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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

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