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• The figure shows a square of side $y$ units divided into a square of side $x$ units and four congruent rectangles. What is the perimeter, in units, of one of the four congruent rectangles? Express your answer in terms of $y$.
  0 bytes (0 words) - 22:13, 4 March 2020

Page text matches

• Pythagorean Theorem
  ...f geometry problems can be solved by building right triangles and applying the Pythagorean Theorem. ...Geometric inequality#Pythagorean_Inequality | Pythagorean Inequality]] and the [[Law of Cosines]].
  5 KB (886 words) - 21:12, 22 January 2024
• 2016 AMC 10A Problems/Problem 7
  ...0, 100, x, 40, 50, 200, 90</math> are all equal to <math>x</math>. What is the value of <math>x</math>? Since <math>x</math> is the mean,
  2 KB (268 words) - 18:19, 27 September 2023
• 2016 AMC 10A Problems/Problem 10
  ...<math>1</math> foot wide on all four sides. What is the length in feet of the inner rectangle? Let the length of the inner rectangle be <math>x</math>.
  2 KB (337 words) - 14:56, 25 June 2023
• 2016 AMC 10A Problems/Problem 11
  Find the area of the shaded region. ...h>, <math>2</math>, and <math>\frac{5}{4}</math>, which add to the area of the shaded region, which is <math>\boxed{6\frac{1}{2}}</math>.
  8 KB (1,016 words) - 00:17, 31 December 2023
• 2016 AMC 10A Problems/Problem 13
  ...moved two seats to the right, Ceci had moved one seat to the left, and Dee and Edie had switched seats, leaving an end seat for Ada. In which seat had Ada ...cupied and the positioning would not work. So, Edie and Dee are in seats 4 and 5. This means that Bea was originally in seat 1. Ceci must have been in sea
  2 KB (402 words) - 14:54, 25 June 2023
• List of mathematics competitions
  ...l links to more helpful pages about mathematics competitions. This is not the place to list individual competitions. ...an mathematics competitions are categorized in two, one for public schools and other for particular schools.
  4 KB (406 words) - 11:20, 26 February 2021
• Alabama mathematics competitions
  ...nd there are a number of quality contests around the state for both middle and high school students. ...298 middle school math forums] where students can discuss contest problems and mathematics.
  3 KB (411 words) - 21:32, 8 December 2014
• MathILy (serious mathematics infused with levity)
  ...r undergraduate math students. The math content covered is undergraduate- and graduate-level. ...3, with MathILy at Bryn Mawr College and MathILy-Er at Arcadia University. The program fee will be \$5300.
  5 KB (706 words) - 23:49, 29 January 2024
• Mathematics, science, and technology scholarships
  This page is for listing of '''Mathematics, science, and technology scholarships''' that are not subject specific. Links to scholar == Mathematics, science, and technology scholarships ==
  7 KB (851 words) - 10:54, 29 January 2022
• Proton Energy Scholarship
  ...of science and technology. They must also be planning further education in the field. ...th>\textdollar</math>100,000, though only one had been originally promised the scholarship.
  894 bytes (123 words) - 15:48, 13 June 2022
• MATHCOUNTS
  ...of Teachers of Mathematics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligib ...]], [[geometry]], [[number theory]], [[probability]], and [[statistics]]. The focus of MATHCOUNTS curriculum is in developing [[mathematical problem solv
  10 KB (1,497 words) - 11:42, 10 March 2024
• ASIA TEAM Problems/Problem 1
  ...er of <math>2n-1</math> is one more than the order of <math>2n</math>, and the answer is <math>\frac{1000}{2}=\boxed{500}</math>.
  227 bytes (40 words) - 05:42, 16 February 2024
• ICTM Math Contest
  ...ade School (Grades 3, 4, 5, 6, 7, 8 + Algebra 1) and High School (Regional and State Finals). ...schools at each grade level will receive certificates and be recognized in the ICTM Bulletin.
  8 KB (1,182 words) - 14:26, 3 April 2024
• Resources for mathematics competitions
  ...Many also have links to books, websites, and other resources relevant to the topic. == Quick Start Video Introduction to the AMC ==
  24 KB (3,269 words) - 00:43, 24 April 2024
• Math books
  ...re recommended by [[Art of Problem Solving]] administrators and members of the [http://aops.com/community AoPS Community]. Levels of reading and math ability are loosely defined as follows:
  24 KB (3,177 words) - 12:53, 20 February 2024
• Physics
  ...is considered to be the most fundamental of all the sciences, and is also the oldest. ...e 19th century is called <strong>Classical Physics</strong>. Physics after the 19th century is know as <strong>Modern Physics</strong>.
  9 KB (1,355 words) - 07:29, 29 September 2021
• Mathematics
  ...mportant to people in most modern technical discplines such as engineering and economics. ...2\,3\,4\,5\,6\,7\,8\,9\,0</math>|right|The ten [[digit]]s making up <br /> the base ten number system.}}
  6 KB (902 words) - 12:53, 3 September 2019
• AMC historical results
  <!-- Post AMC statistics and lists of high scorers here so that the AMC page doesn't get cluttered. --> ...he '''AMC historical results''' page. This page should include results for the [[AIME]] as well. For [[USAMO]] results, see [[USAMO historical results]].
  17 KB (1,921 words) - 13:00, 28 April 2024
• Inequality
  ...] methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in [[mathematics Olympiads]]. Inequalities are arguably a branch of [[elementary algebra]], and relate slightly to [[number theory]]. They deal with [[relations]] of [[var
  12 KB (1,798 words) - 16:20, 14 March 2023
• United States of America Mathematical Talent Search
  ...ics competition]] in which students are challenged to write full solutions and mathematical proofs to exploratory math problems. ...the [[Art of Problem Solving Foundation]] with support and sponsorship by the [[National Security Agency]] (NSA).
  4 KB (613 words) - 13:08, 18 July 2023
• Academic scholarships
  ...ve guides to '''academic scholarships'''. Get started by clicking through the links below by subject area. ...a category rather than a list, please create a new page for that category and list it here.
  3 KB (337 words) - 03:35, 7 September 2020
• AMC 10
  ...on the path towards choosing the team that represents the United States at the [[International Mathematics Olympiad]] (IMO). High scoring AMC 10 and AMC 12 students are invited to take the [[American Invitational Mathematics Examination]] (AIME).
  4 KB (574 words) - 15:28, 22 February 2024
• 1955 AHSME Problems/Problem 38
  ...numbers <math>29, 23, 21</math>, and <math>17</math> are obtained. One of the original integers is: ...ath> to be the four numbers. In order to satisfy the following conditions, the system of equation should be constructed. (It doesn't matter which variable
  1 KB (200 words) - 23:35, 28 August 2020
• American Invitational Mathematics Examination
  ...on the path toward choosing the team that represents the United States at the [[International Mathematics Olympiad]] (IMO). While most AIME participants ...America Junior Mathematics Olympiad (USAJMO) for qualification from taking the AMC 10.
  8 KB (1,057 words) - 12:02, 25 February 2024
• American Mathematics Competitions
  ...bbreviation for American Math Contest, used to refer to the AMC 8, AMC 10, and AMC 12. * [[AMC 8]] &mdash; for students grades 8 and under.
  5 KB (696 words) - 03:47, 24 December 2019
• United States of America Mathematical Olympiad
  ...on the path toward choosing the team that represents the United States at the [[International Mathematics Olympiad]] (IMO). ...). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC and of the recent expansion of USAMO participants from around 250 to around 500.
  6 KB (869 words) - 12:52, 20 February 2024
• American Regions Mathematics League
  The '''American Regions Math League''' (ARML) is a [[mathematical problem solvi ...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
  Team selection for the [[American Regions Mathematics League]] varies from team to team. ...ge Xinke's AoPS account "hurdler", if you are interested in trying out for the Alabama team.
  21 KB (3,500 words) - 18:41, 23 April 2024
• Proof writing
  ...a proof, they intend a different meaning to how the word is understood by the wider population. Students who spend time studying maths can develop proof- ...good proof are precision, accuracy, and clarity. A single word can change the intended meaning of a proof, so it is best to be as precise as possible.
  3 KB (502 words) - 18:16, 18 January 2016
• Community Service Scholarships
  ...rship programs in other countries. Just make that reorganization as clear and as clean as possible. ...[[Mathew Crawford]] using the email crawford@artofproblemsolving.com with the scholarship listing you wish to contribute.
  3 KB (350 words) - 01:18, 19 June 2016
• California MathCounts
  ...nts]] team is made up of the top four students during the written round of the state competition. ...won eight national championships (1986, 1992, 2000, 2002, 2003, 2010, 2011 and 2014), more than any other team.
  644 bytes (87 words) - 02:05, 25 March 2015
• Mu Alpha Theta
  ...itions]] and each year a national Mu Alpha Theta convention is held during the summer that includes a large competition. == The Competition ==
  4 KB (632 words) - 17:09, 11 October 2020
• Wilson's Theorem
  ...ath> if and only if <math>p</math> is prime. It was stated by John Wilson. The French mathematician Lagrange proved it in 1771. ...entary 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
  ...uare of any real number is nonnegative. Its name comes from its simplicity and straightforwardness. ...>x<0</math>, then <math>x^2 = (-x)(-x) > 0,</math> again by the closure of the set of positive numbers under multiplication.
  3 KB (560 words) - 22:51, 13 January 2024
• Heron's Formula
  ...formula | formula]] for finding the [[area]] of a [[triangle]] given only the three side lengths. ...s <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
  ...s and multiplied by [[coefficient]]s from a predetermined [[set]] (usually the set of integers; [[rational]], [[real]] or [[complex]] numbers; but in [[ab * <math>4x^2 + 6x - 9</math>, in the variable <math>x</math>
  6 KB (1,100 words) - 01:44, 17 January 2024
• Simon's Favorite Factoring Trick
  ==The General Statement== ...s the product of the variables, <math>66x</math> and <math>-88y</math> are the variables in linear terms.
  7 KB (1,107 words) - 07:35, 26 March 2024
• Factoring
  ...r]] or [[polynomial]]) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious. ==Differences and Sums of Powers==
  3 KB (532 words) - 22:00, 13 January 2024
• Proof by contradiction
  ...ue usually works well on problems where not a lot of information is known, and thus we can create some using proof by contradiction. ===Proof that the square root of 2 is irrational===
  2 KB (374 words) - 14:01, 21 August 2022
• Mathematical problem solving
  ...thinking. Mathematical [[problem solving]] involves using all the tools at one's disposal to attack a problem in a new way. ...eresting example of this kind of thinking is the calculation of the sum of the [[series]] <math>\frac11 + \frac14 + \frac19 + \cdots + \frac{1}{n^2} + \cd
  2 KB (314 words) - 06:45, 1 May 2014
• Pigeonhole Principle
  ...if <math>n+1</math> or more pigeons are placed into <math>n</math> holes, one hole must contain two or more pigeons. This seemingly trivial statement may ...> boxes and <math>n>k</math>, then at least one box must contain more than one ball.
  11 KB (1,985 words) - 21:03, 5 August 2023
• Quadratic equation
  ...one of three main strategies: [[factoring]], [[completing the square]] and the [[quadratic formula]]. The purpose of factoring is to turn a general quadratic into a product of [[bin
  2 KB (264 words) - 12:04, 15 July 2021
• 1965 AHSME Problems/Problem 40
  ...<math>x</math> such that <math>P = x^4 + 6x^3 + 11x^2 + 3x + 31</math> is the square of an integer. Then <math>n</math> is: ...ginning of <math>P</math>, we notice <math>x^4+6x^3</math>, which gives us the idea to use <math>(x^2+3x)^2=x^4+6x^3+9x^2</math>.
  3 KB (571 words) - 00:42, 22 October 2021
• Minnesota MathCounts
  ...held every year at the Crowne Plaza Minneapolis West Hotel in Plymouth on the second weekend of March (Friday-Saturday). == Past Winners and Schools ==
  995 bytes (131 words) - 18:02, 12 March 2023
• Combinatorics
  ...ive]] formula for the [[Fibonacci numbers]], and so too methods of solving the [[Rubiks cube]]. Mathematicians who spend their careers studying combinator ...mathematics, notably in [[theoretical computer science]], [[statistics]], and various fields of science.
  1 KB (208 words) - 02:12, 4 October 2020
• Principle of Inclusion-Exclusion
  ...the size of each set, and the size of all possible [[intersection]]s among the sets. ...t and undercount, in the end making sure every element is counted once and only once. In particular, memorizing a formula for PIE is a bad idea for problem
  9 KB (1,703 words) - 07:25, 24 March 2024
• Combination
  ...objects are chosen is irrelevant. We are generally concerned with finding the number of combinations of size <math>r</math> from an original set of size ...pe! It serves as a great introductory video to combinations, permutations, and counting problems in general! [https://bit.ly/CombinationsAndPermutations P
  4 KB (615 words) - 11:43, 21 May 2021
• Cauchy-Schwarz Inequality
  ...s. In high-school competitions, its applications are limited to elementary and linear algebra. ...auchy-Schwarz forms the foundation for inequality problems in intermediate and olympiad competitions. It is particularly crucial in proof-based contests.
  13 KB (2,048 words) - 15:28, 22 February 2024
• Factorial
  ...mportant function in [[combinatorics]] and [[analysis]], used to determine the number of ways to arrange objects. ...{i=1}^n i</math>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>.
  10 KB (809 words) - 16:40, 17 March 2024
• Discriminant
  ...ive, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. ...(x)=a_nx^n+a_{n-1}x^{n-1}+...+a_0</math> of degree <math>n</math> with all the coefficients being real. But for polynomials of degree 4 or higher it can b
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  '''Menelaus' theorem''' deals with the [[collinearity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. ...<math>AC</math>, and <math>R</math> on the intersection of <math>PQ</math> and <math>AB</math>, then
  5 KB (804 words) - 03:01, 12 June 2023
• ARML historical results
  This is a list of historical results from the [[American Regions Mathematics League]]. * 2022: Luke Robitaille ([[Texas ARML]]) and Chris Qiu ([[Lehigh Valley ARML]])
  19 KB (2,632 words) - 14:31, 12 June 2022
• Fermat's Little Theorem
  ...ld study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. If <math>{a}</math> is an [[integer]], <math>{p}</math> is a [[prime number]] and <math>{a}</math> is not [[divisibility|divisible]] by <math>{p}</math>, the
  16 KB (2,675 words) - 10:57, 7 March 2024
• Euler's Totient Theorem
  ...are relatively prime to <math>n</math>. If <math>{a}</math> is an integer and <math>m</math> is a positive integer [[relatively prime]] to <math>a</math> ...h> is prime. For this reason it is also known as Euler's generalization or the Fermat-Euler theorem.
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  ...y on Intermediate level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]]. ...ality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says:
  7 KB (1,296 words) - 14:22, 22 October 2023
• Algebra
  ...name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic manipulation]]. ...and high school algebra. [[Group]]s, [[ring]]s, [[field]]s, [[module]]s, and [[vector space]]s are common objects of study in higher algebra.
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...gebraic number theory include the [[Birch and Swinnerton-Dyer Conjecture]] and [[Fermat's Last Theorem]]. ...ies of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].
  5 KB (849 words) - 16:14, 18 May 2021
• 1959 IMO Problems/Problem 2
  ...<math>A=\sqrt{2}</math>, (b) <math>A=1</math>, (c) <math>A=2</math>, where only non-negative real numbers are admitted for square roots? The square roots imply that <math>x\ge \frac{1}{2}</math>.
  3 KB (466 words) - 12:04, 12 April 2024
• Euler's totient function
  ...)</math> applied to a [[positive integer]] <math>n</math> is defined to be the number of positive integers less than or equal to <math>n</math> that are [ ...e_2} \cdots {p}_m^{e_m}</math>, one can compute <math>\phi(n)</math> using the formula <cmath>\phi(n)= n\left(1-\frac{1}{p_1} \right) \left(1-\frac{1}{p_2
  5 KB (898 words) - 19:12, 28 January 2024
• Probability
  ...dox]] and the [[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 i
  4 KB (588 words) - 12:47, 2 October 2022
• Contest strategies
  ...ting a good night sleep can help refresh the brain, so it can be ready for the test. ...ant but don't forget to leave room to exercise. Exercise can help refresh the brain, allowing problem solvers to prepare for tests more effectively.
  3 KB (538 words) - 13:13, 16 January 2021
• Rearrangement Inequality
  ...>{b_k}</math> is also greater than or equal to exactly <math>{i}</math> of the other members <math>B</math>). ...possible interpretation of the rearrangement inequality is that sometimes, the greedy algorithm works.
  5 KB (804 words) - 13:54, 26 January 2023
• Divisibility rules
  ...her [[integer]]s. All of these rules apply for [[Base number| base-10]] ''only'' -- other bases have their own, different versions of these rules. === Divisibility Rule for 2 and Powers of 2 ===
  8 KB (1,315 words) - 18:18, 2 March 2024
• Orthic triangle
  ...d <math>C</math>, respectively. Then, <math>\triangle DEF</math> is called the '''orthic triangle''' of <math>\triangle ABC</math>. ...ABC</math> is either acute or obtuse each carry different characteristics and must be handled separately.
  8 KB (1,408 words) - 11:54, 8 December 2021
• Triangle
  ...In [[physics]], triangles are noted for their durability, since they have only three [[vertex|vertices]] around with to distort. Triangles are split into six categories; three by their [[angle]]s and three by their side lengths.
  4 KB (628 words) - 17:17, 17 May 2018
• Greatest common divisor
  This video comprehensively explains everything related to the GCD: https://youtu.be/HboSeb_gQH8 ...o or more [[integer]]s is the largest integer that is a [[divisor]] of all the given numbers.
