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• Pythagorean Theorem
  ...n Theorem is one of the most frequently used theorems in [[geometry]], and is one of the many tools in a good geometer's arsenal. A very large number of This is generalized by the [[Geometric inequality#Pythagorean_Inequality | Pythagor
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• 2005 AMC 12A Problems/Problem 16
  ...s</math> is tangent to both axes and to the second and third circles. What is <math>r/s</math>? dotfactor=3;
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• 2016 AMC 10A Problems/Problem 3
  ...so we can write <math>\$12.50\cdot (4+3)=\$ 87.50.</math> Then the answer is <math>\boxed{\text{(C)}}.</math>
  1 KB (176 words) - 10:58, 16 June 2023
• 2016 AMC 10A Problems/Problem 4
  ...o <math>\tfrac{x}{y}</math>. What is the value of <math>\text{rem} (\tfrac{3}{8}, -\tfrac{2}{5} )</math>? ...xtbf{(B) } -\frac{1}{40} \qquad \textbf{(C) } 0 \qquad \textbf{(D) } \frac{3}{8} \qquad \textbf{(E) } \frac{31}{40}</math>
  2 KB (257 words) - 10:57, 16 June 2023
• 2016 AMC 10A Problems/Problem 5
  A rectangular box has integer side lengths in the ratio <math>1: 3: 4</math>. Which of the following could be the volume of the box? ...ath>x \cdot 3x \cdot 4x =12x^3</math>. If <math>x=2</math>, then <math>12x^3 = 96 \implies \boxed{\textbf{(D) } 96.}</math>
  1 KB (184 words) - 13:58, 22 August 2023
• 2016 AMC 10A Problems/Problem 6
  .../math>. Star adds her numbers and Emilio adds his numbers. How much larger is Star's sum than Emilio's? ...2 appears 3 times as a units digit, the answer is <math>10\cdot 10+1\cdot 3=\boxed{\textbf{(D) }103.}</math>
  967 bytes (143 words) - 03:18, 27 June 2023
• 2016 AMC 10A Problems/Problem 7
  ...h>60, 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 9
  ...ow, and so on up to <math>N</math> coins in the <math>N</math>th row. What is the sum of the digits of <math>N</math>? ...\frac{63\cdot 64}{2}=2016,</math> we have <math>N=63,</math> so our answer is <math>\boxed{\textbf{(D) } 9}.</math>
  2 KB (315 words) - 15:34, 18 June 2022
• 2016 AMC 10A Problems/Problem 10
  ...the two shaded regions is <math>1</math> foot wide on all four sides. What is the length in feet of the inner rectangle? filldraw(rectangle((2,2),(5,3)),white);
  2 KB (337 words) - 14:56, 25 June 2023
• 2016 AMC 10A Problems/Problem 11
  label("$4$",(8,3),dir(0)); <math>\textbf{(A)}\ 4\dfrac{3}{5} \qquad \textbf{(B)}\ 5\qquad \textbf{(C)}\ 5\dfrac{1}{4} \qquad \textbf
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• 2016 AMC 10A Problems/Problem 12
  ...bout the probability <math>p</math> that the product of the three integers is odd? ...rac{1}{3}\qquad\textbf{(D)}\ p=\dfrac{1}{3}\qquad\textbf{(E)}\ p>\dfrac{1}{3}</math>
  2 KB (297 words) - 14:54, 25 June 2023
• 2016 AMC 10A Problems/Problem 13
  <math>\textbf{(A) }1 \qquad \textbf{(B) } 2 \qquad \textbf{(C) } 3 \qquad \textbf{(D) } 4\qquad \textbf{(E) } 5</math> ...This means that Bea was originally in seat 1. Ceci must have been in seat 3 to keep seat 1 open, which leaves seat 2.
  2 KB (402 words) - 14:54, 25 June 2023
• List of United States middle school mathematics competitions
  * [[AMC 8]] hosted by the [[American Mathematics Competitions]] is a very large middle school math contest taken in-school. ([http://www.maa.o *[http://www.imc-impea.org IMC-IMPEA] is an offline/online math contest for all grades level. The contest offers ind
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• MathILy (serious mathematics infused with levity)
  ...ors are graduate or undergraduate math students. The math content covered is undergraduate- and graduate-level. Our 2024 programs will be taking place online from June 30-August 3, with MathILy at Bryn Mawr College and MathILy-Er at Arcadia University. Th
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• MATHCOUNTS
  ...ics], and others including Art of Problem Solving, the focus of MATHCOUNTS is on mathematical problem solving. Students are eligible for up to three year ...>Countdown</u>: 0.5 (School/Chapter), 1 (State/National)<br><u>Sprint</u>: 1-1.5 (School/Chapter), 2-2.5 (State/National)<br><u>Target:</u> 1.5 (School),
  10 KB (1,497 words) - 11:42, 10 March 2024
• ASIA TEAM Problems/Problem 2
  ...</math>, and <math>2013_{10}=133131</math>, so the answer is <math>1+3+3+1+3+1=\boxed{12}</math>.
