B) } \frac{1}{6}\qquad \textbf{(C) } \frac{1
  1 KB (211 words) - 04:32, 4 November 2022
• 1991 AIME Problems/Problem 9
  Substituting <math>\cot x = \frac{1}{\tan x} = \frac{308}{435}</math> yields a [[quadratic equation]]: <math>0 = 435y^2 - 616y - 435 = (15y - 29)(29y + 15 Make the substitution <math>u = \tan \frac x2</math> (a substitution commonly used in calculus). By the half-angle identity for tan
  10 KB (1,590 words) - 14:04, 20 January 2023
• 1991 AIME Problems/Problem 8
  For how many real numbers <math>a</math> does the [[quadratic equation]] <math>x^2 + ax + 6a=0</math> have on ...ratic, and since <math>x_1,x_2</math> are integers, <math>a</math> must be an integer. Applying the [[quadratic formula]],
  2 KB (310 words) - 11:25, 13 June 2023
• 1991 AIME Problems/Problem 2
  pair A=(0,0),B=(4,0),C=(4,3),D=(0,3); D(A--B--C--D--cycle);
  4 KB (595 words) - 12:51, 17 June 2021
• 1991 AIME Problems/Problem 1
  ...5) = 0</math>. Either <math>a = 16</math> and <math>b = 55</math> or <math>a = 55</math> and <math>b = 16</math>. For the first case, it is easy to see ...to <math>(x + y)xy = 880 = 2^4 \cdot 5 \cdot 11</math>. The only set with a factor of <math>11</math> is <math>(5,11)</math>, and checking shows that i
  4 KB (628 words) - 22:05, 7 June 2021
• 1992 AIME Problems/Problem 12
  ...rnately take bites from a 5-by-7 grid of [[unit square]]s. To take a bite, a player chooses one of the remaining [[square (geometry) | squares]], then r ...and 7 that go across, with the shape on the right "carved" out by the path a possible subset.
  2 KB (443 words) - 22:41, 22 December 2021
• 1992 AIME Problems/Problem 11
  Let <math>l</math> be a line that makes an angle of <math>\theta</math> with the positive <math>x</math>-axis. Let <ma ...also be <math>\theta - \frac{\pi}{70}</math>. Thus, <math>l'</math> makes an angle of <math>\frac{\pi}{70}-\left(\theta-\frac{\pi}{70}\right) = \frac{\p
  2 KB (404 words) - 19:24, 4 July 2013
• 1992 AIME Problems/Problem 10
  ...1</math>, inclusive. What is the integer that is nearest the area of <math>A</math>? ...q 1</math> we have the inequality <cmath>0\leq a,b \leq 40</cmath>which is a square of side length <math>40</math>.
  2 KB (323 words) - 12:05, 16 July 2019
• 1992 AIME Problems/Problem 7
  ...and <math>BCD^{}_{}</math> of tetrahedron <math>ABCD^{}_{}</math> meet at an angle of <math>30^\circ</math>. The area of face <math>ABC^{}_{}</math> is {{AIME box|year=1992|num-b=6|num-a=8}}
  800 bytes (114 words) - 17:40, 14 March 2017
• 1992 AIME Problems/Problem 5
  ...rm <math>0.abcabcabc\ldots=0.\overline{abc}</math>, where the digits <math>a^{}_{}</math>, <math>b^{}_{}</math>, and <math>c^{}_{}</math> are not necess ...ethod in which repeating decimals are normally converted to fractions with an example:
  2 KB (277 words) - 20:45, 4 March 2024
• 1992 AIME Problems/Problem 3
  ...she has won by the total number of matches she has played. At the start of a weekend, her win ratio is exactly <math>.500</math>. During the weekend, sh ...>n < \frac{988}{6} = 164.\overline{6}.</math> Since <math>n</math> must be an integer, the largest possible value for <math>n</math> is <math>\boxed{164}
  2 KB (251 words) - 08:05, 2 January 2024
• 1992 AIME Problems/Problem 2
  A [[positive integer]] is called ascending if, in its [[decimal representatio ..., since the only position for 0 is at the leftmost end of the number, i.e. a leading 0), there is exactly one ascending number with those digits.