  2 KB (288 words) - 22:40, 26 January 2021
• Overcounting
  ...a role in incredibly important counting tools such as [[combinations]] and the [[Principle of Inclusion-Exclusion]]. ...twice. A number that is divisible by both 2 and 3 must be divisible by 6, and there are 16 such numbers. Thus, there are <math>50+33-16=\boxed{67}</math>
  4 KB (635 words) - 12:19, 2 January 2022
• Constructive counting
  ...unts the total possibilities of each step and assembles these to enumerate the full set. ...[casework]] and [[complementary counting]], constructive counting is among the most fundamental techniques in counting. Familiarity with constructive coun
  12 KB (1,896 words) - 23:55, 27 December 2023
• Jensen's Inequality
  ...unction]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge 0</math> satisfy <math>a_1+\dots+a_n=1</math>. We only prove the case where <math>F</math> is concave. The proof for the other case is similar.
  3 KB (623 words) - 13:10, 20 February 2024
• Geometric probability
  ==Geometric Counting and Probability== ...fact that probability is the ratio of the number of successful outcomes to the number of total outcomes.
  1 KB (175 words) - 23:50, 18 November 2023
• Casework
  ...totals of each part. Casework is a very general problem-solving approach, and as such has wide applicability. Here are some examples that demonstrate casework in action. Unlike the selections in this article, most problems cannot be completely solved throu
  5 KB (709 words) - 10:28, 19 February 2024
• Euclidean algorithm
  ...CD) of two elements of a [[Euclidean domain]], the most common of which is the [[nonnegative]] [[integer]]s <math>\mathbb{Z}{\geq 0}</math>, without [[fac ==Main idea and Informal Description==
  6 KB (924 words) - 21:50, 8 May 2022
• Conic section
  ...clear from their definition, the conic sections are all [[plane curve]]s, and every conic section can be described in [[Cartesian coordinates]] by a [[po All conic sections fall into the following categories:
  5 KB (891 words) - 01:14, 9 January 2023
• Generating function
  ...e generating function) is <math>c_0 + c_1 x + c_2 x^2 + \cdots </math> and the sequence is <math>c_0, c_1, c_2,\ldots</math>. This function can be described as the number of ways we can get <math>{k}</math> heads when flipping <math>n</mat
  4 KB (659 words) - 12:54, 7 March 2022
• Binomial Theorem
  ...'' states that for [[real]] or [[complex]] <math>a</math>, <math>b</math>, and [[non-negative]] [[integer]] <math>n</math>, ...h> is expanded and like terms are collected are the same as the entries in the <math>n</math>th row of [[Pascal's Triangle]].
  5 KB (935 words) - 13:11, 20 February 2024
• Prime number
  ...nteger]] <math>p>1</math> whose only positive [[divisor | divisors]] are 1 and itself. ...ime nor [[composite number|composite]] because it is its only factor among the [[natural number|natural numbers]].
  6 KB (985 words) - 12:38, 25 February 2024
• Function
  ...ule that it takes its input value, and squares it to get an output value. One can call this function <math>f</math>. ...math>A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if
  10 KB (1,761 words) - 03:16, 12 May 2023
• Complementary counting
  .... A large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ...</math>. In most instances, though, <math>A</math> is obvious from context and is committed from mention.
  8 KB (1,192 words) - 17:20, 16 June 2023
• Bijection
  ...|two-sided inverse]] exactly when it is a bijection between its [[domain]] and [[range]]. ...when dealing with arguments concerning [[infinite]] sets or in permutation and probability.
  1,016 bytes (141 words) - 03:39, 29 November 2021
• Combinatorial identity
  ...]]s <math>k</math> and <math>n</math>. Here, <math>\binom{n}{k}</math> is the binomial coefficient <math>\binom{n}{k} = {}_nC_k = C_k^n</math>. ...ys to choose <math>k-1</math> things from <math>n-1</math> things added to the number of ways to choose <math>k</math> things from <math>n-1</math> things
  12 KB (1,993 words) - 23:49, 19 April 2024
• Integer
  ...b</math> are all integers (that is, the integers are closed under addition and multiplication), but their quotient <math>a\div b</math> may or may not be For a more simple and straightforward definition, an integer is a number that is '''not''' a [[de
  2 KB (296 words) - 15:04, 5 August 2022
• Prime factorization
  ...ve integer]] has a unique prime factorization, up to changing the order of the terms. The form of a prime factorization is
  3 KB (496 words) - 22:14, 5 January 2024
• Composite number
  ...ifferent from 1 and itself. Some composite numbers are <math>4=2^2</math> and <math>12=2\times 6=3\times 4</math>. Composite numbers '''atleast have 2 di ...even [[prime number]], three is the only multiple of three that is prime, and so on.
  6 KB (350 words) - 12:58, 26 September 2023
• Circle
  ...nter]] and the distance from the center to a point on the circle is called the [[radius]]. [[Image:circle1.PNG|thumb|right|The radius and center of a circle.]]
  9 KB (1,555 words) - 20:05, 2 November 2023
• Ellipse
  ...distances from <math>P</math> to two fixed [[focus|foci]] is a constant. (The equivalence of these two definitions is a non-trivial fact.) ...to the axis of the the cone, or (in the second definition) the two foci of the ellipse coincide.
  5 KB (892 words) - 21:52, 1 May 2021
• Base numbers
  ...l]], or base-10, number system. To help explain what this means, consider the number 2746. This number can be rewritten as <math>2746_{10}=2\cdot10^3+7\ ...math>'s, the third digit tells us there are seven <math>10^2</math>'s, and the fourth digit tells us there are two <math>10^3</math>'s.
  4 KB (547 words) - 17:23, 30 December 2020
• Base numbers/Common bases
  ...is [[phinary]], which is base [[phi]]; others include "[[Fibonacci base]]" and base negative two. ...avorite among computer programmers. It has just two digits: <math>0</math> and <math>1</math>.
  2 KB (351 words) - 10:39, 1 October 2015
• Henstock-Kurzweil integral
  ...ple and elegant idea. It was developed independently by [[Ralph Henstock]] and [[Jaroslav Kurzweil]]. ...</math> is ''Generalized Riemann Integrable'' on <math>[a,b]</math> if and only if, <math>\forall\epsilon>0</math>, there exists a [[gauge]] <math>\delta:[
  2 KB (401 words) - 09:46, 31 January 2018
• Mathematical Kangaroo
  ...coring students per grade at the national level. High scoring students in the US that are ranked high within their state will also be awarded as a state ==Registration and Eligibility==
  6 KB (936 words) - 15:38, 22 February 2024
• Least common multiple
  ...gers has an [[infinite]] number of [[common multiple]]s, but only one LCM. The LCM of a set of numbers <math>\{a_1,a_2,\cdots,a_n\}</math> is conventional ...4 and 6. We would begin by listing the multiples of 4 and 6 until we find the smallest number in both lists, as shown below.
  2 KB (383 words) - 10:49, 4 September 2022
• Math Day at the Beach
  '''Math Day at the Beach''' is a [[mathematical problem solving]] festival for Southern Califo ...nd team competition. Teams represent high schools and have 6 members each. The competition takes place on a Saturday in March.
  4 KB (644 words) - 12:56, 29 March 2017
• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician Ptolemy. The inequality states that in for four points <math>A, B, C, D </math> in the plane,
  3 KB (602 words) - 09:01, 7 June 2023
• Pi
  ...act, [[transcendental number]], as proved by Lindemann in 1882) denoted by the Greek letter <math>\pi </math>. ...on]]al approximations for pi are <math>\frac{22}{7} \approx 3.14285</math> and <math>\frac{355}{113} \approx 3.1415929</math>.
  8 KB (1,469 words) - 21:11, 16 September 2022
• Phi
  ...iting, often to represent the constant <math>\frac{1+\sqrt{5}}{2}</math>. (The Greek letter [[Tau]] (<math>\tau</math>) was also used for this purpose in ...e terms of the [[Fibonacci sequence]], as well as the positive solution of the [[quadratic equation]] <math>x^2-x-1=0</math>.
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• Fibonacci sequence
  ...h equal to 1 and each subsequent term is the sum of the two preceding it. The first few terms are <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...</math>. ...> and <math>F_n=F_{n-1}+F_{n-2}</math> for <math>n \geq 3</math>. This is the simplest nontrivial example of a [[linear recursion]] with constant coeffic
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• Muirhead's Inequality
  The inequality is easier to understand given an example. Since the sequence <math>(5,1)</math> majorizes <math>(4,2)</math> (as <math>5>4, 5+1 ...erify that an inequality ''can'' be proved with AM-GM before demonstrating the full AM-GM proof.
  8 KB (1,346 words) - 12:53, 8 October 2023
• Cardinality
  ...of <math>\{1, \{2, 3\}, \{1, 2, 3\}\}</math> is 3, and the cardinality of the [[empty set]] is 0. ...is <math>|\{3, 4\}| = 2</math>. Sometimes, the notations <math>n(A)</math> and <math>\# (A)</math> are used.
  2 KB (263 words) - 00:54, 17 November 2019
• Injection
  ...or which every value in the [[range]] is the image of exactly one value in the [[domain]]. .../math> and <math>|Y|\leq|Z|</math> implies <math>|X|\leq|Z|</math> because the composition of injections is again an injection.
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• Polygon
  ...ine segment]]s. There are two types of polygons: [[convex polygon|convex]] and [[concave polygon|concave]]. ...ular]] or irregular. A polygon is regular if all sides are the same length and all angles are [[congruent]].
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• Writing scholarships
  ...rship programs in other countries. Just make that reorganization as clear and as clean as possible. Additions to this list are welcomed and encouraged. Please don't be stingy about letting students in on how they c
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• 2011 AMC 10B Problems/Problem 22
  ...posite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? .... Take a cross section of the pyramid through the apex and two points from the base that are opposite to each other. Place it in two dimensions.
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• Semiperimeter
  ...of the [[perimeter]], or <math>\frac{P}{2}</math>, where <math>P</math> is the total perimeter of a figure. It is typically denoted <math>s</math>. ...s [[inradius]] (that is, the [[radius]] of the [[circle]] [[inscribed]] in the triangle).
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• Vector
  ...t. Physicists will often refer to a vector as "a quantity with a direction and magnitude." For Euclidean geometers, a vector is essentially a directed lin ...s within angle brackets or parentheses, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
  7 KB (1,265 words) - 13:22, 14 July 2021
• Law of Cosines
  ...le. All triangles with two sides and an include angle are [[congruent]] by the Side-Angle-Side congruence postulate. ...f measure <math>A</math>, <math>B</math> and <math>C</math>, respectively, the Law of Cosines states:
  6 KB (1,003 words) - 09:11, 7 June 2023
• Trigonometric identities
  ...es. Trigonometric functions have an abundance of identities, of which only the most widely used are included in this article. The Pythagorean identities state that
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• Irrational number
  An '''irrational number''' is a [[real number]] that cannot be expressed as the [[ratio]] of two [[integer]]s. Equivalently, an irrational number, when ex ...untable]], one can say that the irrational numbers make up "almost all" of the real numbers.
  3 KB (368 words) - 19:26, 6 June 2015
• Complex plane
  ...nt two units to the right and one unit down from the origin corresponds to the complex number <math>2 - i</math>.
  892 bytes (134 words) - 16:52, 3 September 2017
• Rational number
  A '''rational number''' is a [[number]] that can be represented as the [[ratio]] of two [[integer]]s. ...ver, any rational number satisfies exactly one of the last two conditions. The same remark holds if "decimal" is replaced with any other [[number base|bas
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• Cis
  ...> is made up by the first letter of <math>\cos</math>, <math>i</math>, and the first letter of <math>\sin</math>. Once one gets used to the notation, it is almost always preferred to write <math>re^{i\theta}</math>
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• Angle bisector
  ...and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</math>: ...ined by <math>\angle A + \text{ external }\angle A = 180^\circ</math>, and the two angle bisectors are perpendicular to each other.
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• The Art and Craft of Problem Solving
  ...rally thought of as a good mathematical book for when one is finished with the Art of Problem Solving.
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• Science
  ...eories and mathematical models based on [[statistics|statistical methods]] and disciplined experimentation. ...e key distinguishing factor between science and [[pseudo-science]] is that the scientist will be skeptical of data from poorly-designed experiments.
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• Geometric sequence
  ...utive terms is constant. This constant is called the '''common ratio''' of the sequence. ...math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between consecutive terms varies.
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• Arithmetic sequence
  ...terms is constant. This constant is called the '''common difference''' of the sequence. ...> and <math>4, 12, 36, 108, \ldots</math> are not arithmetic sequences, as the difference between consecutive terms varies.
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• Fermat's Last Theorem
  ...<math>a,b,c,n</math> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...or in general for any number that is a power greater than the second to be the sum of two like powers. ''I have discovered a truly marvelous demonstration
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• 2005 USAMO Problems/Problem 4
  ...gth <math>k_i</math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...is ''stable'' if it can be placed so that all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
  7 KB (1,276 words) - 20:51, 6 January 2024
• Geometry
  ...etic]] (which eventually branched into [[number theory]] and [[algebra]]). The geometry usually studied is ...ormally outlined by the Greek [[mathematician]] [[Euclid]] in his book ''[[The Elements]]''.
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• Diophantine equation
  ...parametric form]] is used to express the relation between the variables of the equation. Diophantine equations are named for the ancient Greek/Alexandrian mathematician Diophantus.
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• Area
  ...fers to the size of the region that a two-[[dimension]]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. ...elementary means. One can find the area of even more complex regions via the use of [[calculus]].
  6 KB (1,181 words) - 22:37, 22 January 2023
• Fraction
  ...Most commonly, we consider [[rational number]]s, those fractions which are the ratio of two [[integer]]s or [[decimal]]s. ...ed an [[improper fraction]]. A fraction where the numerator is the same as the denominator such as <math>\frac{3}{3}</math> is always equal to <math>1</ma
  3 KB (432 words) - 19:34, 11 June 2020
• Pascal's triangle
  '''Pascal's triangle''' is a triangle which contains the values from the [[binomial expansion]]; its various properties play a large role in [[combi ...xample, because <math>\sum_{k=0}^{n}{{n \choose k}}=2^n</math>, the sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.
  5 KB (838 words) - 17:20, 3 January 2023
• Exponentiation
  ...th>, where <math>b</math> is the exponent (or power) and <math>a</math> is the [[base]]. ...second operation performed if a equation has [[parentheses]] or the first one performed when there is no parentheses.
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• Rational approximation
  ...e may have to use larger and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how well can we approximate <math>x</mat ...ne integer. Choosing <math>p</math> to be that integer, we immediately get the result.
  7 KB (1,290 words) - 12:18, 30 May 2019
• Trigonometry
  ...ons'''. It is a fundamental branch of mathematics, and its discovery paved the way towards countless famous results. ...nctions appear frequently on contests. These are solved by clever usage of the trigonometric functions' countless [[Trigonometric identities | identities]
  8 KB (1,217 words) - 20:15, 7 September 2023
• Euler's number
  ..._1^x \frac{1}{t} \, dt</math>. It has been shown to be both [[irrational]] and [[transcendental number|transcendental]]. ==Euler's Number as the Base of Logarithms and Exponential Functions==
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• Exponential function
  ...''[[e]]''. It is a very important function in [[analysis]], both [[real]] and [[complex]]. == General Info and Definitions ==
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• Mathematical Olympiad Summer Program
  ...m is free to participants, invitations are limited to the top finishers on the [[USAMO]]. ...The results of the USAMO and the TST are weighted equally when selecting the US IMO team.
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• Complex analysis
  ...the calculus of [[real number]]s, but, amazingly, this turns out to be not the case. There are many pathological functions of a real variable that cannot ...ath> be a [[simple closed Jordan curve]]. Then for any <math>z_0</math> in the interior of <math>\Gamma</math>, we have
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• Harmonic series
  ...monic series''' is a [[series]] whose terms involve the [[reciprocal]]s of the [[positive integer]]s. The the most basic harmonic series is the infinite sum
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• Goldbach Conjecture
  ...ry [[even integer]] greater than two is the sum of two [[prime number]]s. The conjecture has been tested up to 400,000,000,000,000. ...h's conjecture is one of the oldest unsolved problems in [[number theory]] and in all of mathematics.
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• Twin Prime Conjecture
  The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) t ...a adopted from the proof of [[Dirichlet's Theorem]]. If one can show that the sum
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• Riemann Hypothesis
  ...ots</math>. These are called the trivial zeros. This hypothesis is one of the seven [http://www.claymath.org/millennium-problems/millennium-prize-problem ...}(x)=\int_2^x \frac{1}{\ln t}\; dt</math>. Then an equivalent statement of the Riemann hypothesis is that <math>\pi(x)=\mathrm{Li}(x)+O(x^{1/2}\ln(x))</ma
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• Quadratic residues
  Let <math>a</math> and <math>m</math> be [[integer]]s, with <math>m\neq 0</math>. We say that <mat ...quadratic\ residue\ modulo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</ma
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• Power of a Point Theorem
  ...the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. There are three possibilities as displayed in the figures below.
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• International Mathematical Olympiad
  ...l international scientific competitions. Each year, countries from around the world send a team of 6 students to compete in a grueling competition. == Format of the Competition ==
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• Sam Vandervelde
  ...COUNTS]]. 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
  ...of solutions of [[polynomial]] equations by means of [[abstract algebra]], and in particular [[ring theory]]. Algebraic geometry is most easily done over ...\ldots,X_n]</math> be the [[polynomial ring]] in <math>n</math> variables, and let <math>I</math> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=
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• Prime Number Theorem
  The '''Prime Number Theorem''' (PNT) is one of the most possibly the most famous major result in all of number theory, with
  10 KB (1,729 words) - 19:52, 21 October 2023
• ...\dotsc, z_1, z_2, \dotsc, z_n</math> are [[nonnegative]] [[real number]]s and <math>\lambda_a, \lambda_b, \dotsc, \lambda_z</math> are nonnegative reals ...d <math>\lambda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]].
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• Infinite
  ...if there is a [[surjection]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. * A set is infinite if it can be put into [[bijection]] with one of its proper [[subset]]s.