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• ICTM Math Contest
  ...ers of Mathematics offers two areas of math contests: Grade School (Grades 3, 4, 5, 6, 7, 8 + Algebra 1) and High School (Regional and State Finals). ...Committee offers in-school contests at six different grade levels (grades 3-8). The season consists of three contests to be given at your school. Each
  8 KB (1,182 words) - 14:26, 3 April 2024
• Resources for mathematics competitions
  * The [http://www.kalva.demon.co.uk/ Kalva site] is one of the best resources for math problems on the planet. (Currently offli * [https://brilliant.org/ Brilliant] is a website where one can solve problems to gain points and go to higher leve
  24 KB (3,269 words) - 22:58, 18 March 2024
• William Lowell Putnam Mathematical Competition
  The '''William Lowell Putnam Mathematical Competition''' is a highly challenging, proof-oriented [[mathematics competition]] for underg ...f|difficulty=7 - 9|breakdown=<u>Problem A/B, 1/2</u>: 7<br><u>Problem A/B, 3/4</u>: 8<br><u>Problem A/B, 5/6</u>: 9}}
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• Mathematics
  '''Mathematics''' is the [[science]] of structure and change. Mathematics is important to the other sciences because it provides rigourous methods for d ==Overview=={{asy image|<math>1\,2\,3\,4\,5\,6\,7\,8\,9\,0</math>|right|The ten [[digit]]s making up <br /> the b
  6 KB (902 words) - 12:53, 3 September 2019
• AMC historical results
  This is the '''AMC historical results''' page. This page should include results for *Mean: 68.3
  17 KB (1,921 words) - 11:32, 13 April 2024
• Inequality
  ...with [[optimization]] methods. While most of the subject of inequalities is often left out of the ordinary educational track, they are common in [[math ...f <math>a</math> is greater than <math>b</math>, that is, <math>a-b</math> is positive.
  12 KB (1,798 words) - 16:20, 14 March 2023
• United States of America Mathematical Talent Search
  The '''United States of America Mathematical Talent Search''' ('''USAMTS''') is a [[mathematics competition]] in which students are challenged to write ful The USAMTS is administered by the [[Art of Problem Solving Foundation]] with support and
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• AMC 10
  ...rican Mathematics Contest 10''' ('''AMC 10'''), along with the [[AMC 12]], is one of the first exams in the series of exams used to challenge bright stud ...rican Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC.
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• AMC 12
  The '''American Mathematics Contest 12''' ('''AMC 12''') is the first exam in the series of exams used to challenge bright students, gr ...rican Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC!
  4 KB (520 words) - 12:11, 13 March 2024
• 1955 AHSME Problems/Problem 38
  ...21</math>, and <math>17</math> are obtained. One of the original integers is: ...ystem of equation should be constructed. (It doesn't matter which variable is which.)
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• American Invitational Mathematics Examination
  The '''American Invitational Mathematics Examination''' ('''AIME''') is the second exam in the series of exams used to challenge bright students on ...matical Association of America]] (MAA). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC!