  2 KB (336 words) - 05:18, 4 November 2022
• 2006 AMC 10B Problems/Problem 19
  A circle of radius <math>2</math> is centered at <math>O</math>. Square <math pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B;
  5 KB (873 words) - 15:39, 29 May 2023
• 2006 AMC 10B Problems/Problem 20
  In rectangle <math>ABCD</math>, we have <math>A=(6,-22)</math>, <math>B=(2006,178)</math>, <math>D=(8,y)</math>, for some i <math> \mathrm{(A) \ } 4000\qquad \mathrm{(B) \ } 4040\qquad \mathrm{(C) \ } 4400\qquad \math
  4 KB (594 words) - 15:45, 30 July 2023
• 2006 AMC 10B Problems/Problem 21
  ...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? <math> \mathrm{(A) \ } \frac{4}{63}\qquad \mathrm{(B) \ } \frac{1}{8}\qquad \mathrm{(C) \ } \
  3 KB (484 words) - 19:09, 15 October 2023
• 2006 AMC 10B Problems/Problem 23
  A triangle is partitioned into three triangles and a quadrilateral by drawing two lines from vertices to their opposite sides. T pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A);
  5 KB (861 words) - 00:53, 25 November 2023
• 2006 AMC 10B Problems/Problem 25
  ...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! <math> \mathrm{(A) \ } 4\qquad \mathrm{(B) \ } 5\qquad \mathrm{(C) \ } 6\qquad \mathrm{(D) \
  5 KB (878 words) - 14:39, 3 December 2023
• 1993 AIME Problems/Problem 15
  Let <math>\overline{CH}</math> be an altitude of <math>\triangle ABC</math>. Let <math>R\,</math> and <math>S\,< pair A,B,C,H;
  3 KB (449 words) - 21:39, 21 September 2023
• 1993 AIME Problems/Problem 11
  ...Bonnie play this game several times with the stipulation that the loser of a game goes first in the next game. Suppose that Alfred goes first in the fir ...ath>\frac{1}{2^n}</math>. The first person wins if the coin lands heads on an odd numbered flip. So, the probability of the first person winning the game
  7 KB (1,058 words) - 20:57, 22 December 2020
• 1993 AIME Problems/Problem 4
  ...,b,c,d)\,</math> with <math>0 < a < b < c < d < 500\,</math> satisfy <math>a + d = b + c\,</math> and <math>bc - ad = 93\,</math>? ...>(k-c)c - a(k-a) = (a-c)(a+c-k) = (c-a)(d-c) = 93</math>. Hence <math>(c - a,d - c) = (1,93),(3,31),(31,3),(93,1)</math>.
  8 KB (1,343 words) - 16:27, 19 December 2023
• 1994 AIME Problems/Problem 14
  ...ith angle of incidence <math>\alpha=19.94^\circ\,</math> and reflects with an equal angle of reflection as shown. The light beam continues its path, ref pair B = MP("B",(0,0),NW), C = MP("C",D((1,0))), A = MP("A",expi(alpha * pi/180),N); path r = C + .4 * expi(beta * pi/180) -- C - 2*ex
  2 KB (303 words) - 00:03, 28 December 2017
• 1994 AIME Problems/Problem 13
  ...xt{cis}\left(\frac {(2k + 1)\pi}{10}\right)</math> where <math>k</math> is an integer between <math>0</math> and <math>9</math>. ...where <math>\omega = e^{i(\pi n/5+\pi/10)}</math> where <math>n</math> is an integer.
  3 KB (375 words) - 23:46, 6 August 2021
• 1994 AIME Problems/Problem 12
  ...ctangular field measures <math>24</math> meters by <math>52</math> meters. An agricultural researcher has 1994 meters of fence that can be used for inter ...partitioned square has a side length of <math>1</math> (just so we can get a clear image of what the formula will look like). The amount of internal fen
  3 KB (473 words) - 17:06, 1 January 2024
• 1994 AIME Problems/Problem 10
  In triangle <math>ABC,\,</math> angle <math>C</math> is a right angle and the altitude from <math>C\,</math> meets <math>\overline{AB ...form <math>29^2 x</math> and <math>29 x^2</math>, respectively, where x is an integer.
  3 KB (534 words) - 16:23, 26 August 2018
• 1994 AIME Problems/Problem 9
  ...matched tiles are placed in a bag. The player randomly draws tiles one at a time from the bag and retains them, except that matching tiles are put asid *Case 1. We draw a pair on the first two cards. The second card is the same as the first with
  3 KB (589 words) - 14:18, 21 July 2019
• 1994 AIME Problems/Problem 8
  ...ath>, <math>(a,11)\,</math>, and <math>(b,37)\,</math> are the vertices of an equilateral triangle. Find the value of <math>ab\,</math>. ...<math>b+37i</math> is then a rotation of <math>60</math> degrees of <math>a+11i</math> about the origin, so:
  5 KB (788 words) - 13:53, 8 July 2023
• 1994 AIME Problems/Problem 7
  For certain ordered pairs <math>(a,b)\,</math> of [[real number]]s, the system of equations ...pair <math>(x,y)\,</math> of integers. How many such ordered pairs <math>(a,b)\,</math> are there?
  3 KB (442 words) - 19:51, 8 January 2024
• 1994 AIME Problems/Problem 6
  ...of the six extremal lines into disjoint unit regular triangles, and forms a series of unit regular triangles along the edge of the hexagon. ...the six equilateral triangles composing the regular hexagon to the area of a unit regular triangle is just <math>\left(\frac{20/\sqrt{3}}{2/\sqrt{3}}\ri
  4 KB (721 words) - 16:14, 8 March 2021
• 1994 AIME Problems/Problem 4
  ...x+1}</math> for some <math>x\in\mathbb{Z}</math>, then <math>\lfloor\log_2{a}\rfloor=\log_2{2^{x}}=x</math>. ...th>. So the sum of <math>\lfloor\log_2{a}\rfloor</math> for all such <math>a</math> is <math>x\cdot2^x</math>.