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• Euler's Polyhedral Formula
  ...h> and <math>F</math> denote the number of [[vertex|vertices]], [[edge]]s, and [[face]]s, respectively. Then <math>V-E+F=2</math>. Apply Euler's Polyhedral Formula on the following polyhedra:
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• Physics forums
  ...re great places to ask questions about physics, practice teaching physics, and to socialize with other physics students. Here are a few popular online ph ...csforums.com/ Physics Forums] is one of the most popular science forums on the internet.
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• Rhombus
  ...uent (geometry) | congruent]]. It is a special type of [[parallelogram]], and its properties (aside from those properties of parallelograms) include: * Its diagonals divide the figure into 4 congruent [[triangle]]s.
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• Kite
  ==Definition and Usage== ...), and the kite (if the members of each pair are adjacent to each other). The properties of kites include:
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• Trapezoid
  A '''trapezoid''' is a cool and pretty geometric figure that lies in a plane. It is also a type of [[quadri ...apezoid. (Careful: some authors insist that a trapezoid must have exactly one pair of parallel sides.)
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• Mock AMC
  ...s, the administrator may ask other people to sign up to write problems for the contest. ...members will make Mock AMCs in any given year, but there probably will be one.
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• Permutation
  ...inear ordering) of the <math>r</math> objects. There are <math>r!</math> (the [[factorial]] of <math>r</math>) permutations of a set with <math>r</math> ...n of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.
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• Algebraic topology
  ...al space]], we can associate various algebraic objects, such as [[group]]s and [[ring]]s. ...[0,1]\to X</math> with <math>g(a,0)=p(a)</math>, <math>g(a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now de
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• Riemann zeta function
  The '''Riemann zeta function''' is a function very important in [[number theory]]. In particular, the [[Riemann Hypothesis]] is a conjecture
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• Modular arithmetic/Introduction
  ...ics of modular arithmetic while presenting a progression of more difficult and more interesting problems that are easily solved using modular arithmetic. ...lock as an example, except let's replace the <math>12</math> at the top of the clock with a <math>0</math>.
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• Uncountable
  ...[[countably infinite]]. The most common example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. ...an [[indirect proof]] here. This is one of the most famous indirect proofs and was first given by [[Georg Cantor]].
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• Modular arithmetic/Intermediate
  ...dulo'' <math>n</math>, or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...|Diophantine equations]], testing whether certain large numbers are prime, and even some problems in cryptology.
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• Right triangle
  ...use is labeled <math>c</math>. The other two sides are called the legs of the triangle. ...s, the field of [[trigonometry]] arises from the study of right triangles, and nearly all [[trigonometric identities]] can be deduced from them.
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• 2006 AIME I Problems
  ...math> \overline{CD}, AB=18, BC=21, </math> and <math> CD=14. </math> Find the perimeter of <math> ABCD. </math> ...> S </math> be the sum of the elements of <math> \mathcal{A}. </math> Find the number of possible values of <math> S. </math>
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• 2006 AIME I Problems/Problem 15
  ...th> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible value of <math> |x_1+x_2+\cdots+x_{2006}|. </math> .../math>, <math>b_2 = 0</math>, <math>b_3 = -1</math>, <math>b_4 = 0</math>, and so on until <math>b_{1962} = 0</math>, after which we let <math>b_k = b_{k
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• 2006 AIME I Problems/Problem 14
  ...t{n}\rfloor. </math> (The notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>) ...<math>BC</math>. Let <math>h</math> be the distance from <math>T</math> to the triangle <math>SBC</math> (<math>h</math> is what we want to find).
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• 2006 AIME I Problems/Problem 13
  ...r <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </mat ...are divisible by <math>2^{n-1}</math> but not by <math>2^n, \ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are
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• 2006 AIME I Problems/Problem 10
  ...e expressed in the form <math> ax=by+c, </math> where <math> a, b, </math> and <math> c </math> are positive integers whose [[greatest common divisor]] is ...on each side. The line passes through <math>\left(1,\frac 12\right)</math> and <math>\left(\frac 32,2\right)</math>, which can be easily solved to be <mat
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• 2006 AIME I Problems/Problem 8
  ...al{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</math>. Let <math>x</math> denote the common side length of the rhombi.
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• 2006 AIME I Problems/Problem 7
  ...th> B </math> is 11/5. Find the ratio of shaded region <math> D </math> to the area of shaded region <math> A. </math> Note that the apex of the angle is not on the parallel lines. Set up a [[coordinate proof]].
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• 2006 AIME I Problems/Problem 6
  ...ath> where <math> a, b, c </math> are distinct [[digit]]s. Find the sum of the elements of <math> \mathcal{S}. </math> ...h> times, and the sum of the digits, 0 through 9, is 45. So the sum of all the numbers is <math>\frac{45\times72\times111}{999}= \boxed{360} </math>.
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• 2006 AIME I Problems/Problem 4
  ...l representation of the product <math> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>. ...ssion. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.
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• 2021 JMC 10
  What is the value of <cmath>\dfrac{20}{2\cdot1} - \dfrac{2+0}{2/1}?</cmath> ...>|\pi - |e - | e - \pi|||</math> be expressed in terms of <math>\pi</math> and <math>e?</math>
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• 2006 AMC 12B Problems
  For real numbers <math>x</math> and <math>y</math>, define <math>x\spadesuit y = (x + y)(x - y)</math>. What i ...points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
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• 2006 AMC 12A Problems
  ...each. How many dollars will it cost to purchase <math>5</math> sandwiches and <math>8</math> sodas? The ratio of Mary's age to Alice's age is <math>3:5</math>. Alice is <math>30</
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• 2005 AMC 12A Problems
  Two is <math>10 \%</math> of <math>x</math> and <math>20 \%</math> of <math>y</math>. What is <math>x - y</math>? ...= 3</math> and <math>bx - 10 = - 2</math> have the same solution. What is the value of <math>b</math>?
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  ...ves <math>8</math> of the <math>25</math> problems unanswered, how many of the remaining problems must she answer correctly in order to score at least <ma ...randdaughters, and no great-granddaughters. How many of Bertha's daughters and grand-daughters have no daughters?
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  ...he sum of the first <math>2003</math> even counting numbers and the sum of the first <math>2003</math> odd counting numbers? ...rt for away games. If the total cost is &#36;2366, how many members are in the League?
  13 KB (1,955 words) - 21:06, 19 August 2023
• 2002 AMC 12A Problems
  Compute the sum of all the roots of ...3, giving an answer of 43. What would her answer have been had she worked the problem correctly?
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2000 AMC 12 Problems
  ...th>I \cdot M \cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? ...d of the second day, <math>32</math> remained. How many jellybeans were in the jar originally?
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2001 AMC 12 Problems
  ...numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then each of the resulting numbers is doubled. What is the sum of the final two
  13 KB (1,957 words) - 12:53, 24 January 2024
• 2002 AMC 12B Problems
  ...l of whose digits are distinct. The number <math>M</math> does not contain the digit What is the value of
  10 KB (1,547 words) - 04:20, 9 October 2022
• 2003 AMC 12B Problems
  Which of the following is the same as ...nd 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
  ...fth practice she made 48 free throws. How many free throws did she make at the first practice? ...hough not necessarily in that order. What is the maximum possible value of the result?
  13 KB (2,049 words) - 13:03, 19 February 2020
• 2005 AMC 12B Problems
  ...price of five for <math>2</math> dollars. They sell all the candy bars at the price of two for <math>1</math> dollar. What was their profit, in dollars? A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What i
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 9
  How many even three-digit integers have the property that their digits, all read from left to right, are in strictly in ...}{2}=10</math> and <math>\binom{7}{2}=21</math> choices for <math>a</math> and <math>b</math>. Thus there are altogether <math>3+10+21=\boxed{34}</math> s
  3 KB (409 words) - 17:10, 30 April 2024
• 2006 AMC 12B Problems/Problem 10
  ...length of the third side is 15. What is the greatest possible perimeter of the triangle? ...s length <math>7</math>, the third side has length <math>15</math>, and so the perimeter is <math>21+7+15=43 \Rightarrow \boxed{\text {(A)}}</math>.
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• 2006 AMC 12B Problems/Problem 13
  ...math> ABCD</math> is 24, and <math> \angle BAD = 60^\circ</math>. What is the area of rhombus <math> BFDE</math>? ...h>. One-third the area of <math>ABCD</math> is equal to <math>8</math>. So the answer is <math>\boxed{\text{C}}</math>.
  3 KB (447 words) - 03:49, 16 January 2021
• 2006 AMC 12B Problems/Problem 15
  ...\overline{BC}</math> are common external tangents to the circles. What is the area of hexagon <math> AOBCPD</math>? ...bisecting <math>DP</math>. The length <math>OP</math> is <math>4+2</math> and <math>HP</math> has a length of <math>2</math>, so by pythagorean's, <math>
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• 2006 AMC 12B Problems/Problem 17
  ...:6</math>. What is the probability of rolling a total of <math>7</math> on the two dice? The probability of getting an <math>x</math> on one of these dice is <math>\frac{x}{21}</math>.
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• 2006 AMC 12B Problems/Problem 18
  ...t the origin and takes a ten-step path, how many different points could be the final point? ...rity]] of exactly one coordinate. Hence after <math>10</math> steps we can only be in locations <math>(x,y)</math> where <math>x+y</math> is even. It can e
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• 2006 AMC 12B Problems/Problem 19
  ...digits just happen to be my age." Which of the following is not the age of one of Mr. Jones's children? First, The number of the plate is divisible by <math>9</math> and in the form of
  4 KB (696 words) - 09:47, 10 August 2015
• 2006 AMC 12B Problems/Problem 20
  ...x</math> be chosen at random from the interval <math>(0,1)</math>. What is the probability that Here <math>\lfloor x\rfloor</math> denotes the greatest integer that is less than or equal to <math>x</math>.
  3 KB (485 words) - 14:09, 21 May 2021
• 2006 AMC 12B Problems/Problem 21
  ...>ab\pi</math> where <math>2a</math> and <math>2b</math> are the lengths of the axes.) ...of the ellipse on which the foci lie have length <math>2a</math>, and let the other axis have length <math>2b</math>. We have
  2 KB (339 words) - 13:15, 12 July 2015
• 2006 AMC 12B Problems/Problem 22
  ...e integers and <math>m</math> is not divisible by <math>10</math>. What is the smallest possible value of <math>n</math>? The power of <math>10</math> for any factorial is given by the well-known algorithm
  5 KB (881 words) - 15:52, 23 June 2021
• 2006 AMC 12B Problems/Problem 23
  .../math> have length <math>s=\sqrt{a+b\sqrt{2}}</math>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? Using the Law of Cosines on <math>\triangle PBC</math>, we have:
  7 KB (1,169 words) - 14:04, 10 June 2022
• 2006 AMC 12A Problems/Problem 10
  {{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #10]] and [[2006 AMC 10A Problems/Problem 10|2006 AMC 10A #10]]}} The perfect squares that are less than or equal to <math>120</math> are <math>\
  1 KB (167 words) - 23:23, 16 December 2021
• 2006 AMC 12A Problems/Problem 11
  {{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #11]] and [[2006 AMC 10A Problems/Problem 11|2008 AMC 10A #11]]}} Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>?
  898 bytes (133 words) - 17:52, 3 July 2013
• 2006 AMC 12A Problems/Problem 14
  {{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #14]] and [[2006 AMC 10A Problems/Problem 22|2006 AMC 10A #22]]}} ...id with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?
  3 KB (442 words) - 03:13, 8 August 2022
• 2006 AMC 12A Problems/Problem 17
  ...math> as <math>A</math>. Segment <math>AF</math> is tangent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>? ...th>E</math> are [[collinear]] and contain a diagonal of <math>ABCD</math>. The [[Pythagorean theorem]] results in
  6 KB (958 words) - 23:29, 28 September 2023
• 2006 AMC 12A Problems/Problem 20
  {{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #20]] and [[2006 AMC 10A Problems/Problem 25|2006 AMC 10A #25]]}} ...ll choices are independent. What is the probability that after seven moves the bug will have visited every vertex exactly once?
  5 KB (908 words) - 19:23, 22 September 2022
• 2006 AMC 12A Problems/Problem 24
  The expression is simplified by expanding it and combining like terms. How many terms are in the simplified expression?
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  ...pty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> have the following two properties? ...finding. We give a somewhat more general attack, based on the solution to the following problem:
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 25
  ...re <math>x</math>, <math>y</math>, and <math>z</math> are each chosen from the set <math>\{0,1,2\}</math>. How many [[equilateral]] [[triangle]]s all have ...8 unit cubes, and then the entire cube (green triangle), giving us 9 cubes and <math>9 \cdot 8 = 72</math> equilateral triangles.
  4 KB (498 words) - 00:46, 4 August 2023
• 2005 AMC 12B Problems/Problem 2
  {{Duplicate|[[2005 AMC 12B Problems|2005 AMC 12B #2]] and [[2005 AMC 10B Problems|2005 AMC 10B #2]]}} A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What i
  1 KB (145 words) - 13:56, 14 December 2021
• 2005 AMC 12B Problems/Problem 3
  {{duplicate|[[2005 AMC 12B Problems|2005 AMC 12B #3]] and [[2005 AMC 10B Problems|2005 AMC 10B #5]]}} ...the CDs. What fraction of her money will she have left after she buys all the CDs?
  978 bytes (156 words) - 14:14, 14 December 2021
• 2005 AMC 12B Problems/Problem 6
  {{duplicate|[[2005 AMC 12B Problems|2005 AMC 12B #6]] and [[2005 AMC 10B Problems|2005 AMC 10B #10]]}} ...h> such that <math>B</math> lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>?
  2 KB (299 words) - 15:29, 5 July 2022
• 2005 AMC 12B Problems/Problem 11
  {{duplicate|[[2005 AMC 12B Problems|2005 AMC 12B #11]] and [[2005 AMC 10B Problems|2005 AMC 10B #15]]}} ...th> twenties. Two bills are drawn at random without replacement. What is the probability that their sum is &#36;<math>20</math> or more?
  4 KB (607 words) - 21:01, 20 May 2023
• 2005 AMC 12B Problems/Problem 15
  ...o of them are the same. Which of the following is '''not''' included among the eight digits? ...st value <math>(e+f+g+h)</math> can have is <math>(1+2+3+4)=10</math>, and the greatest value is <math>(6+7+8+9)=30</math>. Therefore, <math>(e+f+g+h)</ma
  2 KB (411 words) - 21:02, 21 December 2020
• 2005 AMC 12B Problems/Problem 16
  ...coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains these eight spheres? ...om the origin to any other point on the spheres is strictly smaller. Thus, the answer is <math>\boxed{\mathrm{D}}</math>.
  2 KB (364 words) - 04:54, 16 January 2023
• 2005 AMC 12B Problems/Problem 20
  Let <math>a,b,c,d,e,f,g</math> and <math>h</math> be distinct elements in the set <math>\{-7,-5,-3,-2,2,4,6,13\}.</math> What is the minimum possible value of <math>(a+b+c+d)^{2}+(e+f+g+h)^{2}?</math>
  3 KB (463 words) - 19:28, 6 November 2022
• 2005 AMC 12B Problems/Problem 22
  ...nce of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule ..._{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</math>. How many possible values are there for <math>z_{0}
  4 KB (660 words) - 17:40, 24 January 2021
• 2005 AMC 12B Problems/Problem 24
  ...</math> and <math>n</math> are relatively prime positive integers. What is the value of <math>m + n</math>? ...ween the first two is <math>2</math>, and <math>A</math> is the point with the least <math>y</math>-coordinate.
  4 KB (761 words) - 09:10, 1 August 2023
• 2005 AMC 12B Problems/Problem 25
  ...equal [[probability]]. What is the probability that no two ants arrive at the same vertex? ...ways ants can do their desired migration, and then multiple this number by the probability that each case occurs.
  10 KB (1,840 words) - 21:35, 7 September 2023
• Root (polynomial)
  ...s or any abstract [[field]] is a value <math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. ...math>; that is, counting any double roots twice, triple roots three times, and so on, there are in fact exactly <math>n</math> complex roots of <math>P(x)
  8 KB (1,427 words) - 21:37, 13 March 2022
• 2006 AMC 10A Problems
  ...lar 2 </math> each. How many dollars will it cost to purchase 5 sandwiches and 8 sodas? The ratio of Mary's age to Alice's age is <math>3:5</math>. Alice is <math>30</
  13 KB (2,028 words) - 16:32, 22 March 2022
• 2006 AMC 10A Problems/Problem 11
  Which of the following describes the graph of the equation <math>(x+y)^2=x^2+y^2</math>? ...ines} \qquad \textbf{(D) } \textrm{a\,circle} \qquad \textbf{(E) } \textrm{the\,entire\,plane} </math>
  820 bytes (123 words) - 08:05, 17 December 2021
• 2006 AMC 10A Problems/Problem 15
  ...h> meters, starting on the same radial line as Odell. How many times after the start do they pass each other? ...will meet exactly twice per lap (once at the starting point, the other at the half-way point). Thus, there are <math>\frac{30}{\frac{2\pi}{5}} \approx 23
  3 KB (532 words) - 17:49, 13 August 2023
• 2006 AMC 10A Problems/Problem 21
  How many four-digit positive integers have at least one digit that is a <math>2</math> or a <math>3</math>? ...can find this by finding the total number of <math>4</math>-digit integers and subtracting off those which do not have any <math>2</math>s or <math>3</mat
  3 KB (525 words) - 20:25, 30 April 2024
• 2006 AMC 10A Problems/Problem 24
  ...faces of a unit cube are joined to form a regular [[octahedron]]. What is the volume of this octahedron? ...o [[square pyramid]]s by cutting it along a [[plane]] [[perpendicular]] to one of its internal diagonals.