  8 KB (1,057 words) - 12:02, 25 February 2024
• 1954 AHSME Problems/Problem 46
  dotfactor=3; pair A=(-3*sqrt(3)/32,9/32), B=(3*sqrt(3)/32, 9/32), C=(0,9/16);
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• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...t points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
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• United States of America Mathematical Olympiad
  The '''United States of America Mathematical Olympiad''' ('''USAMO''') is the third test in a series of exams used to challenge bright students on th ...rican Mathematics Competitions]] (AMC). [[Art of Problem Solving]] (AoPS) is a proud sponsor of the AMC and of the recent expansion of USAMO participant
  6 KB (869 words) - 12:52, 20 February 2024
• Joining an ARML team
  ...e Spring Semester to determine the team each year. The 6 practices include 3 individual tests to help determine the team and some lectures on certain ma ...ent process of selecting team members has yet to be decided upon. The team is organized by and practices at the San Diego Math Circle (SDMC), and most of
  21 KB (3,500 words) - 18:41, 23 April 2024
• AoPS Online School
  ...hosts classes for outstanding middle and high school students. The school is also accredited by the Western Association of Schools and Colleges. Each of ...ine School/Intermediate Algebra | Intermediate Algebra]] (formerly Algebra 3) — [https://artofproblemsolving.com/school/course/catalog/intermediate-al
  8 KB (965 words) - 03:41, 17 September 2020
• Wilson's Theorem
  ...)! + 1</math> is divisible by <math>p</math> if and only if <math>p</math> is prime. It was stated by John Wilson. The French mathematician Lagrange prov ...h> is composite. Then <math>p</math> has a factor <math>d > 1</math> that is less than or equal to <math>p-1</math>. Then <math>d</math> divides <math>
  4 KB (639 words) - 01:53, 2 February 2023
• Trivial Inequality
  ...ity''' is an [[inequality]] that states that the square of any real number is nonnegative. Its name comes from its simplicity and straightforwardness. ...al inequality is one of the most commonly used theorems in mathematics. It is very well-known and does not require proof.
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• Arithmetic mean
  The '''arithmetic mean''' of a [[set]] of numbers (or variables) is the sum of all the numbers, divided by the number of numbers - the [[averag is the arithmetic mean of the <math>{n}</math> numbers <math>x_1,x_2,\ldots,x_
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• Completing the square
  The idea of '''completing the square''' is to add something to an equation to make that equation a [[perfect square]]. ...math> was added to this, then we would have a [[perfect square]], <math>(x-3)^2=x^2-6x+9</math>. To do this, add <math>7</math> to each side of the equ
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• Heron's Formula
  '''Heron's Formula''' (sometimes called Hero's formula) is a [[mathematical formula | formula]] for finding the [[area]] of a [[triang ...serve as a reason for why the area <math>A</math> is never imaginary. This is equivalent of ending at step <math>4</math> in the proof and distributing.
  4 KB (675 words) - 00:05, 22 January 2024
• Polynomial
  ...abstract algebra]] often an arbitrary [[field]]). Note that a [[constant]] is also a polynomial. * <math>x^3 + 3x^2y + 3xy^2 + y^3</math>, in the variables <math>x</math> and <math>y</math>
  6 KB (1,100 words) - 01:44, 17 January 2024
• Simon's Favorite Factoring Trick
  ...3333</cmath>where <math>23333</math> is the constant term, <math>xy</math> is the product of the variables, <math>66x</math> and <math>-88y</math> are th ...>a</math> are integer constants, and the coefficient of xy must be 1(If it is not 1, then divide the coefficient off of the equation.). According to Simo
  7 KB (1,107 words) - 07:35, 26 March 2024
• Factoring
  ...mathematical toolbox. To factor, or to break an expression into factors, is to write the expression (often an [[integer]] or [[polynomial]]) as a produ This leads to the difference of cubes factorization, <cmath>a^3-b^3=(a-b)(a^2+ab+b^2)</cmath>
  3 KB (532 words) - 22:00, 13 January 2024
• Mathematical problem solving
  ...ehind The [[Art of Problem Solving]] as well as many [[math competitions]] is the use of creative methods to solve problems. In a way, students are disco An interesting example of this kind of thinking is the calculation of the sum of the [[series]] <math>\frac11 + \frac14 + \fra
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• Pigeonhole Principle
  ...principle'''. A common phrasing of the principle uses balls and boxes and is that if <math>n</math> balls are to be placed in <math>k</math> boxes and < An intuitive proof of the pigeonhole principle is as follows: suppose for contradiction that there exists a way to place <mat
  11 KB (1,985 words) - 21:03, 5 August 2023
• 1965 AHSME Problems/Problem 40
  ...+ 11x^2 + 3x + 31</math> is the square of an integer. Then <math>n</math> is: \textbf{(B) }\ 3 \qquad
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• Geometric mean
  ...while the geometric mean of the numbers <math>b</math> and <math>c</math> is the number <math>g</math> such that <math>g\cdot g = b\cdot c</math>. ...nd 2 is <math>\sqrt[4]{6\cdot 4\cdot 1 \cdot 2} = \sqrt[4]{48} = 2\sqrt[4]{3}</math>.