  2 KB (264 words) - 13:33, 11 August 2018
• 1995 AIME Problems/Problem 15
  ...in, one will encounter a run of <math>5</math> heads before one encounters a run of <math>2</math> tails. Given that <math>p_{}</math> can be written i ...'s, the total probability is that <math>3/2</math> of it has to start with an <tt>H</tt>.
  6 KB (979 words) - 13:20, 11 April 2022
• 1995 AIME Problems/Problem 14
  ...s]] <math>42</math>, two [[chord]]s of length <math>78</math> intersect at a point whose distance from the center is <math>18</math>. The two chords di ...), D=E+48*expi(7*pi/6), A=E+30*expi(5*pi/6), C=E+30*expi(pi/6), F=foot(O,B,A);
  3 KB (484 words) - 13:11, 14 January 2023
• 1995 AIME Problems/Problem 8
  Thus, for a given value of <math>y</math>, we need the number of multiples of <math>y(y ...rime factors, it follows that <math>\frac{x-y}{y(y+1)}</math> must also be an integer.
  4 KB (646 words) - 17:37, 1 January 2024
• 1995 AIME Problems/Problem 3
  ...ing at <math>(0,0),</math> an object moves in the [[coordinate plane]] via a sequence of steps, each of length one. Each step is left, right, up, or do It takes an even number of steps for the object to reach <math>(2,2)</math>, so the num
  3 KB (602 words) - 23:15, 16 June 2019
• 1996 AIME Problems/Problem 14
  ...[[solid]] is made by gluing together <math>1\times 1\times 1</math> cubes. An internal [[diagonal]] of this solid passes through the interiors of how man ...oordinates of the diagonally opposite corner of the rectangle, where <math>a, b, c \in \mathbb{Z_{+}}</math>.
  5 KB (923 words) - 21:21, 22 September 2023
• 1996 AIME Problems/Problem 12
  ...her way to think about this is that the total sum is 330. Because you have a total of <math>10 \cdot 9</math> possible "pairs" or "differences", the ave ...ers. We get <math>\frac{11}{3}</math> and <math>\frac{22}{3}</math> giving an absolute difference of <math>\frac{11}{3}</math>. Therefore, the average/ex
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 10
  ...e that if we have <math>\alpha</math> equal to both the sine and cosine of an angle, we can use the sine sum formula (and the cosine sum formula on the d {{AIME box|year=1996|num-b=9|num-a=11}}
  4 KB (503 words) - 15:46, 3 August 2022
• 1996 AIME Problems/Problem 7
  ...r schemes are equivalent if one can be obtained from the other by applying a [[rotation]] in the plane board. How many inequivalent color schemes are po ...e painted yellow. There are four possible ways to rotate each board. Given an arbitrary pair of yellow squares, these four rotations will either yield tw
  4 KB (551 words) - 11:44, 26 June 2020
• 1996 AIME Problems/Problem 6
  ...robability that the tournament will produce neither an undefeated team nor a winless team, where <math>m</math> and <math>n</math> are relatively prime ...}{2} = 10</math> games in total, and every game can either end in a win or a loss. Therefore, there are <math>2^{10} = 1024</math> possible outcomes.
  3 KB (461 words) - 00:33, 16 May 2024
• 1996 AIME Problems/Problem 5
  ...x^3+rx^2+sx+t=0</math> are <math>a+b</math>, <math>b+c</math>, and <math>c+a</math>. Find <math>t</math>. ...al <math>P(x) = x^3+3x^2+4x-11 = (x-a)(x-b)(x-c) = 0</math>, we have <math>a + b + c = s = -3</math>, <math>ab + bc + ca = 4</math>, and <math>abc = 11<
  3 KB (585 words) - 22:08, 19 November 2022
• 1996 AIME Problems/Problem 4
  ...th>x</math> centimeters directly above an upper [[vertex]], the cube casts a shadow on the horizontal surface. The area of the shadow, which does not in {{AIME box|year=1996|num-b=3|num-a=5}}
  2 KB (257 words) - 17:50, 4 January 2016
• 1996 AIME Problems/Problem 3
  ...ir product will contain <math>(n+1)^2</math> terms, as each term will have an unique power of <math>x</math> or <math>y</math> and so none of the terms w This is equivalent to starting off with a <math>x^0y^0</math> terms and choosing between <math>4</math> options <math
  3 KB (515 words) - 04:29, 27 November 2023
• 1997 AIME Problems/Problem 15
  ...ngle lies outside <math>ABCD</math>. The maximum possible [[area]] of such a triangle can be written in the form <math>p\sqrt{q}-r</math>, where <math>p ...{2} - \frac {a\sqrt {3}}{2}\right) + \left(\frac {11\sqrt {3}}{2} + \frac {a}{2}\right)i = b + 10i</math>.
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 13
  ...th>b</math> is not divisible by the square of any prime number. Find <math>a+b</math>. ...hing won't be too bad of an idea. Since <math>f(x) = f(-x)</math>, there's a symmetry about all four [[quadrant]]s, so just consider the first quadrant.