  2 KB (292 words) - 10:19, 19 December 2021
• Set
  ...hall present just a brief discussion of the most common properties of sets and operations related to them. ...misconception is that a set can have multiple indistinct elements, such as the following: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually
  11 KB (2,021 words) - 00:00, 17 July 2011
• 1991 AJHSME Problems
  Which of the "checkerboards" illustrated below CANNOT be covered exactly and completely by a whole number of non-overlapping dominoes? ...olumn and the smallest in its row? (Columns go up and down, rows go right and left.)
  17 KB (2,246 words) - 13:37, 19 February 2020
• 1990 AJHSME Problems/Problem 1
  ...ained by placing each of the six digits <math>4,5,6,7,8,9</math> in one of the six boxes in this addition problem? Let the two three-digit numbers be <math>\overline{abc}</math> and <math>\overline{def}</math>. Their [[sum]] is equal to <math>100(a+d)+10(b
  1 KB (191 words) - 17:12, 29 October 2016
• 2022 USAJMO Problems/Problem 1
  ...nite geometric sequence of integers <math>g_1,g_2,\cdots</math> satisfying the following properties? ...h that there 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
• 3D Geometry
  ...ll, on the other hand, is three-dimensional because it not only has length and width, but also depth. ...2D. We can do this by looking at certain cross-section(s) of the diagram one at a time.
  2 KB (263 words) - 12:29, 30 December 2023
• 2005 AIME II Problems/Problem 2
  ...t one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are [[relatively prime]] [[integer]]s, find <math> m+n. </ ...st and second person all get a roll of each type, since then the rolls for the third person are determined.
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 4
  Find the number of [[positive integer]]s that are divisors of at least one of <math> 10^{10},15^7,18^{11}. </math> ...which divide two or more of our three numbers. Thus, we must subtract off the divisors of their pair-wise [[greatest common divisor]]s.
  3 KB (377 words) - 18:36, 1 January 2024
• 2005 AIME II Problems/Problem 13
  ...distinct integer solutions <math> n_1 </math> and <math> n_2, </math> find the product <math> n_1\cdot n_2. </math> ...ne <math>Q(x)=P(x)-x+7</math>, noting that 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,
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems/Problem 6
  ...and number 6 retain their original positions. Find the number of cards in the magical stack in which card number 131 retains its original position. ...in front of card 131. This suggests that <math>n = 131 + 65 = 196</math>; the total number of cards is <math>196 \cdot 2 = \boxed{392}</math>.
  2 KB (384 words) - 00:31, 26 July 2018
• 2005 AIME II Problems
  ...ber of possible sets of 6 cards that can be drawn from the deck is 6 times the number of possible sets of 3 cards that can be drawn. Find <math> n. </math ...t one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are relatively prime integers, find <math> m+n. </math>
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 9
  ...t)^n = \cos nt + i \sin nt</math> for all [[real number]]s <math>t</math> and all [[integer]]s <math>n</math>. So, we'd like to somehow convert our give ...math>, it must certainly hold for <math>t = \frac{\pi}2 - u</math>. Thus, the question is equivalent to asking for how many [[positive integer]]s <math>n
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ...th> are [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...<math>AG = BG</math>). Then <math>\tan \angle EOG = \frac{x}{450}</math>, and <math>\tan \angle FOG = \frac{y}{450}</math>.
  13 KB (2,080 words) - 21:20, 11 December 2022
• Problem solving
  ...permeate the studies of [[mathematics]], [[science]], and [[technology]]. The human processes involved in problem solving are often studied by [[cognitiv ...g something. By the end of MOP, I had learned a somewhat unsettling truth. The others knew fewer tricks than I did, not more. They didn’t even have form
  6 KB (1,039 words) - 17:43, 30 July 2018
• 2005 AIME I Problems
  ...he region inside circle <math> C </math> and outside of the six circles in the ring. Find <math> \lfloor K \rfloor. </math> ...h> For how many values of <math> k </math> does <math> S_k </math> contain the term 2005?
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 4
  ...mation with 7 more rows than columns, there are no members left over. Find the maximum number of members this band can have. ...4 = 17^2 + 5</math>, so this number works and no larger number can. Thus, the answer is <math>\boxed{294}</math>.
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  ...face to face. Find the number of possible distinguishable arrangements of the 8 coins. ...parts to this problem: one is the color (gold vs silver), and the other is the orientation.
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 9
  ...> r </math> are distinct [[prime number | prime]]s and <math> a,b, </math> and <math> c </math> are [[positive integer]]s, find <math> a+b+c+p+q+r. </math We can consider the orientation of each of the individual cubes independently.
  4 KB (600 words) - 21:44, 20 November 2023
• 2005 AIME I Problems/Problem 10
  ...| median]] to side <math> BC </math> has [[slope]] <math> -5. </math> Find the largest possible value of <math> p+q. </math> ...math>-5 = \frac{q - \frac{39}{2}}{p - \frac{35}{2}}</math>. Cross multiply and simplify to yield that <math>-5p + \frac{35 \cdot 5}{2} = q - \frac{39}{2}<
  5 KB (852 words) - 21:23, 4 October 2023
• 2005 AIME I Problems/Problem 11
  ...ained in a [[square (geometry) | square]] whose sides have length 8. Given the maximum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math ...t is arranged so that the center of the semicircle lies on one diagonal of the square.
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME I Problems/Problem 13
  A particle moves in the [[Cartesian plane]] according to the following rules: # From any lattice point <math> (a,b), </math> the particle may only move to <math> (a+1,b), (a,b+1), </math> or <math>(a+1,b+1). </math>
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME II Problems/Problem 15
  ..._1. </math> Given that <math> m^2=\frac pq, </math> where <math> p </math> and <math> q </math> are relatively prime integers, find <math> p+q. </math> Rewrite the given equations as <math>(x+5)^2 + (y-12)^2 = 256</math> and <math>(x-5)^2 + (y-12)^2 = 16</math>.
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2004 AIME I Problems/Problem 15
  ...ve integers <math>x</math> such that <math>d(x)=20</math>. Find the sum of the distinct prime factors of <math>m</math>. ...of <math>n</math>. Using [[complementary counting]], we see that there are only <math>2^9 - 1</math> ways.
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 13
  ...t <math> a_1 + a_2 + a_3 + a_4 + a_5 = m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m+n. < ...o work with directly, but there is one obvious transformation to make: sum the [[geometric series]]:
  2 KB (298 words) - 20:02, 4 July 2013
• 2004 AIME I Problems/Problem 12
  ...integers, find <math> m+n. </math> The notation <math> [z] </math> denotes the [[floor function|greatest integer]] that is less than or equal to <math> z. ...quence]] <math>\frac{1}{2} , \frac{1}{8}, \frac{1}{32}, \cdots</math>, and the <math>y</math> interval is given by <math>\frac{4}{5} , \frac{4}{125}, \fra
  2 KB (303 words) - 22:28, 11 September 2020
• 2004 AIME I Problems/Problem 11
  ...h> k</math>. Given that <math> k=\frac m n, </math> where <math> m </math> and <math> n </math> are [[relatively prime]] [[positive integer]]s, find <math ...the cone. Using the [[Pythagorean Theorem]], we get <math>\ell = 5</math> and <math>A = 24\pi</math>.
  5 KB (839 words) - 22:12, 16 December 2015
• 2004 AIME I Problems/Problem 10
  ...h diagonal <math> AC </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. Find <math> m + n. ...h>. We want to find the area of the [[right triangle]] with [[hypotenuse]] one unit away from <math>\overline{AC}</math>. Let this triangle be <math>A'B'C
  5 KB (836 words) - 07:53, 15 October 2023
• 2004 AIME I Problems/Problem 9
  ...ath> can be written in the form <math> m/n, </math> where <math> m </math> and <math> n</math> are [[relatively prime]] positive integers. Find <math> m+n ...so <math>U_1</math> is similar to <math>ABC</math>. Thus <math>U_1</math>, and hence <math>U_2</math>, are <math>3-4-5\,\triangle</math>s.
  4 KB (618 words) - 20:01, 4 July 2013
• 2004 AIME I Problems/Problem 4
  ...e line segments in set <math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k</math>. Find <math> 100k</math>. ...h>(0,0)</math> to <math>\left(\frac{x}{2},\frac{y}{2}\right)</math>. Using the [[distance formula]] we see that <math>d=\sqrt{\left(\frac{x}{2}\right)^2+\
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 6
  ...math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 have four distinct digits? ...ses: one in which zero is one of the digits and one in which it is not. In the latter case, suppose we pick digits
  3 KB (562 words) - 18:12, 4 March 2022
• 2004 AIME I Problems/Problem 7
  ...C </math> be the [[coefficient]] of <math> x^2 </math> in the expansion of the product <math> (1 - x)(1 + 2x)(1 - 3x)\cdots(1 + 14x)(1 - 15x). </math> Fin It is clear that the coefficient of <math>x</math> in <math>P(x)</math> is <math>-1 + 2 - 3 + \l
  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> line segments <math> P_1P_2, P_2P_3,\ldots, P_nP_ * the points <math> P_1, P_2,\ldots, P_n </math> are coplanar and no three of them are collinear,
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  ...egers in decreasing order when read from left to right. What is the sum of the possible remainders when <math> n </math> is divided by 37? ...lue of the difference between the greatest element of <math> A </math> and the greatest element of <math> B </math> is 99. Find <math> m. </math>
  9 KB (1,434 words) - 13:34, 29 December 2021
• 2004 AIME II Problems/Problem 15
  ...the square that was originally the <math>942</math>nd square counting from the left? ..._{n, k}</math> the 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
  ...f <math> n </math> is it possible to insert <math> + </math> signs so that the resulting expression has value <math> 7000 </math>? ...math>, <math>a + 11b + 111c = 1000</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
  ...ct square | square]] of any [[prime number | prime]], and <math> k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> Let the radius of the center circle be <math>r</math> and its center be denoted as <math>O</math>.
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 11
  ...from the vertex is <math>375\sqrt{2}.</math> Find the least distance that the fly could have crawled. ...counterclockwise. The circumference of the base is <math>C=1200\pi</math>. The sector's radius (cone's sweep) is <math>R=\sqrt{r^2+h^2}=\sqrt{600^2+(200\s
  2 KB (268 words) - 22:20, 23 March 2023
• 2004 AIME II Problems/Problem 10
  ...divisible by <math>9</math> is <math> p/q, </math> where <math> p </math> and <math> q </math> are relatively prime positive integers. Find <math> p+q. < ...se <math>2^{40}</math> has 41) with leading zeroes if necessary. Therefore the number of sets where there are exactly two 1’s in this binary representat
  7 KB (1,091 words) - 18:41, 4 January 2024
• 2004 AIME II Problems/Problem 8
  ...[[prime factorization]] of 2004 is <math>2^2\cdot 3\cdot 167</math>. Thus the prime factorization of <math>2004^{2004}</math> is <math>2^{4008}\cdot 3^{2 ...exponents]] of the prime factors in its prime factorization. For example, the number of divisors of <math>2004=2^2\cdot 3^1\cdot 167^1</math> is <math>(2
  2 KB (353 words) - 18:08, 25 November 2023
• 2004 AIME II Problems/Problem 6
  ...the [[ratio]] <math> 3: 2: 1, </math>what is the least possible total for the number of bananas? ...y got <math>\frac{1}{8}b_1 + \frac{1}{4}b_2 + \frac{11}{24}b_3</math>, and the third monkey got <math>\frac{1}{8}b_1 + \frac{3}{8}b_2 + \frac{1}{12}b_3</m
  6 KB (950 words) - 14:18, 15 January 2024
• 2004 AIME II Problems/Problem 5
  ...must be hired after three-quarters of the work has been completed so that the entire project can be completed on schedule or before? ...must travel more than <math>1565</math> mph to make it on time. Therefore the company needs to add <math>1566-800 = \boxed{766}</math> more workers.
  4 KB (592 words) - 19:02, 26 September 2020
• 2004 AIME II Problems/Problem 4
  ...numbers with only a single digit. We have nine of these for each length, and four lengths, so 36 total numbers. ...'s count those with two distinct digits. We handle the cases "0 included" and "0 not included" separately.
  3 KB (508 words) - 01:16, 19 January 2024
• 2004 AIME II Problems/Problem 3
  ...e visible, exactly <math>231</math> of the 1-cm cubes cannot be seen. Find the smallest possible value of <math> N. </math> ...Then the extra layer makes the entire block <math>4\times8\times12</math>, and <math>N= \boxed{384}</math>.
  2 KB (377 words) - 11:53, 10 March 2014
• 2004 AIME II Problems/Problem 2
  ...tion, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relatively prime]] [[positive integer]]s, find <math ...ot \frac{28}{153} = \frac{14}{323}</math>. The same calculation works for the blue candies.
  2 KB (330 words) - 13:42, 1 January 2015
• 2004 AIME II Problems
  ...f </math> is divisible by the square of any prime. Find the remainder when the product <math> abcdef </math> is divided by 1000. ...tion, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers, find <math> m+n. <
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1983 AIME Problems
  ...itive number such that <math>\log_xw=24</math>, <math>\log_y w = 40</math> and <math>\log_{xyz}w=12</math>. Find <math>\log_zw</math>. ...ine the [[minimum]] value taken by <math>f(x)</math> for <math>x</math> in the [[interval]] <math>p \leq x\leq15</math>.
  7 KB (1,104 words) - 12:53, 6 July 2022
• 1984 AIME Problems
  ...th>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ldots+a_{98}=137</math>. The integer <math>n</math> is the smallest positive multiple of <math>15</math> such that every digit of <mat
  6 KB (933 words) - 01:15, 19 June 2022
• 1986 AIME Problems
  What is the sum of the solutions to the equation <math>\sqrt[4]{x} = \frac{12}{7 - \sqrt[4]{x}}</math>? Evaluate the product <cmath>\left(\sqrt{5}+\sqrt{6}+\sqrt{7}\right)\left(\sqrt{5}+\sqrt{
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  ...dition <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to <math>1 ...sphere of radius 19 with center <math>(-2,-10,5),</math> and the other on the sphere of radius 87 with center <math>(12,8,-16)</math>?
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...locks are redesigned so that sets of as many as nine buttons or as few as one button could serve as combinations. How many additional combinations would ...er <math>k</math>, let <math>f_1(k)</math> denote the square of the sum of the digits of <math>k</math>. For <math>n \ge 2</math>, let <math>f_n(k) = f_1
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  ...convex polygons of three or more sides can be drawn using some (or all) of the ten points as vertices? Suppose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1990 AIME Problems
  ...are | square]] nor the [[perfect cube | cube]] of a positive integer. Find the 500th term of this sequence. Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>.
  6 KB (870 words) - 10:14, 19 June 2021
• 1992 AIME Problems
  Find the sum of all positive rational numbers that are less than 10 and that have denominator 30 when written in lowest terms. ...ascending if, in its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  How many even integers between 4000 and 7000 have four different digits? ...of this tour, how many miles was he from his starting point at the end of the <math>40^{\mbox{th}}_{}</math> day?
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  ...less than a perfect square. What is the remainder when the 1994th term of the sequence is divided by 1000? ...be written in the form <math>m + \sqrt{n}\,</math>, where <math>m\,</math> and <math>n\,</math> are integers. Find <math>m + n\,</math>.
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1995 AIME Problems
  ...}</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</ma Find the last three digits of the product of the positive roots of
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1996 AIME Problems
  ...n any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Find <math>x</math>. ...many positive integers <math>n</math> is it true that <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
  ...egers between 1 and 1000, inclusive, can be expressed as the difference of the squares of two nonnegative integers? ...r</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1998 AIME Problems
  ...multiple]] of the positive integers <math>6^6</math> and <math>8^8</math>, and <math>k</math>? ...x,y)</math> of positive integers that satisfy <math>x \le 2y \le 60</math> and <math>y \le 2x \le 60</math>.
  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 a ...ygons. The slope of the line is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers. Find <math>m+n.<
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  ...of any two positive integers, at least one of these two integers contains the digit <math>0</math>. ...let <math>E</math> be the reflection of <math>D</math> across the y-axis. The area of pentagon <math>ABCDE</math> is <math>451</math>. Find <math>u + v</
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  Find the sum of all positive two-digit integers that are divisible by each of their ...h> is <math>27</math> more than the mean of <math>\mathcal{S}</math>. Find the mean of <math>\mathcal{S}</math>.
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  ...-right as it does right-to-left) is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math ...n as <math>\frac{1}{2}\left(\sqrt{p}-q\right)</math>, where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>.
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  where <math> k </math> and <math> n </math> are positive integers and <math> n </math> is as large as possible, find <math> k + n. </math> ...radius 100 can be expressed as <math> m/n, </math> where <math> m </math> and <math> n </math> are relatively prime positive integers. Find <math> m + n.
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  The number can be written as <math>\frac mn</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</m
  6 KB (947 words) - 21:11, 19 February 2019
• 2002 AIME II Problems
  ...h>y</math> is the number formed by reversing the digits of <math>x</math>; and <math>z=|x-y|</math>. How many distinct values of <math>z</math> are possib ...,8,1)</math>, and <math>R=(11,3,9)</math>. What is the [[surface area]] of the cube?
  7 KB (1,177 words) - 15:42, 11 August 2023
• 2003 AIME II Problems
  ...times their sum, and one of the integers is the sum of the other two. Find the sum of all possible values of <math>N</math>. ...he greatest integer multiple of 8, whose digits are all different. What is the remainder when <math>N</math> is divided by 1000?
  7 KB (1,127 words) - 09:02, 11 July 2023
• 1983 AIME Problems/Problem 1
  ...ve number such that <math>\log_x w = 24</math>, <math>\log_y w = 40</math> and <math>\log_{xyz} w = 12</math>. Find <math>\log_z w</math>. ...hm]]ic notation doesn't tell us much, so we'll first convert everything to the equivalent exponential forms.
  4 KB (642 words) - 03:14, 17 August 2022
• 1983 AIME Problems/Problem 5
  ...th> is <math>7</math> and the sum of the cubes is <math>10</math>. What is the largest real value that <math>x + y</math> can have? One way to solve this problem is by [[substitution]]. We have
  4 KB (672 words) - 10:17, 17 March 2023
• 1983 AIME Problems/Problem 7
  ...is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...probability that none of the three knights are sitting next to each other and subtracting it from <math>1</math>.