  2 KB (282 words) - 22:04, 11 July 2008
• Principle of Inclusion-Exclusion
  ...is counted once and only once. In particular, memorizing a formula for PIE is a bad idea for problem solving. Here, we will illustrate how PIE is applied with various numbers of sets.
  9 KB (1,703 words) - 07:25, 24 March 2024
• Combination
  ...om a set of <math>n</math> where the order in which the objects are chosen is irrelevant. We are generally concerned with finding the number of combinat This video is a great introduction to permutations, combinations, and constructive counti
  4 KB (615 words) - 11:43, 21 May 2021
• 2020 CIME II Problems/Problem 10
  ...htarrow (a-1)(b-1)=2</math> from whence we have <math>(a,b,c)\in\{(2,3,1),(3,2,1)\}</math>. ...c|a+b</math>; hence <math>a+b</math> is a multiple of <math>c</math> which is no more than <math>2c+6</math>. It follows that <math>a+b\in\{c,2c,3c,4c,5c
  2 KB (332 words) - 09:37, 30 December 2021
• Cauchy-Schwarz Inequality
  ...Bunyakovsky–Schwarz Inequality''' or informally as '''Cauchy-Schwarz''', is an [[inequality]] with many ubiquitous formulations in abstract algebra, ca ...tion 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
  The '''factorial''' is an important function in [[combinatorics]] and [[analysis]], used to determ ...h>. Alternatively, a [[recursion|recursive definition]] for the factorial is <math>n!=n \cdot (n-1)!</math>.
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• Discriminant
  ...negative, the equation has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. We know that the discriminant of a polynomial is the product of the squares of the differences of the polynomial roots <math
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• Menelaus' theorem
  It is named after Menelaus of Alexandria. ...gle ABC</math>, where <math>P</math> is on <math>BC</math>, <math>Q</math> is on the extension of <math>AC</math>, and <math>R</math> on the intersection
  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 ...ards. One indvididual [need name] from Taiwan would have placed in the top 3 students overall on the individual round tiebreaker but was not considered
  19 KB (2,632 words) - 14:31, 12 June 2022
• Fermat's Little Theorem
  ...if they have a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. ...n [[integer]], <math>{p}</math> is a [[prime number]] and <math>{a}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equi
  16 KB (2,675 words) - 10:57, 7 March 2024
• Parabola
  A '''parabola''' is a type of [[conic section]]. A parabola is a [[locus]] of points that are equidistant from a point (the [[focus]]) and ...: <math>y = a{x}^2+b{x}+c</math> where a, b, and c are [[constant]]s. This is useful for manipulating the polynomial.
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• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...me 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>, then <math>{a}^{
  3 KB (542 words) - 17:45, 21 March 2023
• Geometric inequality
  A '''geometric inequality''' is an [[inequality]] involving various measures ([[angle]]s, [[length]]s, [[ar ...e]] triangle is greater than the length of the third side. This inequality is particularly useful and shows up frequently on Intermediate level geometry
  7 KB (1,296 words) - 14:22, 22 October 2023
• Brahmagupta's Formula
  '''Brahmagupta's Formula''' is a [[formula]] for determining the [[area]] of a [[cyclic quadrilateral]] gi ...formula which Brahmagupta derived for the area of a general quadrilateral is
  3 KB (465 words) - 18:31, 3 July 2023
• Ptolemy's theorem
  ...tween the side lengths and the diagonals of a [[cyclic quadrilateral]]; it is the [[equality condition | equality case]] of [[Ptolemy's Inequality]]. Pto ...\angle ABC+m\angle ADC=180^\circ .</math> However, <math>\angle ADP</math> is also supplementary to <math>\angle ADC,</math> so <math>\angle ADP=\angle A
  7 KB (1,198 words) - 20:39, 9 March 2024
• Elementary symmetric sum
  An '''elementary symmetric sum''' is a type of [[summation]]. ...leq n</math>). For example, if <math>n = 4</math>, and our set of numbers is <math>\{a, b, c, d\}</math>, then:
  2 KB (275 words) - 12:51, 26 July 2023
• Number theory/Advanced
  ...ory from the perspective of [[abstract algebra]]. In particular, heavy use is made of [[ring theory]] and [[Galois theory]]. Algebraic methods are partic ...erties of prime numbers. The most famous problem in analytic number theory is the [[Riemann Hypothesis]].