  7 KB (1,225 words) - 19:56, 4 August 2021
• 1997 AIME Problems/Problem 12
  ...math>f</math> defined by <math>f(x)= \frac{ax+b}{cx+d}</math>, where <math>a</math>,<math>b</math>,<math>c</math> and <math>d</math> are nonzero real nu ...in the domain. Substituting the function definition, we have <math>\frac {a\frac {ax + b}{cx + d} + b}{c\frac {ax + b}{cx + d} + d} = x</math>, which r
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1997 AIME Problems/Problem 11
  Using the identity <math>\sin a + \sin b = 2\sin \frac{a+b}2 \cos \frac{a-b}{2}</math> <math>\Longrightarrow \sin x + \cos x</math> <math>= \sin x + A slight variant of the above solution, note that
  10 KB (1,514 words) - 14:35, 29 March 2024
• 1997 AIME Problems/Problem 9
  ...angle a^2\rangle</math>, and <math>2<a^2<3</math>. Find the value of <math>a^{12}-144a^{-1}</math>. ...g at the properties of the number, it is immediately guess-able that <math>a = \phi = \frac{1+\sqrt{5}}2</math> (the [[phi|golden ratio]]) is the answer
  4 KB (586 words) - 21:53, 30 December 2023
• 1997 AIME Problems/Problem 6
  ...1</math>, <math>A_n</math>, and <math>B</math> are consecutive vertices of a regular polygon? ...her regular polygon have <math>m</math> sides. Using the interior angle of a regular polygon formula, we have <math>\angle A_2A_1A_n = \frac{(n-2)180}{n
  3 KB (497 words) - 00:39, 22 December 2018
• 1997 AIME Problems/Problem 5
  ...>r</math> by a fraction whose numerator is 1 or 2 and whose denominator is an integer. The closest such fraction to <math>r</math> is <math>\frac 27.</ma {{AIME box|year=1997|num-b=4|num-a=6}}
  1 KB (208 words) - 11:46, 4 June 2021
• 1997 AIME Problems/Problem 2
  The nine horizontal and nine vertical lines on an <math>8\times8</math> checkerboard form <math>r</math> [[rectangle]]s, of w To determine the two horizontal sides of a rectangle, we have to pick two of the horizontal lines of the checkerboard,
  3 KB (416 words) - 21:09, 27 October 2022
• 1998 AIME Problems/Problem 15
  ...rdered pair of distinct positive integers. A proper sequence of dominos is a list of distinct dominos in which the first coordinate of each pair after t We can draw a comparison between the domino a set of 40 points (labeled 1 through 40) in which every point is connected w
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 14
  An <math>m\times n\times p</math> rectangular box has half the volume of an <math>(m + 2)\times(n + 2)\times(p + 2)</math> rectangular box, where <math ...th>98</math>. We now check that <math>130</math> is optimal, setting <math>a=m-2</math>, <math>b=n-2</math> in order to simplify calculations. Since
  2 KB (390 words) - 21:05, 29 May 2023
• 1998 AIME Problems/Problem 13
  If <math>\{a_1,a_2,a_3,\ldots,a_n\}</math> is a [[set]] of [[real numbers]], indexed so that <math>a_1 < a_2 < a_3 < \cdots ...; hence we can just append a <math>9</math> to any of those subsets to get a new one.
  2 KB (384 words) - 19:02, 20 October 2023
• 1998 AIME Problems/Problem 12
  ...h>c</math> is not divisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>? ...)^{2}}{k^{2}} = \frac {4}{7 - 3\sqrt {5}} = 7 + 3\sqrt {5}</math> so <math>a^{2} + b^{2} + c^{2} = 7^{2} + 3^{2} + 5^{2} = \boxed{083}</math>.
  5 KB (876 words) - 20:27, 9 June 2022
• 1998 AIME Problems/Problem 11
  ...,</math> and <math>\overline{CD},</math> and <math>\overline{AD}</math> is an interior diagonal. Points <math>P, Q,</math> and <math>R</math> are on <ma triple A=(0,0,0),B=(20,0,0),C=(20,0,20),D=(20,20,20);
  7 KB (1,084 words) - 11:48, 13 August 2023
• 1998 AIME Problems/Problem 10
  ...math>c</math> is not divisible by the square of any [[prime]]. Find <math>a + b + c</math>. ...a ninth one; instead the circles will overlap since the middle sphere has a larger radius and will sort of “bulge” out.
  3 KB (496 words) - 13:02, 5 August 2019
• 1998 AIME Problems/Problem 7
  ...<math>n= {50\choose3}</math>. Computing this and dividing by 100 gives us an answer of <math>\boxed{196}</math>. ...mber of stones in the box remains odd, because (an odd #) + (an even #) = (an odd #).