  9 KB (1,392 words) - 20:37, 19 January 2024
• 1983 AIME Problems/Problem 10
  The numbers <math>1447</math>, <math>1005</math> and <math>1231</math> have something in common: each is a <math>4</math>-digit ...ther three digits can be <math>1</math>. This means the possible forms for the number are
  5 KB (855 words) - 20:26, 14 January 2023
• 1983 AIME Problems/Problem 11
  ...<math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? ...is <math>3\sqrt{2}</math>. We apply the Pythagorean Theorem to deduce that the height is <math>6</math>.
  5 KB (865 words) - 21:11, 6 February 2023
• 1983 AIME Problems/Problem 13
  ...sums for <math>n=7</math>.<!-- don't remove the following tag, for PoTW on the Wiki front page--></onlyinclude> ...hen we take an alternating sum, each number in <math>S</math> ends up with the opposite sign of each corresponding element of <math>S\cup \{7\}</math>.
  5 KB (894 words) - 22:02, 5 April 2024
• 1983 AIME Problems/Problem 14
  ...<math>QP</math> and <math>PR</math> have equal length. Find the square of the length of <math>QP</math>. ...the Midpoint Theorem to the trapezoid made by dropping perpendiculars from the centers onto <math>QR</math>.
  13 KB (2,149 words) - 18:44, 5 February 2024
• 1983 AIME Problems/Problem 15
  ...expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? ...<math>P</math> are (1) on opposite sides of <math>\ell</math>. and (2) at the same distance from <math>\ell</math>.
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 1
  ..._3\ldots</math> is an [[arithmetic progression]] with common difference 1, and <math>a_1+a_2+a_3+\ldots+a_{98}=137</math>. ...the value of <math>a_1</math>, then use that to calculate <math>a_2</math> and sum another arithmetic series to get our answer.
  4 KB (576 words) - 21:03, 23 December 2023
• 1984 AIME Problems/Problem 3
  ...]s <math>4</math>, <math>9</math>, and <math>49</math>, respectively. Find the area of <math>\triangle ABC</math>. ...responding side on the large triangle is <math>12x</math>, and the area of the triangle is <math>12^2 = \boxed{144}</math>.
  4 KB (726 words) - 13:39, 13 August 2023
• 1984 AIME Problems/Problem 4
  ...ed, the average of the remaining numbers drops to <math>55</math>. What is the largest number that can appear in <math>S</math>? ...th>S</math> has <math>n</math> numbers other than the <math>68,</math> and the sum of these numbers is <math>s.</math>
  2 KB (319 words) - 03:38, 16 January 2023
• 1984 AIME Problems/Problem 6
  ...e three circles to the other side of it. What is the [[absolute value]] of the [[slope]] of this line? ...s [[diameter]] and splits the circle into two equal areas. For the rest of the problem, we do not have to worry about that circle.
  6 KB (1,022 words) - 19:29, 22 January 2024
• 1984 AIME Problems/Problem 8
  ...circ</math> and <math>180^\circ</math> in the [[complex plane]]. Determine the degree measure of <math>\theta</math>. ...nth degree [[roots of unity]]). Now we simply need to find the root within the desired range that satisfies our original equation <math>x^6 + x^3 + 1 = 0<
  3 KB (430 words) - 19:05, 7 February 2023
• 1984 AIME Problems/Problem 10
  ...here <math>c</math> is the number of correct answers and <math>w</math> is the number of wrong answers. (Students are not penalized for problems left unan Let Mary's score, number correct, and number wrong be <math>s,c,w</math> respectively. Then
  7 KB (1,163 words) - 23:53, 28 March 2022
• 1984 AIME Problems/Problem 11
  ...in lowest terms be the [[probability]] that no two birch trees are next to one another. Find <math>m+n</math>. ...you also multiply the denominator by the number of ways to arrange the oak and maple trees, making them cancel out.)
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1984 AIME Problems/Problem 14
  What is the largest even integer that cannot be written as the sum of two odd composite numbers? ...e that the numbers <math>9</math>, <math>15</math>, <math>21</math>, ... , and in general <math>9 + 6n</math> for nonnegative <math>n</math> are odd compo
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1985 AIME Problems/Problem 14
  ...gainst the other nine of the ten). What was the total number of players in the tournament? ...e of 9. Thus we must have <math>n > 10</math>, so <math>n = 15</math> and the answer is <math>15 + 10 = \boxed{25}</math>.
  5 KB (772 words) - 22:14, 18 June 2020
• 1985 AIME Problems/Problem 12
  ...tex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of <math>n</math>. ...ive integers <math>k,</math> 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>
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 10
  How many of the first 1000 [[positive integer]]s can be expressed in the form ...>x</math> is a [[real number]], and <math>\lfloor z \rfloor</math> denotes the greatest [[integer]] less than or equal to <math>z</math>?
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 6
  ...ior point. The [[area]]s of four of these triangles are as indicated. Find the area of triangle <math>ABC</math>. ...ases. Moreover, the two pairs of bases are actually the same, and thus in the same ratio. As a result, we have:
  5 KB (789 words) - 03:09, 23 January 2023
• 1986 AIME Problems/Problem 15
  ...ng the lines <math>y=x+3</math> and <math>y=2x+4</math> respectively, find the area of triangle <math>ABC</math>. ...of the respective medians; in other words, <math>\tan \theta_1 = 1</math>, and <math>\tan \theta_2 =2</math>.
  11 KB (1,722 words) - 09:49, 13 September 2023
• 1986 AIME Problems/Problem 13
  ...will contain exactly two <tt>HH</tt>, three <tt>HT</tt>, four <tt>TH</tt>, and five <tt>TT</tt> subsequences? ...l ignore them for now. However, adding <tt>HT</tt> or <tt>TH</tt> switches the last coin. <tt>H</tt> switches to <tt>T</tt> three times, but <tt>T</tt> sw
  3 KB (445 words) - 19:44, 8 January 2023
• 1986 AIME Problems/Problem 12
  ...ppose no two disjoint subsets of <math>S</math> have the same sum. What is the largest sum a set <math>S</math> with these properties can have? By using the greedy algorithm, we obtain <math>\boxed{061}</math>, with <math>S=\{ 15,14
  2 KB (364 words) - 19:41, 1 September 2020
• 1986 AIME Problems/Problem 11
  ...here <math>y=x+1</math> and the <math>a_i</math>'s are [[constant]]s. Find the value of <math>a_2</math>. ...he denominator reduces the degrees in the numerator by <math>1</math>). By the [[Binomial Theorem]], this is <math>(-1) \cdot (-1)^{15}{18 \choose 3} = \b
  6 KB (872 words) - 16:51, 9 June 2023
• 1986 AIME Problems/Problem 10
  ...ify the original number, <math>(abc)</math>. Play the role of the magician and determine <math>(abc)</math> if <math>N= 3194</math>. Let <math>m</math> be the number <math>100a+10b+c</math>. Observe that <math>3194+m=222(a+b+c)</math>
  3 KB (565 words) - 16:51, 1 October 2023
• 1986 AIME Problems/Problem 9
  ...[[segment]]s are drawn through <math>P</math> [[parallel]] to the sides of the [[triangle]]. If these three segments are of an equal length <math>d</math> ...e ABC \sim \triangle DPD' \sim \triangle PEE' \sim \triangle F'PF</math>). The remaining three sections are [[parallelogram]]s.
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1986 AIME Problems/Problem 7
  ...s which are [[exponent|powers]] of 3 or sums of distinct powers of 3. Find the <math>100^{\mbox{th}}</math> term of this sequence. ...e get <math>1100100</math>. However, we must change it back to base 10 for the answer, which is <math>3^6 + 3^5 + 3^2 = 729 + 243 + 9 = \boxed {981}</math
  5 KB (866 words) - 00:00, 22 December 2022
• 1986 AIME Problems/Problem 6
  ...ing in an incorrect sum of <math>1986_{}^{}</math>. What was the number of the page that was added twice? ...arge, disregard it for now and solve <math>\frac{n(n+1)}{2} = 1986</math>. The positive root for <math>n \approx \sqrt{3972} \approx 63</math>. Quickly te
  2 KB (336 words) - 14:13, 6 September 2020
• 1986 AIME Problems/Problem 4
  ...2</math>, <math>x_3</math>, <math>x_4</math>, and <math>x_5</math> satisfy the system of equations below. ...fourth given equation gives <math>x_4 = 17</math> and subtracting it from the fifth given equation gives <math>x_5 = 65</math>, so our answer is <math>3\
  1 KB (212 words) - 16:25, 17 November 2019
• 1986 AIME Problems/Problem 3
  If <math>\tan x+\tan y=25</math> and <math>\cot x + \cot y=30</math>, what is <math>\tan(x+y)</math>? Since <math>\cot</math> is the reciprocal function of <math>\tan</math>:
  3 KB (545 words) - 23:44, 12 October 2023
• 1987 AIME Problems/Problem 13
  ..._n</math>, with its current predecessor and exchanging them if and only if the last term is smaller. ..., 9, 8, 7 is transformed into the sequence 1, 8, 7, 9 by one bubble pass. The numbers compared at each step are underlined.
  3 KB (514 words) - 21:27, 31 December 2023
• 1987 AIME Problems/Problem 11
  ...le value of <math>k</math> for which <math>3^{11}</math> is expressible as the sum of <math>k</math> consecutive [[positive integer]]s. Let us write down one such sum, with <math>m</math> terms and first term <math>n + 1</math>:
  3 KB (418 words) - 18:30, 20 January 2024
• 1987 AIME Problems/Problem 10
  ...it time) is three times Bob's walking speed, how many steps are visible on the escalator at a given time? (Assume that this value is constant.) ...steps be <math>x</math>, the speed of the escalator be <math>e</math> and the speed of Bob be <math>b</math>.
  7 KB (1,187 words) - 16:21, 27 January 2024
• 1987 AIME Problems/Problem 8
  What is the largest positive integer <math>n</math> for which there is a unique integer Multiplying out all of the [[denominator]]s, we get:
  2 KB (393 words) - 16:59, 16 December 2020
• 1987 AIME Problems/Problem 7
  ...e integers for which <math>[a,b] = 1000</math>, <math>[b,c] = 2000</math>, and <math>[c,a] = 2000</math>. ...th>(0, 3, 4), (1, 3, 4), (2, 3, 4), (3, 3, 4), (3, 2, 4), (3, 1, 4)</math> and <math>(3, 0, 4)</math>.
  3 KB (547 words) - 22:54, 4 April 2016
• 1987 AIME Problems/Problem 6
  ...Find the [[length]] of <math>AB</math> (in cm) if <math>BC = 19</math> cm and <math>PQ = 87</math> cm. ...{4}(XY + 87)</math>. This number is also equal to one quarter the area of the entire rectangle, which is <math>\frac{19\cdot AB}{4}</math>, so we have <m
  3 KB (530 words) - 07:46, 1 June 2018
• 1987 AIME Problems/Problem 3
  ...s equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? ...proper divisors of <math>n</math>. A number <math>n</math> is ''nice'' in one of two instances:
  3 KB (511 words) - 09:29, 9 January 2023
• 1987 AIME Problems/Problem 2
  ...of [[radius]] 19 with [[center]] <math>(-2,-10,5)</math> and the other on the sphere of radius 87 with center <math>(12,8,-16)</math>? ...ssible distance would be the sum of the two radii and the distance between the two centers, making it <math>19 + 87 + 31 = \boxed{137}</math>.
  697 bytes (99 words) - 18:46, 14 February 2014
• 1987 AIME Problems/Problem 1
  ...tion]] <math>m+n</math> in base <math>10</math> requires no carrying. Find the number of simple ordered pairs of non-negative integers that sum to <math>1 ...in <math>1492</math> (the values of <math>n</math> are then fixed). Thus, the number of [[ordered pair]]s will be <math>(1 + 1)(4 + 1)(9 + 1)(2 + 1) = 2\
  1 KB (191 words) - 14:42, 17 September 2016
• 1988 AIME Problems/Problem 15
  ...are nine letters to be typed during the day, and the boss delivers them in the order <math>1, 2, 3, 4, 5, 6, 7, 8, 9</math>. ...orders are possible? (That there are no letters left to be typed is one of the possibilities.)
  7 KB (1,186 words) - 10:16, 4 June 2023
• 1988 AIME Problems/Problem 14
  ...ath>y = 2x</math>. Let the [[equation]] of <math>C^*</math> be written in the form Find the product <math>bc</math>.
  4 KB (700 words) - 17:21, 3 May 2021
• 1988 AIME Problems/Problem 11
  ...line <math>L</math> in the [[complex plane]] is called a mean [[line]] for the [[point]]s <math>w_1, w_2, \dots, w_n</math> if <math>L</math> contains poi .../math>, there is a unique mean line with <math>y</math>-intercept 3. Find the [[slope]] of this mean line.
  2 KB (422 words) - 00:22, 6 September 2020
• 1988 AIME Problems/Problem 10
  ...any [[segment]]s joining vertices of the polyhedron lie in the interior of the polyhedron rather than along an [[edge]] or a [[face]]? The polyhedron described looks like this, a truncated cuboctahedron.
  5 KB (811 words) - 19:10, 25 January 2021
• 1988 AIME Problems/Problem 8
  ...math>, defined on the set of ordered pairs of positive integers, satisfies the following properties: <cmath> f(x, x) = x,\; f(x, y) = f(y, x), {\rm \ and\ } (x+y)f(x, y) = yf(x, x+y). </cmath>
  4 KB (538 words) - 13:24, 12 October 2021
• 1988 AIME Problems/Problem 6
  ...c sequences. Find the number that must occupy the vacant square marked by the asterisk (*). ...ates of the square at the bottom left be <math>(0,0)</math>, the square to the right <math>(1,0)</math>, etc.
  5 KB (878 words) - 23:06, 20 November 2023
• 1988 AIME Problems/Problem 4
  What is the smallest possible value of <math>n</math>? ...\frac{19}{20} + \dots - \frac{19}{20}\right| = 19 + 0 = 19</math>, and so the equation can hold for <math>n = \boxed{020}</math>.
  2 KB (394 words) - 10:21, 27 January 2024
• 1988 AIME Problems/Problem 3
  ...he fact that <math>k \log_a x = \log_a x^k</math>. On the 3rd step, we use the [[change of base formula]], which states <math>\log_a b = \frac{\log_k b}{\ ...rt this expression into one which has a uniform base. Let's scale down all the powers of 8 to 2.
  3 KB (481 words) - 21:52, 18 November 2020
• 1989 AIME Problems/Problem 15
  .../math>, <math>PD=6</math>, <math>PE=3</math>, and <math>CF=20</math>, find the area of <math>\triangle ABC</math>. ...if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</math> intersects line <math>YZ</math> at point <math>L</math
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 14
  ...math> using the integers <math>0,1,2,\ldots,n^2</math> as digits. That is, the equation ...eger <math>m</math> and digits <math>a_0,a_1,\ldots,a_m</math> chosen from the set <math>\{0,1,2,\ldots,n^2\}</math>, with <math>a_m\ne 0</math>. We write
  2 KB (408 words) - 17:28, 16 September 2023
• 1989 AIME Problems/Problem 13
  ...bers of <math>S</math> differ by <math>4</math> or <math>7</math>. What is the largest number of [[element]]s <math>S</math> can have? ...which rules out <math>5,8</math>. Now we can take at most one from each of the pairs: <math>[2,9]</math>, <math>[3,7]</math>, <math>[4,11]</math>, <math>[
  2 KB (274 words) - 04:07, 17 December 2023
• 1989 AIME Problems/Problem 11
  ...\rfloor</math>? (For real <math>x</math>, <math>\lfloor x\rfloor</math> is the [[floor function|greatest integer]] less than or equal to <math>x</math>.) .../math> times. We let the arithmetic mean be <math>M</math>, and the sum of the numbers <math>\neq x</math> be <math>S</math>. Then
  5 KB (851 words) - 18:01, 28 December 2022
• 1989 AIME Problems/Problem 9
  ...ositive integer such that <cmath>133^5+110^5+84^5+27^5=n^{5}.</cmath> Find the value of <math>n</math>. Taking the given equation modulo <math>2,3,</math> and <math>5,</math> respectively, we have
  6 KB (874 words) - 15:50, 20 January 2024
• 1989 AIME Problems/Problem 8
  Find the value of <math>16x_1+25x_2+36x_3+49x_4+64x_5+81x_6+100x_7</math>. Note that each given equation is of the form <cmath>f(k)=k^2x_1+(k+1)^2x_2+(k+2)^2x_3+(k+3)^2x_4+(k+4)^2x_5+(k+5)^2
  8 KB (1,146 words) - 04:15, 20 November 2023
• 1989 AIME Problems/Problem 7
  ...mbers <math>36</math>, <math>300</math>, and <math>596</math>, one obtains the squares of three consecutive terms of an arithmetic series. Find <math>k</m Call the terms of the [[arithmetic progression]] <math>a,\ a + d,\ a + 2d</math>, making their sq
  2 KB (239 words) - 16:09, 2 June 2023
• 1989 AIME Problems/Problem 5
  ...twice. Let <math>\frac ij</math>, in lowest terms, be the probability that the coin comes up heads in exactly <math>3</math> out of <math>5</math> flips. ...ut terms, we get <math>1 - h = 2h</math>, so <math>h = \frac{1}{3}</math>. The answer we are looking for is <math>{5\choose3}(h)^3(1-h)^2 = 10\left(\frac{
  2 KB (258 words) - 00:07, 25 June 2023
• 1989 AIME Problems/Problem 2
  ...nvex polygon]]s of three or more sides can be drawn using some (or all) of the ten points as [[vertex | vertices]]? ...> have 1 member and <math>{10 \choose 2} = 45</math> have 2 members. Thus the answer is <math>1024 - 1 - 10 - 45 = \boxed{968}</math>.