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• 1959 IMO Problems/Problem 2
  For what real values of <math>x</math> is Since the term inside the square root is a perfect square, and by factoring 2 out, we get
  3 KB (466 words) - 12:04, 12 April 2024
• Harmonic mean
  ...math>n</math> [[positive]] [[real number]]s <math> x_1, x_2... x_n </math> is defined to be: <math> \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_ ...ate <math>\frac 3{\frac 13 + \frac 16 - \frac 12} = \frac 30</math>, which is obviously problematic.
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• LaTeX
  ...d [[math|mathematical]] and scientific writing. <math>\text{\LaTeX}</math> is very handy for producing equations such as <cmath>1+2+3+4+5+\sin \pi = \frac{5\cdot 6}{2}+0=15.</cmath>
  1 KB (164 words) - 19:09, 14 February 2024
• North Carolina MathCounts
  In the North Carolina MathCounts State Competition, the Countdown Round is unofficial in that it doesn't affect individual results. * 1987 - Ashley Reiter (3), Stephen London (41), Tim Ross (37), Ghene Faulcon, Coach: Caroline Wolfe
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• Divisibility
  In [[number theory]], '''divisibility''' is the ability of a number to evenly divide another number. The study of divis ...th>a</math> is a '''multiple''' of <math>b</math>, and that <math>a</math> is '''divisible''' or '''evenly divisible''' by <math>b</math>.
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• Real number
  ...s that are not real are <math>\ 3i</math>, <math>\ 3+2.5i</math>, <math>\ 3+2i+2j+k</math>, i.e. [[complex number]]s, and [[quaternion]]s. The set of real numbers, denoted by <math>\mathbb{R}</math>, is a subset of [[complex number]]s(<math>\mathbb{C}</math>). Commonly used sub
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• Divisibility rules
  ..., in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. === Divisibility Rule for 3 and 9===
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• Induction
  ...rks for <math>n=1+1=2</math>, which in turn means it works for <math>n=2+1=3</math>, and so on. ...e. If a problem asks you to prove something for all integers greater than 3, you can use <math>n=4</math> as your base case instead. You might have to
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• Triangle
  A '''triangle''' is a type of [[polygon]]. {{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}
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• Greatest common divisor
  ...common factor''')) of two or more [[integer]]s is the largest integer that is a [[divisor]] of all the given numbers. The GCD is sometimes called the '''greatest common factor''' ('''GCF''').
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• Overcounting
  ...otal count via subtraction or division. The idea of strategic overcounting is fundamental to [[combinatorics]] and plays a role in incredibly important c An example of a classic problem is as follows:
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• Constructive counting
  In [[combinatorics]], '''constructive counting''' is a [[counting]] technique that involves constructing an item belonging to a ...fundamental techniques in counting. Familiarity with constructive counting is essential in combinatorics, especially in intermediate competitions.
  12 KB (1,896 words) - 23:55, 27 December 2023
• Jensen's Inequality
  '''Jensen's Inequality''' is an inequality discovered by Danish mathematician Johan Jensen in 1906. If <math>{F}</math> is a concave function, we have:
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• Casework
  ...parts individually, then adding together the totals of each part. Casework is a very general problem-solving approach, and as such has wide applicability ...e, most problems cannot be completely solved through casework. However, it is crucial as an intermediate step across all of mathematics, not just in comp
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• Euclidean algorithm
  ...(GCD) of two elements of a [[Euclidean domain]], the most common of which is the [[nonnegative]] [[integer]]s <math>\mathbb{Z}{\geq 0}</math>, without [ The basic idea is to repeatedly use the fact that <math>\gcd({a,b}) \equiv \gcd({b,a - b})</m
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• Generating function
  ...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>. ...n}=2^n</math>(let <math>{x}=1</math>), also <math>{n \choose 1}+{n \choose 3}+\cdots={n \choose 0}+{n \choose 2}+\cdots</math>.
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• Counting number
  ...at we count numbers of objects using positive integers (for example, <math>3</math> pencils). These are just the numbers in the set of {1,2,3,4,..}
  429 bytes (61 words) - 01:10, 20 February 2023
• Binomial Theorem
  ...efficient]]. In other words, the coefficients when <math>(a + b)^n</math> is expanded and like terms are collected are the same as the entries in the <m For example, <math>(a + b)^5 = a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a b^4 + b^5</math>, with coefficients <math>1 = \binom{5}{0}</math>, <m
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• Prime number
  A '''prime number''' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | div ...fined as being neither prime 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
• Recursion
  ...f the sequence in terms of previous values: <math>F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8</math>, and so on. Often, it is convenient to convert a recursive definition into a closed-form definition.