  5 KB (684 words) - 11:41, 13 August 2023
• 1998 AIME Problems/Problem 5
  ...ually not that difficult. <math>\frac {k(k-1)}2</math> always evaluates to an integer ([[triangular number]]), and the [[cosine]] of <math>n\pi</math> wh If we group the terms in pairs, we see that we need a formula for <math>-\frac{(n)(n-1)}{2} + \frac{(n+1)(n)}{2} = \left(\frac n2
  1 KB (225 words) - 02:20, 16 September 2017
• 1998 AIME Problems/Problem 4
  ...ums those three values. The [[probability]] that all three players obtain an [[odd]] sum is <math>m/n,</math> where <math>m</math> and <math>n</math> ar ...tiles: that is, he can either have three odd tiles, or two even tiles and an odd tile. Thus, since there are <math>5</math> odd tiles and <math>4</math
  5 KB (917 words) - 02:37, 12 December 2022
• 1998 AIME Problems/Problem 2
  <div style="text-align:center;"><math>A = I + \frac B2 - 1</math></div> ..., whose area is <math>\frac 12</math> of the 30 by 30 [[square]] it is in. A simple way to see this is to note that the two triangles outside of the qua
  6 KB (913 words) - 16:34, 6 August 2020
• 1998 AIME Problems/Problem 1
  ...ax(24,a)}3^{\max(6,b)}</math>, and <math>b = 12</math>. Since <math>0 \le a \le 24</math>, there are <math>\boxed{25}</math> values of <math>k</math>. At this point, we realize that <math>k=2^a\cdot3^b</math>, as any other prime factors would be included in the <math>\
  2 KB (289 words) - 22:50, 23 April 2024
• 1999 AIME Problems/Problem 15
  ...The vertices of its midpoint triangle are the [[midpoint]]s of its sides. A triangular [[pyramid]] is formed by folding the triangle along the sides of ..."\(D\)",foot(A,B,C),NE);label("\(E\)",foot(B,A,C),SW);label("\(F\)",foot(C,A,B),NW);label("\(P\)",P,NW);label("\(Q\)",Q,NE);label("\(R\)",R,SE);</asy><a
  7 KB (1,169 words) - 15:28, 13 May 2024
• 1999 AIME Problems/Problem 14
  pair A=(0,0),B=(13,0),C=IP(circle(A,15),circle(B,14)); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle);
  7 KB (1,184 words) - 13:25, 22 December 2022
• 1999 AIME Problems/Problem 10
  ...equally likely. The [[probability]] that some three of the segments form a [[triangle]] whose vertices are among the ten given points is <math>m/n,</m ...<math>{10\choose3}</math> sets of 3 points which form triangles. However, a fourth distinct segment must also be picked. Since the triangle accounts fo
  3 KB (524 words) - 17:25, 17 July 2023
• 1999 AIME Problems/Problem 9
  ...is [[equidistant]] from that point and the [[origin]]. Given that <math>|a+bi|=8</math> and that <math>b^2=m/n,</math> where <math>m_{}</math> and <ma ...= (a-b)</math> and <math>y = (a+b)</math>, we get <math>2a = 1 \Rightarrow a = \frac 12</math>.
  6 KB (1,010 words) - 19:01, 24 May 2023
• 1999 AIME Problems/Problem 5
  ...ng the number up a ten, a hundred, etc. If we take <math>T(a999)</math> as an example, <cmath>|(a + 1) + 0 + 0 + 1 - (a + 9 + 9 + 9)| = |2 - 9(3)|</cmath>
  1 KB (170 words) - 19:40, 4 July 2013
• 1999 AIME Problems/Problem 3
  ...[[positive integer]]s <math>n</math> for which <math>n^2-19n+99</math> is a [[perfect square]]. Because <math>n</math> is an integer, this means <math>4x^2-35=q^2</math> for some nonnegative integer <
  2 KB (296 words) - 01:18, 29 January 2021
• 1999 AIME Problems/Problem 1
  Find the smallest prime that is the fifth term of an increasing [[arithmetic sequence]], all four preceding terms also being [[p ...h>6</math>, we find that <math>5,11,17,23</math>, and <math>29</math> form an [[arithmetic sequence]]. Thus, the answer is <math>029</math>.
  2 KB (332 words) - 13:22, 3 August 2020
• 2000 AIME I Problems/Problem 15
  A stack of <math>2000</math> cards is labelled with the integers from <math>1 ...r logic as Solution 1, we find that 1999 has position <math>1024</math> in a <math>2048</math> card stack, where the fake cards towards the front.
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 14
  ...r</math> times as large as angle <math>APQ,</math> where <math>r</math> is a positive real number. Find <math>\lfloor 1000r \rfloor</math>. pair A,B,C,P,Q;
  8 KB (1,275 words) - 03:04, 27 February 2022
• 2000 AIME I Problems/Problem 11
  .../math> be the sum of all numbers of the form <math>a/b,</math> where <math>a</math> and <math>b</math> are [[relatively prime]] positive [[divisor]]s of ...here <math>-3 \le x,y \le 3</math>. Thus every number in the form of <math>a/b</math> will be expressed one time in the product
  4 KB (667 words) - 13:58, 31 July 2020
• 2000 AIME I Problems/Problem 7
  ...xpand this to get <math>xyz+xy+yz+xz+x+y+z+1 = 145k</math>, and if we make a substitution for <math>xyz</math>, and rearrange the terms, we get <math>xy ...to get <math>144k = 36</math>, so <math>k = \frac{1}{4}</math> which gives an answer of <math>1+4 = \boxed{005}</math>
  5 KB (781 words) - 15:02, 20 April 2024
• 2000 AIME I Problems/Problem 6
  ...must be an integer (this is because <math>\sqrt{y} - \sqrt{x} = 2</math>, an integer). Then <math>\sqrt{x} = 997</math>, and we continue this downward u .../math> = <math>x</math> and <math>b^2</math> = <math>y</math>, where <math>a</math> and <math>b</math> are positive.