  911 bytes (135 words) - 08:30, 27 October 2018
• 1989 AIME Problems/Problem 1
  Note that the four numbers to multiply are symmetric with the center at <math>29.5</math>. Multiply the symmetric pairs to get <math>31\cdot 28=868</math> and <math>30\cdot 29=870</math>.
  4 KB (523 words) - 00:12, 8 October 2021
• 1990 AIME Problems/Problem 13
  ...\le 4000\}</math>. Given that <math>9^{4000}_{}</math> has 3817 [[digit]]s and that its first (leftmost) digit is 9, how many [[element]]s of <math>T_{}^{ '''Lemma''': For all positive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 12
  ...h> where <math>a^{}_{}</math>, <math>b^{}_{}</math>, <math>c^{}_{}</math>, and <math>d^{}_{}</math> are positive integers. Find <math>a + b + c + d^{}_{} ...rawing the [[altitude]] of those [[triangle]]s and then solving will yield the respective lengths.
  6 KB (906 words) - 13:23, 5 September 2021
• 1990 AIME Problems/Problem 9
  ...}^{}</math> times. Let <math>\frac{i}{j}^{}_{}</math>, in lowest terms, be the [[probability]] that heads never occur on consecutive tosses. Find <math>i+ Clearly, at least <math>5</math> tails must be flipped; any less, then by the [[Pigeonhole Principle]] there will be heads that appear on consecutive tos
  3 KB (425 words) - 19:31, 30 July 2021
• 1990 AIME Problems/Problem 8
  ...column of two targets. A marksman is to break all the targets according to the following rules: 1) The marksman first chooses a column from which a target is to be broken.
  3 KB (491 words) - 04:24, 4 November 2022
• 1990 AIME Problems/Problem 2
  Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</math>. ...sqrt{43} + 3)^2</math>. Repeating this for <math>52-6\sqrt{43}</math>, the only feasible possibility is <math>(\sqrt{43} - 3)^2</math>.
  5 KB (765 words) - 23:00, 26 August 2023
• 2006 AMC 10B Problems/Problem 8
  ...uare of area 40 is inscribed in a semicircle as shown. What is the area of the semicircle? ...icircle and one of the bottom vertices of the square is half the length of the side, which is <math>\sqrt{10}</math>.
  1 KB (176 words) - 13:49, 26 January 2022
• 2006 AMC 10B Problems/Problem 10
  ...third side is <math>15</math>. What is the greatest possible perimeter of the triangle? Let <math>x</math> be the length of the first side.
  900 bytes (132 words) - 13:57, 26 January 2022
• 1991 AIME Problems/Problem 13
  ...red or both are blue. What is the largest possible number of red socks in the drawer that is consistent with this data? ...number of red and blue socks, respectively. Also, let <math>t=r+b</math>. The probability <math>P</math> that when two socks are drawn randomly, without
  7 KB (1,328 words) - 20:24, 5 February 2024
• 2006 AMC 10B Problems/Problem 15
  ...math> is <math>24</math> and <math> \angle BAD = 60^\circ </math>. What is the area of rhombus <math>BFDE</math>? ...eral triangle]]s <math>DAB</math> and <math>DCB</math>. Let the lengths of the sides of rhombus <math>ABCD</math> be <math>s</math>.
  3 KB (445 words) - 22:01, 20 August 2022
• 2006 AMC 10B Problems/Problem 16
  ...y <math>29</math>, <math>2000</math>, occurred on a Sunday. On what day of the week will Leap Day, February <math>29</math>, <math>2020</math>, occur? Therefore, the number of days between Leap Day <math>2004</math> and Leap Day <math>2020</math> is:
  2 KB (336 words) - 02:41, 1 January 2024
• 1991 AIME Problems/Problem 11
  ...</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. We wish to find the radius of one circle, so that we can find the total area.
  4 KB (740 words) - 19:33, 28 December 2022
• 2006 AMC 10B Problems/Problem 17
  ...s the probability that after this process the contents of the two bags are the same? ...lice puts in Bob's bag doesn't matter. [[Without loss of generality]], let the ball Alice puts in Bob's bag be red.
  1 KB (211 words) - 04:32, 4 November 2022
• 1991 AIME Problems/Problem 6
  ...</math>. (For real <math>x^{}_{}</math>, <math>\lfloor x \rfloor</math> is the [[floor function|greatest integer]] less than or equal to <math>x^{}_{}</ma ...3}{100} \le r < \frac{744}{100}</math>, so <math>743\le 100r < 744</math>, and <math>\lfloor 100r \rfloor = \boxed{743}</math>.
  3 KB (447 words) - 17:02, 24 November 2023
• 1991 AIME Problems/Problem 5
  ...s between <math>0</math> and <math>1</math> will <math>20_{}^{}!</math> be the resulting [[product]]? ...half of them will be between <math>0 < \frac{a}{b} < 1</math>. Therefore, the solution is <math>\frac{2^8}{2} = \boxed{128}</math>.
  919 bytes (141 words) - 20:00, 4 July 2022
• 1991 AIME Problems/Problem 4
  How many [[real number]]s <math>x^{}_{}</math> satisfy the [[equation]] <math>\frac{1}{5}\log_2 x = \sin (5\pi x)</math>? The [[range]] of the [[sine]] function is <math>-1 \le y \le 1</math>. It is [[periodic function
  2 KB (300 words) - 16:01, 26 November 2019
• 1991 AIME Problems/Problem 2
  ...e [[diagonal]] <math>\overline {AC}</math>. Find the sum of the lengths of the 335 [[parallel]] segments drawn. ...ind <math>2\sum\limits_{k=1}^{168} a_k-5</math> since we are over counting the diagonal.
  4 KB (595 words) - 12:51, 17 June 2021
• 1991 AIME Problems/Problem 1
  Find <math>x^2+y^2_{}</math> if <math>x_{}^{}</math> and <math>y_{}^{}</math> are positive integers such that ...>32</math>, and so there are no integral solutions for <math>(x,y)</math>. The solution is <math>5^2 + 11^2 = \boxed{146}</math>.
  4 KB (628 words) - 22:05, 7 June 2021
• 1992 AIME Problems/Problem 15
  ...rial tail if there is some positive integer <math>m^{}_{}</math> such that the decimal representation of <math>m!</math> ends with exactly <math>n</math> Let the number of zeros at the end of <math>m!</math> be <math>f(m)</math>. We have <math>f(m) = \left\lfl
  2 KB (358 words) - 01:54, 2 October 2020
• 1992 AIME Problems/Problem 12
  ....</math> (The squares with two or more dotted edges have been removed form the original board in previous moves.) ...Chomp. How many different subsets are there in all? Include the full board and empty board in your count.
  2 KB (443 words) - 22:41, 22 December 2021
• 1992 AIME Problems/Problem 9
  ...th>. Given that <math>AP^{}_{}=\frac mn</math>, where <math>m^{}_{}</math> and <math>n^{}_{}</math> are relatively prime positive integers, find <math>m+n ...th>\angle AXB;</math> let it meet <math>CD</math> at <math>E.</math> Using the angle bisector theorem, we let <math>XB=y(92-x), XA=xy</math> for some <mat
  5 KB (874 words) - 10:27, 22 August 2021
• 1992 AIME Problems/Problem 8
  ...the sequence <math>\Delta(\Delta A^{}_{})</math> are <math>1^{}_{}</math>, and that <math>a_{19}=a_{92}^{}=0</math>. Find <math>a_1^{}</math>. ...m the condition <math>\Delta(\Delta{A}) = 1</math>, we are led to consider the differential equation
  5 KB (778 words) - 21:36, 3 December 2022
• 1992 AIME Problems/Problem 6
  ...n <math>\{1000,1001,1002,\ldots,2000\}</math> is no carrying required when the two integers are added? ...smaller integer be <math>\underline{1ABC},</math> where <math>A,B,</math> and <math>C</math> are digits from <math>0</math> through <math>9.</math>
  3 KB (455 words) - 02:03, 10 July 2021
• 1992 AIME Problems/Problem 3
  ...>.503</math>. What's the largest number of matches she could've won before the weekend began? ...the number of matches won, so that <math>\frac{n}{2n}=\frac{1}{2}</math>, and <math>\frac{n+3}{2n+4}>\frac{503}{1000}</math>.
  2 KB (251 words) - 08:05, 2 January 2024
• 1992 AIME Problems/Problem 2
  ...nding if, in its [[decimal representation]], there are at least two digits and each digit is less than any digit to its right. How many ascending positive ...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 20
  ...78)</math>, <math>D=(8,y)</math>, for some integer <math>y</math>. What is the area of rectangle <math>ABCD</math>? Let the slope of <math>AB</math> be <math>m_1</math> and the slope of <math>AD</math> be <math>m_2</math>.
  4 KB (594 words) - 15:45, 30 July 2023
• 2006 AMC 10B Problems/Problem 25
  ...ppen to be my age." Which of the following is <b><i>not</i></b> the age of one of Mr. Jones's children? ...m</math> be the positive integer seen on the license plate. Since at least one of <math>4</math> or <math>8</math> is contained in <math>S</math>, we have
  5 KB (878 words) - 14:39, 3 December 2023
• 1993 AIME Problems/Problem 14
  ...can be inscribed unstuck in a 6 by 8 rectangle, the smallest perimeter has the form <math>\sqrt{N}\,</math>, for a positive integer <math>N\,</math>. Find ...ve and negative. Then by symmetry, the other rectangle is also centered at the origin, <math>O</math>.
  3 KB (601 words) - 09:25, 19 November 2023
• 1993 AIME Problems/Problem 10
  ...</math> triangular faces and <math>P</math> pentagonal faces meet. What is the value of <math>100P+10T+V</math>? ...is because we are counting twice each edge, since two adjacent faces share one edge). Thus, <math>E=60</math>. Finally, using Euler's formula we have <mat
  4 KB (716 words) - 20:50, 17 April 2022
• 1993 AIME Problems/Problem 9
  ...ome points will not have a label. What is the smallest integer that labels the same point as <math>1993</math>? ...tarting from 1) A number <math>n</math> will only occupy the same point on the circle if <math>\frac12(n)(n + 1)\equiv \frac12(1993)(1994) \pmod{2000}</ma
  3 KB (488 words) - 02:06, 22 September 2023
• 1993 AIME Problems/Problem 8
  ...s <math>\{a, c\},\{b, c, d, e, f\}</math> represents the same selection as the pair <math>\{b, c, d, e, f\},\{a, c\}.</math> ...ble selection is double counted, except the case where both <math>m</math> and <math>n</math> contain all <math>6</math> elements of <math>S.</math> So ou
  9 KB (1,400 words) - 14:09, 12 January 2024
• 1993 AIME Problems/Problem 4
  ...math>0 < a < b < c < d < 500\,</math> satisfy <math>a + d = b + c\,</math> and <math>bc - ad = 93\,</math>? ...tions don't follow <math>a < b < c < d</math>, so we only need to consider the first two solutions.
  8 KB (1,343 words) - 16:27, 19 December 2023
• 1993 AIME Problems/Problem 1
  How many even [[integer]]s between 4000 and 7000 have four different digits? The thousands digit is <math>\in \{4,5,6\}</math>.
  3 KB (440 words) - 21:20, 22 July 2021
• 1994 AIME Problems/Problem 15
  ...>s\,</math> are positive integers and <math>s\,</math> is not divisible by the square of any prime. What is <math>q+r+s\,</math>? ...each of the respective conditions for <math>P</math> is the region inside the (semi)circles with diameters <math>\overline{AB}, \overline{BC}, \overline{
  4 KB (717 words) - 22:20, 3 June 2021
• 1994 AIME Problems/Problem 14
  ...er of times the light beam will bounce off the two line segments. Include the first reflection at <math>C\,</math> in your count. At each point of reflection, we pretend instead that the light continues to travel straight.
  2 KB (303 words) - 00:03, 28 December 2017
• 1994 AIME Problems/Problem 11
  ...How many different tower heights can be achieved using all ninety-four of the bricks? ...the number of changes we can get from <math>0</math>'s, <math>2</math>'s, and <math>5</math>'s.
  4 KB (645 words) - 15:12, 15 July 2019
• 1994 AIME Problems/Problem 9
  ...that the bag will be emptied is <math>p/q,\,</math> where <math>p\,</math> and <math>q\,</math> are relatively prime positive integers. Find <math>p+q.\, ...n it has <math>k</math> pairs in it. Let's consider the possible draws for the first three cards:
  3 KB (589 words) - 14:18, 21 July 2019
• 1994 AIME Problems/Problem 8
  ...d <math>(b,37)\,</math> are the vertices of an equilateral triangle. Find the value of <math>ab\,</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 has at least one solution, and each solution is an ordered pair <math>(x,y)\,</math> of integers. How man
  3 KB (442 words) - 19:51, 8 January 2024
• 1994 AIME Problems/Problem 6
  The graphs of the equations ...ne for <math>k=-10,-9,-8,\ldots,9,10.\,</math> These 63 lines cut part of the plane into equilateral triangles of side length <math>\tfrac{2}{\sqrt{3}}.\
  4 KB (721 words) - 16:14, 8 March 2021
• 1994 AIME Problems/Problem 5
  ...of the non-zero digits of <math>n\,</math>. (If <math>n\,</math> has only one digits, then <math>p(n)\,</math> is equal to that digit.) Let What is the largest prime factor of <math>S\,</math>?
  2 KB (275 words) - 19:27, 4 July 2013
• 1994 AIME Problems/Problem 4
  Find the positive integer <math>n\,</math> for which (For real <math>x\,</math>, <math>\lfloor x\rfloor\,</math> is the greatest integer <math>\le x.\,</math>)
  2 KB (264 words) - 13:33, 11 August 2018
• 1995 AIME Problems/Problem 15
  .../math> can be written in the form <math>m/n</math> where <math>m_{}</math> and <math>n_{}</math> are relatively prime positive integers, find <math>m+n</m ...ld be <tt>T</tt> or the sequence could start with a block of <tt>H</tt>'s, the total probability is that <math>3/2</math> of it has to start with an <tt>H
  6 KB (979 words) - 13:20, 11 April 2022
• 1995 AIME Problems/Problem 12
  ...> Given that <math>\cos \theta=m+\sqrt{n},</math> where <math>m_{}</math> and <math>n_{}</math> are integers, find <math>m+n.</math> // calculate intersection of line and plane
  8 KB (1,172 words) - 21:57, 22 September 2022
• 1995 AIME Problems/Problem 11
  ...</math> into two prisms, one of which is [[similar]] to <math>P_{},</math> and both of which have nonzero volume. Given that <math>b=1995,</math> for how Let <math>P'</math> be the prism similar to <math>P</math>, and let the sides of <math>P'</math> be of length <math>x,y,z</math>, such that <math>x
  2 KB (292 words) - 19:30, 4 July 2013
• 1995 AIME Problems/Problem 10
  ...ger that is not the sum of a positive integral multiple of <math>42</math> and a positive composite integer? ...self. So we make a table, listing all the primes up to <math>42</math> and the numbers that are multiples of <math>42</math> greater than them, until they
  3 KB (436 words) - 19:26, 2 September 2023
• 1995 AIME Problems/Problem 6
  ...to <math>n^2</math>, then one factor per pair is less than <math>n</math>, and so there are <math>\frac{63\times 39-1}{2} = 1228</math> factors of <math>n The number of factors of <math>n</math> less than <math>n</math> is equal to <m
  2 KB (407 words) - 08:14, 4 November 2022
• 1995 AIME Problems/Problem 3
  ...p</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m+n.</mat ...f steps for the object to reach <math>(2,2)</math>, so the number of steps the object may have taken is either <math>4</math> or <math>6</math>.
  3 KB (602 words) - 23:15, 16 June 2019
• 1995 AIME Problems/Problem 2
  Find the last three digits of the product of the [[positive root]]s of ...th>1995^{1+\sqrt{2}/2} \cdot 1995^{1 - \sqrt{2}/2} = 1995^2</math>, making the solution <math>(2000-5)^2 \equiv \boxed{025} \pmod{1000}</math>.
  2 KB (362 words) - 00:40, 29 January 2021
• 1995 AIME Problems/Problem 1
  ...}</math> can be written in the form <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m-n.</ma The sum of the areas of the [[square]]s if they were not interconnected is a [[geometric sequence]]:
  2 KB (302 words) - 19:29, 4 July 2013
• 1996 AIME Problems/Problem 14
  ...nal [[diagonal]] of this solid passes through the interiors of how many of the <math>1\times 1\times 1</math> [[cube (geometry) | cube]]s? ...,c)</math> be the general coordinates 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
  ..._3,\cdots,a_{10}</math> of the integers <math>1,2,3,\cdots,10</math>, form the sum ...can be written in the form <math>\dfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math
  5 KB (879 words) - 11:23, 5 September 2021
• 1996 AIME Problems/Problem 10
  Find the smallest positive integer solution to <math>\tan{19x^{\circ}}=\dfrac{\cos{9 ...] function is <math>180^\circ</math>, and the tangent function is [[one-to-one]] over each period of its domain.
  4 KB (503 words) - 15:46, 3 August 2022
• 1996 AIME Problems/Problem 8
  ...>(x,y)</math> with <math>x<y</math> is the harmonic mean of <math>x</math> and <math>y</math> equal to <math>6^{20}</math>? ...(<math>3^{20}2^{19}</math>). Since <math>x<y</math>, the answer is half of the remaining number of factors, which is <math>\frac{1599-1}{2}= \boxed{799}</
  1 KB (155 words) - 19:32, 4 July 2013
• 1996 AIME Problems/Problem 7
  ...valent if one can be obtained from the other by applying a [[rotation]] in the plane board. How many inequivalent color schemes are possible? ..."font-size:85%">For those symmetric about the center, <br /> there is only one other.</font></td></tr></table></center>
  4 KB (551 words) - 11:44, 26 June 2020
• 1996 AIME Problems/Problem 6
  ...roduce neither an undefeated team nor a winless team, where <math>m</math> and <math>n</math> are relatively prime integers. Find <math>m+n</math>. ...finding the probability that at least one team wins all games or at least one team loses all games.