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• Function
  ...e value in the second. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its inp ...]] between <math>A</math> and <math>B</math>.) We say that <math>f</math> is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \t
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• Complementary counting
  ...ach. A large hint that complementary counting may lead to a quick solution is the phrase "not" or "at least" within a problem statement. ...th>. In most instances, though, <math>A</math> is obvious from context and is committed from mention.
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• Combinatorial identity
  ...ger]]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>. ...number of ways to choose <math>k</math> things from <math>n</math> things is equal to the number of ways to choose <math>k-1</math> things from <math>n-
  12 KB (1,993 words) - 23:49, 19 April 2024
• Integer
  ...<math>a-b</math>, and their product <math>ab</math> are all integers (that is, the integers are closed under addition and multiplication), but their quot ...a more simple and straightforward definition, an integer is a number that is '''not''' a [[decimal]] or a [[fraction]].
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• Prime factorization
  ...ve integer <math>n</math>, the '''prime factorization''' of <math>n</math> is an expression for <math>n</math> as a product of powers of [[prime number]] The form of a prime factorization is
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• Composite number
  ...elf. Some composite numbers are <math>4=2^2</math> and <math>12=2\times 6=3\times 4</math>. Composite numbers '''atleast have 2 distinct [[prime]] [[di ...s the only even [[prime number]], three is the only multiple of three that is prime, and so on.
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• Natural number
  ...gebra]], but usually not in the contexts of [[number theory]]. When there is risk of confusion, mathematicians often resort to less ambiguous notations,
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• Circle
  A '''circle''' is a geometric figure commonly used in Euclidean [[geometry]]. ...d the [[center]] and the distance from the center to a point on the circle is called the [[radius]].
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• Ellipse
  An '''ellipse''' is a type of [[conic section]]. An ellipse is formed by cutting through a [[cone]] at an [[angle]].
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• Base numbers
  ...the number 2746. This number can be rewritten as <math>2746_{10}=2\cdot10^3+7\cdot10^2+4\cdot10^1+6\cdot10^0.</math> ...<math>10^2</math>'s, and the fourth digit tells us there are two <math>10^3</math>'s.
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• Base numbers/Common bases
  ...], and many other kinds of bases. The best known one is [[phinary]], which is base [[phi]]; others include "[[Fibonacci base]]" and base negative two. [[Binary]] is base 2. It's a favorite among computer programmers. It has just two digits
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• 2001 AMC 12 Problems/Problem 1
  ...1 AMC 12 Problems|2001 AMC 12 #1]] and [[2001 AMC 10 Problems|2001 AMC 10 #3]]}} The sum of two numbers is <math>S</math>. Suppose <math>3</math> is added to each number and then
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• 2001 AMC 12 Problems/Problem 2
  ...<math>P(23) = 6</math> and <math>S(23) = 5</math>. Suppose <math>N</math> is a two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit of <math>N</math>?
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• Mathematical Kangaroo
  ...s in grades 1 through 12. The competition consists of a single round that is taken on the same date (third Thursday of March) at a registered center. A ...me state or country, so competitors often register for a testing site that is the closest or most convenient for them despite being outside of the state.
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• American Scholastic Mathematics Association
  ...top eight scorers of each team counted towards the team's total. The test is 35 minutes long and assumes the use of a calculator. Contest #3 - December 12, 2019
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• Least common multiple
  ...y one LCM. The LCM of a set of numbers <math>\{a_1,a_2,\cdots,a_n\}</math> is conventionally represented as <math>[a_1,a_2,\ldots,a_n]</math>. ...a multiple that is common to all of them. This is a tedious method, so it is usually only used when the numbers are small. For example, suppose we wante
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• Math Bee
  '''Math Bee''' is a [[mathematics competition]] for students in grades K through 8 of Indian * Level II: For grades 3, 4, and 5. [[MOEMS]]-type problems can be found.
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• Ptolemy's Inequality
  '''Ptolemy's Inequality''' is a famous inequality attributed to the Greek mathematician Ptolemy. with equality if and only if <math>ABCD</math> is a cyclic quadrilateral with diagonals <math>AC </math> and <math>BD </math>
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• Median of a triangle
  A '''median''' of a [[triangle]] is a [[cevian]] of the triangle that joins one [[vertex]] to the [[midpoint]] In the following figure, <math>AM</math> is a median of triangle <math>ABC</math>.