  6 KB (966 words) - 21:48, 29 January 2024
• 2001 AIME I Problems/Problem 15
  ...1</math> are considered to be consecutive, are written on faces that share an edge is <math>m/n,</math> where <math>m</math> and <math>n</math> are relat ...hedron <math>DBEG</math>, and four are <math>\textit{long}</math>, joining a vertex of one tetrahedron to the diagonally opposite point from the other.
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2001 AIME I Problems/Problem 14
  ...get mail on the same day, but that there are never more than two houses in a row that get no mail on the same day. How many different patterns of mail d ...represent a house that does not receive mail and <math>1</math> represent a house that does receive mail. This problem is now asking for the number of
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2001 AIME I Problems/Problem 13
  .../math>-degree arc, where <math>d < 120.</math> The length of the chord of a <math>3d</math>-degree arc is <math>- m + \sqrt {n}</math> centimeters, whe Note that a cyclic quadrilateral in the form of an isosceles trapezoid can be formed from three chords of three <math>d</math>
  3 KB (561 words) - 19:25, 27 November 2022
• 2001 AIME I Problems/Problem 12
  A [[sphere]] is inscribed in the [[tetrahedron]] whose vertices are <math>A = (6,0,0), B = (0,4,0), C = (0,0,2),</math> and <math>D = (0,0,0).</math> triple A = (6,0,0), B = (0,4,0), C = (0,0,2), D = (0,0,0);
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2014 USAJMO Problems
  ...frac{10a^2-5a+1}{b^2-5b+10},\frac{10b^2-5b+1}{c^2-5c+10},\frac{10c^2-5c+1}{a^2-5a+10}\right )}\leq abc. </cmath> ...riangle{ABC}</math> be a non-equilateral, acute triangle with <math>\angle A=60^\circ</math>, and let <math>O</math> and <math>H</math> denote the circu
  3 KB (600 words) - 16:42, 5 August 2023
• 2001 AIME I Problems/Problem 11
  In a [[rectangle|rectangular]] array of points, with 5 rows and <math>N</math> c It turns out that there exists such an array satisfying the problem conditions if and only if
  3 KB (493 words) - 13:51, 22 July 2020
• 2001 AIME I Problems/Problem 10
  ..., <math>4</math> OEOs, <math>6</math> OOEs, and <math>4</math> OOOs. Doing a sanity check, <math>12 + 8 + 12 + 6 + 8 + 4 + 6 + 4 = 60,</math> which is t ...tters. There are <math>60 \cdot 59</math> total ways to do this, so we get a final answer of <cmath>\dfrac{460}{60 \cdot 59} = \dfrac{23}{3 \cdot 59} =
  8 KB (1,187 words) - 02:40, 28 November 2020
• 2001 AIME I Problems/Problem 8
  ...math> number that is twice <math>N</math>. For example, <math>51</math> is a 7-10 double because its base-<math>7</math> representation is <math>102</ma ...his is because the digits in <math>N</math> ' s base 7 representation make a number with the same digits in base 10 when multiplied by 2)
  3 KB (502 words) - 11:28, 9 December 2023
• 2001 AIME I Problems/Problem 5
  An [[equilateral triangle]] is inscribed in the [[ellipse]] whose equation is Denote the vertices of the triangle <math>A,B,</math> and <math>C,</math> where <math>B</math> is in [[quadrant]] 4 and
  6 KB (1,043 words) - 10:09, 15 January 2024
• 2001 AIME I Problems/Problem 4
  ...and <math>c</math> is not divisible by the square of any prime. Find <math>a+b+c</math>. ...<math>\angle TCA=75^{\circ}</math>, meaning <math>\triangle TAC</math> is an isosceles triangle and <math>AC=24</math>.