  3 KB (461 words) - 01:00, 19 June 2019
• 1996 AIME Problems/Problem 4
  ...does not include the area beneath the cube is 48 square centimeters. Find the greatest integer that does not exceed <math>1000x</math>. ...+ 1 = 49</math>, and so the sides of the shadow are <math>7</math>. Using the similar triangles in blue,
  2 KB (257 words) - 17:50, 4 January 2016
• 1997 AIME Problems/Problem 15
  ...ath>r</math> are positive integers, and <math>q</math> is not divisible by the square of any prime number. Find <math>p+q+r</math>. ...1 + ai)\left(\frac {1}{2} + \frac {\sqrt {3}}{2}i\right) = b + 10i</math>, and expanding we get <math>\left(\frac {11}{2} - \frac {a\sqrt {3}}{2}\right) +
  4 KB (609 words) - 22:49, 17 July 2023
• 1997 AIME Problems/Problem 14
  ...at <math>\sqrt{2+\sqrt{3}}\le\left|v+w\right|</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <m Define <math>\theta = 2\pi/1997</math>. By [[De Moivre's Theorem]] the roots are given by
  5 KB (874 words) - 22:30, 1 April 2022
• 1997 AIME Problems/Problem 13
  ...math>b</math> are positive integers and <math>b</math> is not divisible by the square of any prime number. Find <math>a+b</math>. ...</math>, there's a symmetry about all four [[quadrant]]s, so just consider the first quadrant. We now gather some points:
  7 KB (1,225 words) - 19:56, 4 August 2021
• 1997 AIME Problems/Problem 12
  ...es except <math>\frac{-d}{c}</math>. Find the unique number that is not in the range of <math>f</math>. ...math>f(f(x)) = x</math> for all <math>x</math> in the domain. Substituting the function definition, we have <math>\frac {a\frac {ax + b}{cx + d} + b}{c\fr
  11 KB (2,063 words) - 22:59, 21 October 2023
• 1997 AIME Problems/Problem 10
  ...nted. A set of three cards from the deck is called complementary if all of the following statements are true: ...ch of the three cards has a different shape or all three of the cards have the same shape.
  3 KB (585 words) - 19:37, 25 April 2022
• 1997 AIME Problems/Problem 8
  ...e the property that the sum of the entries in each row is 0 and the sum of the entries in each column is 0? ...h> grids with 2 1's and 2 -1's in each row and column. We do casework upon the first two columns:
  4 KB (638 words) - 16:41, 22 January 2024
• 1997 AIME Problems/Problem 3
  ...t Sarah should have obtained. What is the sum of the two-digit number and the three-digit number? ...izes to <math>(9x-1)\left(y-\dfrac{1000}{9}\right)=\dfrac{1000}{9}</math>, and <math>(9x-1)(9y-1000)=1000</math>.
  2 KB (375 words) - 19:34, 4 August 2021
• 1998 AIME Problems/Problem 15
  ...length of the longest proper sequence of dominos that can be formed using the dominos of <math>D_{40}.</math> ...every point is connected with every other point. The connections represent the dominoes.
  9 KB (1,671 words) - 22:10, 15 March 2024
• 1998 AIME Problems/Problem 13
  ...= - 176 - 64i</math> and <math> S_9 = p + qi,</math> where <math>p</math> and <math>q</math> are integers, find <math>|p| + |q|.</math> ...e 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> and <math>c</math> are integers, and <math>c</math> is not divisible by the square of any [[prime]]. What is <math>a^{2} + b^{2} + c^{2}</math>? ...>AD = \frac {1}{2}AC</math>, <math>\triangle AED \sim \triangle ABC</math> and <math>ED \parallel BC</math>.
  5 KB (876 words) - 20:27, 9 June 2022
• 1998 AIME Problems/Problem 10
  ...ath> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]]. Find <math>a + b + c</math>. ...stead 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 9
  ...ath> are [[positive]] [[integer]]s, and <math>c</math> is not divisible by the square of any [[prime]]. Find <math>a + b + c.</math> ...le arrival times of the mathematicians, while the shaded region represents the arrival times where they meet.
  4 KB (624 words) - 18:34, 18 February 2018
• 1998 AIME Problems/Problem 8
  ...preceding term from the one before that. The last term of the sequence is the first [[negative]] term encounted. What positive integer <math>x</math> pr The best way to start is to just write out some terms.
  2 KB (354 words) - 19:37, 24 September 2023
• 1998 AIME Problems/Problem 7
  Let <math>n</math> be the number of ordered quadruples <math>(x_1,x_2,x_3,x_4)</math> of positive odd == Solution 1 (Clever Stars and Bars Manipulation) ==
  5 KB (684 words) - 11:41, 13 August 2023
• 1998 AIME Problems/Problem 4
  ...ee players obtain an [[odd]] sum is <math>m/n,</math> where <math>m</math> and <math>n</math> are [[relatively prime]] [[positive integer]]s. Find <math> ...order the people pick the tiles; the final answer is the same if we assume the opposite, that order doesn't matter.)
  5 KB (917 words) - 02:37, 12 December 2022
• 1998 AIME Problems/Problem 3
  ...400</math> partitions the plane into several regions. What is the area of the bounded region? The equation given can be rewritten as:
  1 KB (198 words) - 20:13, 23 February 2018
• 1999 AIME Problems/Problem 10
  ...re among the ten given points is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are [[relatively prime]] [[positive]] [[integer]]s. Find ...m triangles. However, a fourth distinct segment must also be picked. Since the triangle accounts for 3 segments, there are <math>45 - 3 = 42</math> segmen
  3 KB (524 words) - 17:25, 17 July 2023
• 1999 AIME Problems/Problem 7
  ...e step, and so are all the other switches whose labels divide the label on the <math>i</math>-th switch. After step 1000 has been completed, how many swi ...ath>\frac{N}{d_{i}}= 2^{9-x_{i}}3^{9-y_{i}}5^{9-z_{i}}</math>. In general, the divisor-count of <math>\frac{N}{d}</math> must be a multiple of 4 to ensure
  3 KB (475 words) - 13:33, 4 July 2016
• 1999 AIME Problems/Problem 4
  ...octagon <math>ABCDEFGH</math> is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are [[relatively prime]] [[positive]] [[integer]]s. Find ..., the area of all <math>8</math> of them is <math>\frac{86}{99}</math> and the answer is <math>\boxed{185}</math>.
  3 KB (398 words) - 13:27, 12 December 2020
• 1999 AIME Problems/Problem 1
  Find the smallest prime that is the fifth term of an increasing [[arithmetic sequence]], all four preceding ter ...1,17,23</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
  ...999,2000.</math> In the original stack of cards, how many cards were above the card labeled <math>1999</math>? ...ath>, meaning that there were <math>\boxed{927}</math> cards are above the one labeled <math>1999</math>.
  15 KB (2,673 words) - 19:16, 6 January 2024
• 2000 AIME I Problems/Problem 12
  ...,</math> what is the largest number of different values that can appear in the list <math>f(0),f(1),f(2),\ldots,f(999)?</math> Since <math>\mathrm{gcd}(1056, 1760) = 352</math> we can conclude that (by the [[Euclidean algorithm]])
  1 KB (238 words) - 18:50, 10 March 2015
• 2000 AIME I Problems/Problem 11
  ...e [[relatively prime]] positive [[divisor]]s of <math>1000.</math> What is the [[floor function|greatest integer]] that does not exceed <math>S/10</math>? ...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
  ...h> Then <math>z + \frac {1}{y} = \frac {m}{n},</math> where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m + n ...s you think symmetry, but actually can be solved easily with substitution, and other normal technniques
  5 KB (781 words) - 15:02, 20 April 2024
• 2000 AIME I Problems/Problem 5
  ...lity that both marbles are white is <math>m/n,</math> where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. What is <math>m .../math>. The [[Principle of Inclusion-Exclusion]] still requires us to find the individual probability of each box.
  7 KB (1,011 words) - 20:09, 4 January 2024
• 2000 AIME I Problems/Problem 1
  ...of any two positive integers, at least one of these two integers contains the digit <math>0</math>. ...the <math>2</math>s and the <math>5</math>s separated, so we need to find the first power of 2 or 5 that contains a 0.
  1 KB (163 words) - 17:44, 16 December 2020
• 2001 AIME I Problems/Problem 15
  ...tten on faces that share an edge is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</ ...opposite point from the other. Notice that each vertex belongs to exactly one long diagonal.
  11 KB (1,837 words) - 18:53, 22 January 2024
• 2001 AIME I Problems/Problem 14
  ...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 delivery are possible? ...d <math>1</math>'s such that there are no two consecutive <math>1</math>'s and no three consecutive <math>0</math>'s.
  13 KB (2,298 words) - 19:46, 9 July 2020
• 2001 AIME I Problems/Problem 13
  ...gree arc is <math>- m + \sqrt {n}</math> centimeters, where <math>m</math> and <math>n</math> are positive integers. Find <math>m + n.</math> ...</math>-degree arcs can be represented as <math>x + 20</math>, as given in the problem.
  3 KB (561 words) - 19:25, 27 November 2022
• 2001 AIME I Problems/Problem 12
  ...h> The [[radius]] of the sphere is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</ ...e insphere must be located at <math>(r,r,r)</math> where <math>r</math> is the sphere's radius.
  6 KB (1,050 words) - 18:44, 27 September 2023
• 2001 AIME I Problems/Problem 10
  ...e also belongs to <math>S</math> is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n.</ ...math>, <math>y</math>, and <math>z</math> coordinates must be even so that the midpoint can have integer coordinates. Therefore,
  8 KB (1,187 words) - 02:40, 28 November 2020
• 2001 AIME I Problems/Problem 9
  ...C</math> can be written in the form <math>m/n</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m+n</ We let <math>[\ldots]</math> denote area; then the desired value is
  4 KB (673 words) - 20:15, 21 February 2024
• 2001 AIME I Problems/Problem 6
  ...may be expressed in the form <math>\frac{m}{n}</math> where <math>m</math> and <math>n</math> are [[relatively prime]] [[positive]] [[integer]]s. Find <ma ...h>b</math>, <math>c</math>, and <math>d</math> corresponds to the value of the roll.
  11 KB (1,729 words) - 20:50, 28 November 2023
• 2001 AIME I Problems/Problem 5
  ...the length of each side is <math>\frac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math ...>A,B,</math> and <math>C,</math> where <math>B</math> is in [[quadrant]] 4 and <math>C</math> is in quadrant <math>3.</math>
  6 KB (1,043 words) - 10:09, 15 January 2024
• 2001 AIME I Problems/Problem 1
  Find the sum of all positive two-digit integers that are divisible by each of their ...b \equiv b \pmod{a}</math>, or <math>a</math> divides into <math>b</math> and <math>b</math> divides into <math>10a</math>. Thus <math>b = a, 2a,</math>
  4 KB (687 words) - 18:37, 27 November 2022
• 2002 AIME I Problems/Problem 15
  ...math>r</math> are positive integers and <math>r</math> is not divisible by the square of any prime, find <math>p + q + r.</math> ...zoid and <math>CDE</math> is an isosceles triangle, we have symmetry about the <math>xz</math>-plane.
  7 KB (1,181 words) - 20:32, 8 January 2024
• 2002 AIME I Problems/Problem 13
  ...math>n</math> are positive integers and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. Substituting the first equation into the second and simplification yields <math>24^2 = 2\left(3AC^2 + 2 \cdot 24^2 - 4 \cdot 18
  6 KB (974 words) - 13:01, 29 September 2023
• 2002 AIME I Problems/Problem 11
  ...th> and <math>n</math> are integers and <math>n</math> is not divisible by the square of any prime. Find <math>m+n</math>. == Solution 1(Easy one) ==
  3 KB (591 words) - 15:11, 21 August 2019
• 2002 AIME I Problems/Problem 9
  Harold, Tanya, and Ulysses paint a very long picket fence. * Harold starts with the first picket and paints every <math>h</math>th picket;
  4 KB (749 words) - 19:44, 25 April 2024
• 2002 AIME I Problems/Problem 8
  Find the smallest integer <math>k</math> for which the conditions are satisfied by more than one sequence.
  1 KB (205 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 6
  The solutions to the system of equations are <math>(x_1,y_1)</math> and <math>(x_2,y_2)</math>. Find <math>\log_{30}\left(x_1y_1x_2y_2\right)</math
  1 KB (194 words) - 19:55, 23 April 2016
• 2002 AIME I Problems/Problem 5
  ...stinct squares in the plane of the dodecagon have at least two vertices in the set <math>\{A_1,A_2,A_3,\cdots,A_{12}\} ?</math> ...y form 6 squares. So in total, we have overcounted by <math>9+6=15</math>, and <math>198-15=\fbox{183}</math>.
  1 KB (220 words) - 20:50, 12 November 2022
• 2002 AIME I Problems/Problem 4
  ...+\cdots+a_{n-1}=\dfrac{1}{29}</math>, for positive integers <math>m</math> and <math>n</math> with <math>m<n</math>, find <math>m+n</math>. Since we need a factor of 29 in the denominator, we let <math>n=29t</math>.* Substituting, we get
  2 KB (320 words) - 07:55, 4 November 2022
• 2002 AIME I Problems/Problem 2
  ...n be written as <math>\dfrac{1}{2}(\sqrt{p}-q)</math> where <math>p</math> and <math>q</math> are positive integers. Find <math>p+q</math>. ...tten as <math>14r</math>, and by the [[Pythagorean Theorem]], we find that the shorter dimension is <math>2r\left(\sqrt{3}+1\right)</math>.
  2 KB (287 words) - 19:54, 4 July 2013
• 2002 AIME I Problems/Problem 1
  ...it does right-to-left) is <math>\dfrac{m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n.</mat ...th>. Similarly, there is a <math>\frac 1{26}</math> probability of picking the three-letter palindrome.
  3 KB (369 words) - 23:36, 6 January 2024
• 2003 AIME I Problems/Problem 10
  ...angle MAC = 7^\circ </math> and <math> \angle MCA = 23^\circ. </math> Find the number of degrees in <math> \angle CMB. </math> ...ide <math>\triangle ABC</math> such that <math>\angle CBN = 7^\circ</math> and <math>\angle BCN = 23^\circ</math>.
  7 KB (1,058 words) - 01:41, 6 December 2022
• 2003 AIME I Problems/Problem 8
  ...and the first and fourth terms differ by <math>30</math>. Find the sum of the four terms. ...ween the first three terms as <math>d</math>. The 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 6
  ...cube is <math> m + \sqrt{n} + \sqrt{p}, </math> where <math> m, n, </math> and <math> p </math> are [[integer]]s. Find <math> m + n + p. </math> ...blique]] to the edges of the cube, whose sides are three face diagonals of the cube.
  3 KB (477 words) - 18:35, 27 December 2021
• 2003 AIME I Problems/Problem 5
  ...math> and <math> p </math> are positive [[integer]]s, and <math> n </math> and <math> p </math> are [[relatively prime]], find <math> m + n + p. </math> ...here]]s (one centered at each [[vertex]] of the large parallelepiped), and the <math>1/4</math> [[cylinder]]s connecting each adjacent pair of spheres.
  2 KB (288 words) - 19:58, 4 July 2013
• 2003 AIME I Problems/Problem 3
  ...list the greater of the set's two elements. Find the sum of the numbers on the list. Order the numbers in the set from greatest to least to reduce error: <math>\{34, 21, 13, 8, 5, 3, 2,
  2 KB (317 words) - 00:09, 9 January 2024
• 2003 AIME I Problems/Problem 2
  ...>100</math> can be expressed as <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relatively prime]] [[positive integer]]s. Find <math ...radius <math>99</math>, then adding the circle of radius <math>98</math>, and so forth.
  4 KB (523 words) - 15:49, 8 March 2021
• 2003 AIME II Problems/Problem 13
  ...tarting vertex on its tenth move is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers, find <math>m + n.</m ...it has <math>\frac{1}{2}</math> chance of reaching the starting vertex in the next move. Thus <math>P_n=\frac{1}{2}(1-P_{n-1})</math>.
  15 KB (2,406 words) - 23:56, 23 November 2023
• 2003 AIME II Problems/Problem 12
  ...s for that candidate. What was the smallest possible number of members of the committee? ...+v_{27}</math> be the total number of votes cast. Our goal is to determine the smallest possible <math>s</math>.
  4 KB (759 words) - 13:00, 11 December 2022
• 2003 AIME II Problems/Problem 10
  ...integer that is not a perfect square. What is the maximum possible sum of the two integers? ...d the other <math>6</math>, which gives <math>b=48</math>. This checks, so the solution is <math>48+108=\boxed{156}</math>.
  1 KB (218 words) - 14:14, 25 June 2021
• 2003 AIME II Problems/Problem 8
  ...16,</math> <math>1848,\ldots,</math> whose terms are formed by multiplying the corresponding terms of two arithmetic sequences. ...40</math>, <math>f(2)=1716</math>, and <math>f(3)=1848</math>. Plugging in the values for x gives us a system of three equations:
  5 KB (793 words) - 15:18, 14 July 2023
• 2003 AIME II Problems/Problem 6
  ...of the union of the two regions enclosed by the triangles <math>ABC</math> and <math>A'B'C'?</math> ...13</math> triangle and a <math>9-12-15</math> triangle "glued" together on the <math>12</math> side, <math>[ABC]=\frac{1}{2}\cdot12\cdot14=84</math>.