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• Pi
  '''Pi''' is an [[irrational number]] (in fact, [[transcendental number]], as proved by ...math>\frac{22}{7} \approx 3.14285</math> and <math>\frac{355}{113} \approx 3.1415929</math>.
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• Fibonacci sequence
  ...s the sum of the two preceding it. The first few terms are <math>1, 1, 2, 3, 5, 8, 13, 21, 34, 55,...</math>. ...ivial example of a [[linear recursion]] with constant coefficients. There is also an explicit formula [[#Binet's formula|below]].
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• Muirhead's Inequality
  The inequality is easier to understand given an example. Since the sequence <math>(5,1)</mat ...lympiad solution; one should use an application of AM-GM instead. Thus, it is suggested that Muirhead be used only to verify that an inequality ''can'' b
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• Cardinality
  ..., 3\}, \{1, 2, 3\}\}</math> is 3, and the cardinality of the [[empty set]] is 0. ...In the above example, the cardinality of <math>\{3, 4\}</math> is <math>|\{3, 4\}| = 2</math>. Sometimes, the notations <math>n(A)</math> and <math>\# (
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• Fundamental Theorem of Calculus
  This section is for people who know what [[integral]]s are but don't know the Fundamental T * Evaluate: <math>\int_2^5 x^3 dx</math> and <math>\int_{.2}^{.4} \cos(x) dx</math>. (The next few questi
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• Polygon
  A '''polygon''' is a closed [[planar figure]] consisting of straight [[line segment]]s. There A polygon can be [[regular polygon| regular]] or irregular. A polygon is regular if all sides are the same length and all angles are [[congruent]].
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• 2011 AMC 10B Problems/Problem 22
  ...opposite face has all its edges on the lateral faces of the pyramid. What is the volume of this cube? ...27} \qquad\textbf{(D)}\ \frac{\sqrt{2}}{9} \qquad\textbf{(E)}\ \frac{\sqrt{3}}{9}</math>
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• 2011 AMC 10B Problems/Problem 4
  ...the costs equally, LeRoy must give Bernardo half of the difference, which is <math>\boxed{\textbf{(C) } \;\frac{B-A}{2}}</math> .... Quickly, we realize the only way they could pay the same amount of money is if they both pay 45 dollars. This means LeRoy must give Bernardo <math>50 -
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• Vector
  ...x). Most generally, but also most abstractly, a vector is any object which is an element of a given vector space. ...es, <math>(x\,\,y\,\,z\,\,...)</math>. The set of vectors over a [[field]] is called a [[vector space]].
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• Logarithm
  ...come down to never having to deal with massive numbers. ex. :<cmath>((((((3^5)^6)^7)^8)^9)^{10})^{11}=\underbrace{1177\ldots 1}_{\text{793549 digits}}< left to right parenthesized exponentiation) is only 7 digits before the decimal point. Comparing the logs of the numbers t
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• Law of Cosines
  The '''Law of Cosines''' is a theorem which relates the side-[[length]]s and [[angle]]s of a [[triangle In the case that one of the angles has measure <math>90^\circ</math> (is a [[right angle]]), the corresponding statement reduces to the [[Pythagorea
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• Schur's Inequality
  ...Inequality''' is an [[inequality]] that holds for [[positive number]]s. It is named for Issai Schur. ...ath>a=b=c</math> or when two of <math>a,b,c</math> are equal and the third is <math>{0}</math>.
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• Trigonometric identities
  ...<math>(\cos (x), \sin (x))</math> is defined to be on the unit circle, it is a distance one away from the origin. Then by the distance formula, <math>\s * <math>\sin 3x = 3\sin x-4\sin^3 x</math>
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• Irrational number
  An '''irrational number''' is a [[real number]] that cannot be expressed as the [[ratio]] of two [[intege ...entury <math>B.C</math>. The Pythagoreans lived by the doctrine that ''all is number'', or that all things could be explained by relationships between nu
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• Complex number
  ...ive]], so this equation has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{- ...= \sqrt{-1} </math> is the [[imaginary unit]]. The set of complex numbers is denoted by <math>\mathbb{C}</math>. The set of complex numbers contains th
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• Angle bisector
  ...math> such that the angle between this line and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</math>: D=(3,4);
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• Without loss of generality
  ...ten abbreviated to WLOG, is a frequently used expression in math. The term is used to indicate that the following proof emphasizes on a particular case, If you use WLOG in a proof and the statement is not necessarily true, points will get marked off. For example, you can't sa
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• Law of Sines
  The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines ...math>, <math>c</math> opposite to <math>C</math>, and where <math>R</math> is the circumradius:
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• Geometric sequence
  ...hat the ratio between any two consecutive terms is constant. This constant is called the '''common ratio''' of the sequence. ...mon ratio <math>-1/2</math>; however, <math>1, 3, 9, -27</math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between c
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• Arithmetic sequence
  ...he difference between any two consecutive terms is constant. This constant is called the '''common difference''' of the sequence. ...ence with common difference <math>1</math> and <math>99, 91, 83, 75</math> is an arithmetic sequence with common difference <math>-8</math>; however, <ma
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• Fermat's Last Theorem
  ...ting that for positive [[integers]] <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> ...vered a truly marvelous demonstration of this proposition that this margin is too narrow to contain.''"