  3 KB (534 words) - 03:22, 23 January 2023
• 2002 AIME I Problems/Problem 15
  ...CD</math> is a square with <math>AB = 12;</math> face <math>ABFG</math> is a trapezoid with <math>\overline{AB}</math> parallel to <math>\overline{GF},< triple A=(-6,-6,0), B = (-6,6,0), C = (6,6,0), D = (6,-6,0), E = (2,0,12), H=(-6+2*s
  7 KB (1,181 words) - 20:32, 8 January 2024
• 2002 AIME I Problems/Problem 14
  ...alues obtained by deleting <math>x</math> from <math>\mathcal{S}</math> is an integer. Given that 1 belongs to <math>\mathcal{S}</math> and that 2002 is {{AIME box|year=2002|n=I|num-b=13|num-a=15}}
  2 KB (267 words) - 19:18, 21 June 2021
• 2002 AIME I Problems/Problem 13
  ...,0), E=(A+B)/2, C=IP(CR(A,3*70^.5),CR(E,27)), D=(B+C)/2, F=IP(circumcircle(A,B,C),E--C+2*(E-C)); ...MP("12",(A+E)/2,SE,f);MP("12",(B+E)/2,f); MP("27",(C+E)/2,SW,f); MP("18",(A+D)/2,SE,f);
  6 KB (974 words) - 13:01, 29 September 2023
• 2002 AIME I Problems/Problem 11
  ...ht path from the time it leaves point <math>A</math> until it next reaches a vertex of the cube is given by <math>m\sqrt{n}</math>, where <math>m</math> When a light beam reflects off a surface, the path is like that of a ball bouncing. Picture that, and also imagine X, Y, and Z coordinates for t
  3 KB (591 words) - 15:11, 21 August 2019
• 2002 AIME I Problems/Problem 7
  ...cimal of <math>\dfrac{10}{7}</math> repeats every 6 digits, we can cut out a lot of 6's from <math>858</math> to reduce the problem to finding the first ...1} \equiv 3 \pmod{7}</math>; in other words, <math>10^{859}</math> leaves a residue of <math>3</math> after division by <math>7</math>. Then the desire
  2 KB (316 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 4
  Since we need a factor of 29 in the denominator, we let <math>n=29t</math>.* Substituting, Since <math>m</math> is an integer, <math>t+1 = 29</math>, so <math>t=28</math>. It quickly follows th
  2 KB (320 words) - 07:55, 4 November 2022
• 2002 AIME I Problems/Problem 2
  ...shows twenty congruent [[circle]]s arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the r ...h>1</math>. Now, we connect the radii of three circles such that they form an equilateral triangle with side length 4. Obviously, the height this triangl
  2 KB (287 words) - 19:54, 4 July 2013
• 2003 AIME I Problems/Problem 15
  ...name of a point represent the mass located there. Since we are looking for a ratio, we assume that <math>AB=120</math>, <math>BC=169</math>, and <math>C First, reflect point <math>F</math> over angle bisector <math>BD</math> to a point <math>F'</math>.
  8 KB (1,382 words) - 14:23, 29 December 2022
• 2003 AIME I Problems/Problem 14
  ...10^k m - nX</math> is an integer and <math>\frac{10^k m - nX}{n}</math> is a fraction between <math>0</math> and <math>1</math>, we can rewrite this as ...te the problem as having the smallest <math>n</math> such that we can find an positive integer <math>m</math> such that <math>0<\frac{m}{n}-\frac{251}{10
  3 KB (477 words) - 14:23, 4 January 2024
• 2003 AIME I Problems/Problem 11
  ...math> and <math> \sin x \cos x </math> are not the lengths of the sides of a triangle. Given that <math> p = d/n, </math> where <math> d </math> is the ...ssions and those three lengths not forming a [[triangle]] is equivalent to a violation of the [[triangle inequality]]
  2 KB (284 words) - 13:42, 10 October 2020
• 2003 AIME I Problems/Problem 10
  pair A=(0,0), B=(2,0), C=(1,Tan(37)), M=IP(A--(2Cos(30),2Sin(30)),B--B+(-2,2Tan(23))); D(MP("A",A)--MP("B",B)--MP("C",C,N)--cycle); D(A--D(MP("M",M))--B); D(C--M);
  7 KB (1,058 words) - 01:41, 6 December 2022
• 2003 AIME I Problems/Problem 9
  An [[integer]] between <math>1000</math> and <math>9999</math>, inclusive, is ...number <math>\overline{abcd}</math>. Then <math>a+b=c+d</math>. Set <math>a+b=x</math>.
  4 KB (696 words) - 11:55, 10 September 2023
• 2003 AIME I Problems/Problem 8
  ...three terms form an [[arithmetic progression]], the last three terms form a [[geometric progression]], and the first and fourth terms differ by <math>3 ...four numbers thus are in the form <math>a,\ a+d,\ a+2d,\ \frac{(a + 2d)^2}{a + d}</math>.