  5 KB (787 words) - 17:38, 30 July 2022
• 2003 AIME II Problems/Problem 5
  ...as exactly one point in common with the log. The number of cubic inches in the wedge can be expressed as <math>n\pi</math>, where n is a positive integer. ...>. (Imagine taking another identical wedge and sticking it to the existing one). Thus, <math>V=\dfrac{6^2\cdot 12\pi}{2}=216\pi</math>, so <math>n=\boxed
  1 KB (204 words) - 17:41, 30 July 2022
• 2003 AIME II Problems/Problem 4
  ...etrahedron to that of the larger is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m+n</math Embed the tetrahedron in 4-space to make calculations easier.
  3 KB (563 words) - 17:36, 30 July 2022
• 2003 AIME II Problems/Problem 3
  ...>B</math>, <math>B</math> is never immediately followed by <math>C</math>, and <math>C</math> is never immediately followed by <math>A</math>. How many se ...letters can follow it, since one of the letters is restricted. Therefore, the number of seven-letter good words is <math>3*2^6=192</math>
  2 KB (336 words) - 17:29, 30 July 2022
• 2003 AIME II Problems/Problem 1
  ...eir [[sum]], and one of the [[integer]]s is the sum of the other two. Find the sum of all possible values of <math>N</math>. ..., \{3, 4\}</math> so <math>a + b</math> is one of <math>13, 8, 7</math> so the sum of all possible values of <math>N</math> is <math>12 \cdot (13 + 8 + 7)
  1 KB (174 words) - 08:56, 11 July 2023
• 2002 AIME II Problems/Problem 15
  ...h>b</math> is not divisible by the square of any prime, and <math>a</math> and <math>c</math> are relatively prime. Find <math>a + b + c</math>. ...e circles are <math>a</math> and <math>b</math> respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  7 KB (1,182 words) - 09:56, 7 February 2022
• 2002 AIME II Problems/Problem 12
  .../math>, and <math>s</math> are primes, and <math>a</math>, <math>b</math>, and <math>c</math> are positive integers. Find <math>\left(p+q+r+s\right)\left( ...od below, computing the number of paths to each point that only move right and up.
  7 KB (1,127 words) - 13:34, 19 June 2022
• 2002 AIME II Problems/Problem 11
  ...math>p</math> are positive integers and <math>m</math> is not divisible by the square of any prime. Find <math>100m+10n+p</math>. ...e <math>x</math>. Then, the common ratio is <math>\frac{1}{8x}</math>, and the first term is <math>8x^2</math>.
  4 KB (696 words) - 16:27, 22 March 2022
• 2002 AIME II Problems/Problem 8
  ...ions for <math>n</math>. (The notation <math>\lfloor x\rfloor</math> means the greatest integer less than or equal to <math>x</math>.) ...implifies to <math>n(n+1)<2002</math>, which is easy to solve by trial, as the solution is obviously <math>\simeq \sqrt{2002}</math>.)
  6 KB (908 words) - 14:22, 14 July 2023
• 2002 AIME II Problems/Problem 7
  Find the smallest positive integer <math>k</math> such that <math>1^2+2^2+3^2+\ldots Since <math>2k+1</math> is always odd, and only one of <math>k</math> and <math>k+1</math> is even, either <math>k, k+1 \equiv 0 \pmod{16}</math>.
  3 KB (403 words) - 12:10, 9 September 2023
• 2002 AIME II Problems/Problem 4
  ...ath>n</math> on each side. The diagram indicates the path of blocks around the garden when <math>n=5</math>. ...ght)</math> square units, where <math>m</math> is a positive integer. Find the remainder when <math>m</math> is divided by <math>1000</math>.
  2 KB (268 words) - 07:28, 13 September 2020
• 2001 AIME II Problems/Problem 15
  ...math>p</math> are positive integers and <math>p</math> is not divisible by the square of any prime. Find <math>m + n + p</math>. ...T(2,0,2), U(8,6,8), V(8,8,6), W(2,2,0)</math>, and the other two faces of the tunnel are congruent to this shape.
  4 KB (518 words) - 15:01, 31 December 2021
• 2001 AIME II Problems/Problem 14
  ...\ldots < \theta_{2n} < 360</math> and angles are measured in degrees. Find the value of <math>\theta_{2} + \theta_{4} + \ldots + \theta_{2n}</math>. <math>z</math> can be written in the form <math> \text{cis\,}\theta</math>. Rearranging, we find that <math> \t
  2 KB (380 words) - 15:03, 22 July 2018
• 2001 AIME II Problems/Problem 13
  ...may be written in the form <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</ma ...ABD \sim \triangle DCE</math>. Hence <math>\angle ADB = \angle DEC</math>, and <math>\triangle BDE</math> is [[isosceles triangle|isosceles]]. Then <math>
  4 KB (743 words) - 03:32, 23 January 2023
• 2001 AIME II Problems/Problem 12
  ...] of <math>P_{3}</math> is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</ma ...trahedra will have volume <math>\left(\frac 12\right)^3 = \frac 18</math>. The total volume added here is then <math>\Delta P_1 = 4 \cdot \frac 18 = \frac
  2 KB (380 words) - 00:28, 5 June 2020
• 2001 AIME II Problems/Problem 11
  ...h more wins than losses is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are relatively prime positive integers. Find <math>m + n</ma ...the probability that Club Truncator does not have the same number of wins and losses.
  3 KB (415 words) - 23:25, 20 February 2023
• 2001 AIME II Problems/Problem 9
  ...ave a 2-by-2 red square is <math>\frac {m}{n}</math>, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m + n ...complementary counting]], counting all of the colorings that have at least one red <math>2\times 2</math> square.
  8 KB (1,207 words) - 20:04, 5 September 2023
• 2001 AIME II Problems/Problem 5
  ...gers, all of whose ten-element subsets have the triangle property. What is the largest possible value of <math>n</math>? ...uild up <math>\mathcal{S}</math> from the smallest possible <math>a</math> and <math>b</math>:
  2 KB (286 words) - 22:32, 5 January 2024
• 2001 AIME II Problems/Problem 3
  x_{4}&=523,\ \text{and}\\ find the value of <math>x_{531}+x_{753}+x_{975}</math>.
  2 KB (300 words) - 01:28, 12 November 2022
• 2000 AIME II Problems/Problem 15
  Find the least positive integer <math>n</math> such that <center><math>\frac 1{\sin We apply the identity
  3 KB (469 words) - 21:14, 7 July 2022
• 2000 AIME II Problems/Problem 14
  ...e expansion of <math>16!-32!+48!-64!+\cdots+1968!-1984!+2000!</math>, find the value of <math>f_1-f_2+f_3-f_4+\cdots+(-1)^{j+1}f_j</math>. ...if <math>32m\le k \le 32m+15</math> for some <math>m=1,2,\ldots,62</math>, and <math>f_k = 0</math> for all other <math>k</math>.
  7 KB (1,131 words) - 14:49, 6 April 2023
• 2000 AIME II Problems/Problem 13
  ...ath> are integers, <math>m</math> and <math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>. We may factor the equation as:{{ref|1}}
  6 KB (1,060 words) - 17:36, 26 April 2024
• 2000 AIME II Problems/Problem 10
  ...math>QD=23</math>, find the [[Perfect square|square]] of the [[radius]] of the circle. ...h>O</math> tangent to the sides and from <math>O</math> to the vertices of the quadrilateral, four pairs of congruent [[right triangle]]s are formed.
  2 KB (399 words) - 17:37, 2 January 2024
• 2000 AIME II Problems/Problem 6
  ...bases and that divides the trapezoid into two regions of equal area. Find the [[ceiling function|greatest integer]] that does not exceed <math>x^2/100</m ...\frac{b+b+100}{2} = b+50</math>. The two regions which the midline divides the trapezoid into are two smaller trapezoids, both with height <math>h/2</math
  3 KB (433 words) - 19:42, 20 December 2021
• 2000 AIME II Problems/Problem 5
  ...is significant, but it is not required that each finger have a ring. Find the leftmost three nonzero digits of <math>n</math>. ...ents split by three dividers. The number of ways to arrange those dividers and balls is just <math>\binom {8}{3}</math>.
  1 KB (184 words) - 21:13, 12 September 2020
• 2000 AIME II Problems/Problem 4
  What is the smallest positive integer with six positive odd integer divisors and twelve positive even integer divisors? We use the fact that the number of divisors of a number <math>n = p_1^{e_1}p_2^{e_2} \cdots p_k^{e_k
  2 KB (397 words) - 15:55, 11 May 2022
• 2000 AIME II Problems/Problem 3
  ...]] that two randomly selected cards also form a pair, where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. Find <math>m + ...> way. Thus, the answer is <math>\frac{54+1}{703} = \frac{55}{703}</math>, and <math>m+n = \boxed{758}</math>.
  1 KB (191 words) - 04:27, 4 November 2022
• 2005 AMC 10B Problems/Problem 3
  ...used. What fraction of the original amount of paint is available to use on the third day? ...=\dfrac{2}{3}-\dfrac{2}{9}=\dfrac{4}{9}</math>. Therefore, the fraction of the original amount of paint that is left is <math>\dfrac{\dfrac{4}{9}}{1}=\box
  1 KB (163 words) - 14:00, 14 December 2021
• 2005 AMC 10B Problems/Problem 12
  ...ice are rolled. What is the probability that the product of the numbers on the top faces is prime? ...inom{12}{1}</math> ways to choose which die will have the prime number, so the probability is <math>\dfrac{3}{6}\times\left(\dfrac{1}{6}\right)^{11}\times
  3 KB (385 words) - 14:03, 16 June 2022
• 2005 AMC 10B Problems/Problem 25
  ...ty that no two elements of <math>B</math> sum to <math>125</math>. What is the maximum possible number of elements in <math>B</math>? ...> pairs, and at most one number from each pair can be included in the set. The total is <math>24 + 38 = \boxed{\textbf{(C)}\ 62}</math>.
  3 KB (517 words) - 19:15, 15 October 2023
• 2006 USAMO Problems/Problem 2
  ...given positive integer <math>k </math> find, in terms of <math>k </math>, the minimum value of <math>N </math> for which there is a set of <math>2k+1 </m Let one optimal set of integers be <math>\{a_1,\dots,a_{2k+1}\}</math> with <math>a
  2 KB (398 words) - 09:48, 5 August 2014
• 2006 USAMO Problems/Problem 3
  ...>. Find all polynomials <math>f</math> with integer coefficients such that the sequence <math>\{ p(f(n^2))-2n) \}_{n \in \mathbb{Z} \ge 0}</math> is bound integer coefficients, suppose further that no prime divides all the
  9 KB (1,699 words) - 13:48, 11 April 2020
• 2006 USAMO Problems/Problem 5
  ...minimum number of jumps needed to reach <math>2^i k</math> is greater than the minimum number of jumps needed to reach <math>2^i</math>. ...et <math>x_{i,k}</math> denote the minimum number of jumps needed to reach the integer <math>n_{i,k} = 2^i k</math>. We must prove that
  7 KB (1,280 words) - 17:23, 26 March 2016
• 2006 USAMO Problems/Problem 4
  ...of <math>n</math>. For <math>p_1+p_2=n</math>, which is only possible in one case, <math>n=4</math>, we consider <math>p_1=p_2=2</math>. ...htarrow n<9</math>. Thus, we need to check for <math>n=1,2,3,5,7</math>. One is included because it is neither prime nor composite.
  3 KB (486 words) - 22:43, 5 August 2014
• 2006 AMC 12A Problems/Problem 5
  {{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #5]] and [[2006 AMC 10A Problems/Problem 5|2006 AMC 10A #5]]}} ...lf. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave
  1 KB (176 words) - 17:57, 16 December 2021
• 2006 AMC 12A Problems/Problem 6
  {{duplicate|[[2006 AMC 12A Problems|2006 AMC 12A #6]] and [[2006 AMC 10A Problems/Problem 7|2006 AMC 10A #7]]}} ...CD</math> is cut into two congruent hexagons, as shown, in such a way that the two hexagons can be repositioned without overlap to form a square. What is
  3 KB (528 words) - 18:14, 16 December 2021
• 2006 AMC 12A Problems/Problem 9
  ...ole number]] of cents. What is the total cost, in cents, of one pencil and one eraser? ...ath>e</math> are [[positive integer]]s, we must have <math>e \geq 1</math> and <math>p \geq 2</math>.
  3 KB (429 words) - 18:14, 26 September 2020
• Diagonal
  ...o interior diagonals, and [[concave]] quadrilaterals have one interior and one exterior diagonal. The number of diagonals of a polygon with <math>n</math> vertices is given by
  2 KB (374 words) - 00:37, 25 January 2015
• Equation
  ...] which states that two [[expression]]s are equal, identical, or otherwise the same. Equations are easily identifiable because they are composed of two ex ...metric figures instead of numbers) and other relationships which fall into the category of [[equivalence relation]]s.
  5 KB (932 words) - 12:57, 26 July 2023
• Domain (function)
  The '''domain''' of a [[function]] is the [[set]] of values on which that function is defined. ...) = x^2</math> can take a wide variety of domains, if we were to assign it one.
  1 KB (228 words) - 23:17, 16 August 2013
• Integration by parts
  ...ation by parts is to replace a difficult [[integral]] with an easier one. The formula is ...> should be chosen such that it has an "easy" (or "easier") [[derivative]] and <math>dv</math> so that it has a easy [[antiderivative]].
  1 KB (235 words) - 17:01, 11 March 2022
• Harvard-MIT Mathematics Tournament
  ...h year. It is composed of two tournaments, the ''February Tournament'' and the ''November Tournament''. ==The February Tournament==
  4 KB (539 words) - 16:58, 19 February 2023
• Intersection
  ...ction of the sets <math>\{1, 2, 3\}</math> and <math>\{1, 3, 5\}</math> is the set <math>\{1, 3\}</math>. ...1}^n A_i = A_1 \cap A_2 \cap \ldots \cap A_n</math> is the intersection of the <math>n</math> sets <math>A_1, A_2, \ldots, A_n</math>.
  1 KB (221 words) - 22:41, 11 April 2019
• Positive number
  ...ive''' [[real number]] (and so also [[rational number]] or [[integer]]) is one which is greater than [[zero (constant) | zero]]. ...a positive number is required to not have a minus sign before the number (only if it is not an equation).
  1 KB (162 words) - 14:04, 16 January 2023
• Binary operation
  ...des it into machine code or binary. Then it returns a bit based on 2 bits, one from each string on their corresponding index. There are several binary ope '''AND'''...
  591 bytes (74 words) - 19:39, 30 April 2024
• Minnesota ARML
  ...n]]. All four teams currently compete in ARML Division A, due to the White and Blue teams recently winning awards as top teams in Division B. ...oah Franske of Edina HS, and distinguished alumni coaches Andy Niedermaier and Matthias Hunt. Other assistant coaches may vary from year to year.
  4 KB (680 words) - 16:45, 10 June 2015
• 2006 AMC 10B Problems
  For real numbers <math>x</math> and <math>y</math>, define <math> x \mathop{\spadesuit} y = (x+y)(x-y) </math>. ...points, and the Cougars won by a margin of 14 points. How many points did the Panthers score?
  14 KB (2,059 words) - 01:17, 30 January 2024
• 2005 AMC 10B Problems
  ...ell all the candy bars at a price of two for &#36;<math>1</math>. What was the profit, in dollars? A positive number <math>x</math> has the property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is
  12 KB (1,874 words) - 21:20, 23 December 2020
• Platonic solid
  ...trahedron]], the regular [[octahedron]], the regular [[dodecahedron]], and the regular [[icosahedron]]. ...has four faces, all of which are [[triangle]]s. It also has four vertices and six edges. Three faces meet at each vertex.
  8 KB (1,168 words) - 22:48, 19 February 2022
• 2000 AMC 12 Problems/Problem 8
  {{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #8]] and [[2000 AMC 10 Problems|2000 AMC 10 #12]]}} ...</math>, and <math>25</math> nonoverlapping unit squares, respectively. If the pattern were continued, how many nonoverlapping unit squares would there be
  7 KB (988 words) - 15:14, 10 April 2024
• 2000 AMC 12 Problems/Problem 4
  {{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #4]] and [[2000 AMC 10 Problems|2000 AMC 10 #6]]}} ...ten [[digit]]s is the last to appear in the units position of a number in the Fibonacci sequence?
  1 KB (144 words) - 09:48, 8 November 2021
• Sigma notation
  ...rite than the explicit sum, sigma notation is also useful in that it shows the general form of each addend. ...math> defined on the integers, we write <math>\sum_{k=m}^n a(k)</math> for the sum <math>a(m)+a(m+1)+a(m+2)+\ldots+a(n-1)+a(n)</math>.
  2 KB (335 words) - 17:17, 8 February 2024
• Furman University Wylie Mathematics Tournament
  ...r schools from South Carolina, North Carolina, Georgia, Tennessee, Alabama and Florida. ...ivisions based on school population. Division I contains the schools with the higher populations.
  2 KB (365 words) - 21:21, 18 March 2017
• University of South Carolina High School Math Contest/1993 Exam/Problem 3
  If 3 circles of radius 1 are mutually tangent as shown, what is the area of the gap they enclose? ...le triangle is <math>\frac{2^2\sqrt{3}}{4}=\sqrt{3}</math>, so the area of the gap is <math>\sqrt{3}-\frac{\pi}{2}</math>, <math>\mathrm{(A)}</math>.
  1 KB (173 words) - 17:09, 4 October 2016
• University of South Carolina High School Math Contest/1993 Exam/Problem 11
  ...e probability that no card gets placed into a box having the same label as the card? ...ent]]s of 4 objects. We can know the formula for derangements or count in one of two ways:
  2 KB (334 words) - 16:27, 25 October 2023
• University of South Carolina High School Math Contest/1993 Exam/Problem 29
  ...have lengths 2, 3, and 4, what is the radius of the circle circumscribing the triangle? ...But it also has area <math>\frac{abc}{4R}</math> (where <math>R</math> is the [[circumradius]]) so <math>R = \frac{2\cdot3\cdot4}{4 (\frac{3}{4}\sqrt{15}
  2 KB (219 words) - 09:57, 31 August 2012

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