  5 KB (921 words) - 23:21, 22 January 2023
• 2003 AIME I Problems/Problem 7
  pair O=(0,0),A=(-15,0),B=(-6,0),C=(15,0),D=(0,8); D(D(MP("A",A))--D(MP("C",C))--D(MP("D",D,NE))--cycle);
  3 KB (490 words) - 18:13, 13 February 2021
• 2003 AIME I Problems/Problem 6
  The sum of the areas of all triangles whose vertices are also vertices of a <math>1</math> by <math>1</math> by <math>1</math> cube is <math> m + \sqrt ...e face diagonal and one space diagonal of the cube) and those which lie in a plane [[oblique]] to the edges of the cube, whose sides are three face diag
  3 KB (477 words) - 18:35, 27 December 2021
• 2003 AIME I Problems/Problem 2
  ...]s with [[radius | radii]] <math> 1, 2, 3, \dots, 100 </math> are drawn in a plane. The interior of the circle of radius <math>1</math> is colored red, For any <math>a</math>, <math>a^{2}-(a-1)^{2}=a^{2}-(a^{2}-2a+1)=2a-1</math>. We can cancel the [[divisor | factor]] of [[pi]] fr
  4 KB (523 words) - 15:49, 8 March 2021
• 2003 AIME II Problems/Problem 13
  ...ne of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bu ...o return to the original vertex, the net number of clockwise steps must be a multiple of 3, i.e., <math>\#CW - \#CCW \equiv 0 \pmod{3}</math>. Since <m
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2003 AIME II Problems/Problem 12
  The members of a distinguished committee were choosing a president, and each member gave one vote to one of the 27 candidates. For ...>. But then <math>s=v_1+\cdots+v_{27}\geq 27\cdot 5 = 135</math>, which is a contradiction. Hence the smallest possible <math>s</math> is at least <math
  4 KB (759 words) - 13:00, 11 December 2022
• 2003 AIME II Problems/Problem 10
  ...The sum of their square roots is the square root of an integer that is not a perfect square. What is the maximum possible sum of the two integers? ...tor <math>2</math>, but solving gives <math>b=196</math>, which is already a perfect square, so we have to keep going. In order to keep both factors eve
  1 KB (218 words) - 14:14, 25 June 2021
• 2003 AIME II Problems/Problem 4
  In a regular tetrahedron the centers of the four faces are the vertices of a smaller tetrahedron. The ratio of the volume of the smaller tetrahedron to ...ratio of the volumes. Let <math>AB=1</math>, for our convenience. Dropping an altitude from <math>W</math> to <math>BC</math>, and calling the foot <math
  3 KB (563 words) - 17:36, 30 July 2022
• 2003 AIME II Problems/Problem 2
  ...st side. Therefore, the last three digits of the greatest number should be an arrangement of the digits <math>0,1,2</math>. Since the number has to be di {{AIME box|year=2003|n=II|num-b=1|num-a=3}}
  1,013 bytes (162 words) - 09:00, 11 July 2023
• 2002 AIME II Problems/Problem 14
  ...is <math>152</math>, and the angle <math>PAM</math> is a [[right angle]]. A [[circle]] of [[radius]] <math>19</math> with center <math>O</math> on <mat ...MPA</math> are similar. Also note that <math>AM = BM</math> by [[power of a point]]. Using the fact that the ratio of corresponding sides in similar tr
  4 KB (658 words) - 19:15, 19 December 2021
• 2002 AIME II Problems/Problem 11
  Two distinct, real, infinite geometric series each have a sum of <math>1</math> and have the same second term. The third term of one ...> Now we have that <math>ar^2 = \frac18</math> and we also have that <math>a+r = 1</math>. We can solve for <math>r</math> and the only appropriate valu
  4 KB (696 words) - 16:27, 22 March 2022
• 2002 AIME II Problems/Problem 10
  While finding the sine of a certain angle, an absent-minded professor failed to notice that his calculator was not in the 1. <math>x+2\pi a = \frac{\pi x}{180}</math> for any integer <math>a</math>
  2 KB (336 words) - 19:30, 24 June 2020
• 2002 AIME II Problems/Problem 8
  Rewriting the given information and simplifying it a bit, we have ...2002}{k+1} \right\rfloor.</math> We seek the smallest such <math>k.</math> A bit of experimentation yields that <math>k=49</math> is the smallest soluti
  6 KB (908 words) - 14:22, 14 July 2023
• 2002 AIME II Problems/Problem 3
  ...th>b - a</math> is the [[Perfect square|square]] of an integer. Find <math>a + b + c.</math> <math>abc=6^6</math>. Since they form an increasing geometric sequence, <math>b</math> is the [[geometric mean]] of
  2 KB (263 words) - 22:50, 5 April 2024
• 2002 AIME II Problems/Problem 2
  Three [[vertex|vertices]] of a [[cube]] are <math>P=(7,12,10)</math>, <math>Q=(8,8,1)</math>, and <math>R= ...h>PQR</math> is an equilateral triangle. Let the side of the cube be <math>a</math>.
  709 bytes (103 words) - 03:32, 6 December 2019
• 2001 AIME II Problems/Problem 15
  ...th>EI = EJ = EK = 2</math>. A solid <math>S</math> is obtained by drilling a tunnel through the cube. The sides of the tunnel are planes parallel to <ma ...ath>E</math>, where the drilling starts, is at <math>(8,8,8)</math>. Using a little visualization (involving some [[similar triangles]], because we have
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 13
  pair A=(0,0), D=(x,0), B=IP(CR(A,8),CR(D,10)), E=(-3x/5,0), C=IP(CR(E,16),CR(D,64/5)); D(MP("A",A)--MP("B",B,NW)--MP("C",C,NW)--MP("D",D)--cycle); D(B--D); D(A--MP("E",E)--B,d);
  4 KB (743 words) - 03:32, 23 January 2023
• 2001 AIME II Problems/Problem 12
  ...an outward-pointing regular tetrahedron that has the midpoint triangle as a face. The [[volume]] of <math>P_{3}</math> is <math>\frac {m}{n}</math>, wh ...hedra. However, the volume of the tetrahedra being constructed decrease by a factor of <math>\frac 18</math>. Thus we have the recursion <math>\Delta P_
  2 KB (380 words) - 00:28, 5 June 2020

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