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• 2005 USAMO Problems/Problem 4
  ...piece of length <math>k_i</math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...all four of the leg ends touch the floor. Note that a cut leg of length 0 is permitted.)
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• 2000 AMC 12 Problems/Problem 1
  ...ger]]s such that the product <math>I \cdot M \cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? ...process on <math>2001</math> to get <math>667 * 3 * 1</math> as our <math>3</math> factors.
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• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or sometimes [[natural number]] or [[ ...a Diophantine equation has infinitely many solutions, [[parametric form]] is used to express the relation between the variables of the equation.
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• Fraction
  A '''fraction''' is the [[ratio]] of two [[number]]s. Most commonly, we consider [[rational nu ...numerator is the same as the denominator such as <math>\frac{3}{3}</math> is always equal to <math>1</math>.
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• Functional equation
  A '''functional equation''', roughly speaking, is an equation in which some of the unknowns to be solved for are [[function]] ...he '''inverse function'''.) Often the inverse of a function <math>f</math> is denoted by <math>f^{-1}</math>.
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• Exponential form
  ...(yes, again!) rewrite <math>z</math> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. D=(1/2,sqrt(3)/2);
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• Floor function
  ...ion is the same as "dropping everything after the decimal point," but this is ''not'' true for negative values. *<math>\lfloor 3.14 \rfloor = 3</math>
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• Pascal's triangle
  '''Pascal's triangle''' is a triangle which contains the values from the [[binomial expansion]]; its v ...n</math>, the sum of the values on row <math>n</math> of Pascal's Triangle is <math>2^n</math>.
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• Exponentiation
  .../math>, where <math>b</math> is the exponent (or power) and <math>a</math> is the [[base]]. ...ed if a equation has [[parentheses]] or the first one performed when there is no parentheses.
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• Trigonometry
  ...gths and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way toward In contest math, trigonometry is an integral subfield of both [[geometry]] and [[algebra]]. Many essential r
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• Mathematical Olympiad Summer Program
  ...especially the [[International Mathematical Olympiad]]. While the program is free to participants, invitations are limited to the top finishers on the [ ...d train the US team for the [[International Mathematical Olympiad]]. This is done at the start of MOP via a [[team selection test]] (TST). The results
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• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...-Arithmetic Mean-Geometric Mean-Harmonic Mean Inequality''' (EM-AM-GM-HM), is an [[inequality]] of the [[root-mean power]], [[arithmetic mean]], [[geomet ...where <math>n_1>1,~~0<n_2<1,~~-1<n_3<0,~~n_4<-1</math>, and <math>n</math> is the root mean power.
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• Harmonic series
  Generally, a '''harmonic series''' is a [[series]] whose terms involve the [[reciprocal]]s of the [[positive inte The the most basic harmonic series is the infinite sum
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• Goldbach Conjecture
  ...proven [[conjecture]] stating that every [[even integer]] greater than two is the sum of two [[prime number]]s. The conjecture has been tested up to 400 Goldbach's conjecture is one of the oldest unsolved problems in [[number theory]] and in all of math
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• Twin Prime Conjecture
  The '''Twin Prime Conjecture''' is a [[conjecture]] (i.e., not a [[theorem]]) that states that there are [[inf One possible strategy to prove the infinitude of twin primes is an idea adopted from the proof of [[Dirichlet's Theorem]]. If one can show
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• Quadratic residues
  ...</math> if there is some integer <math>n</math> so that <math>n^2-a</math> is [[divisibility | divisible]] by <math>m</math>. ...modulo\ }\ p, \\ -1 & \mathrm{if }\ p\nmid a\ \mathrm{ and }\ a\ \mathrm{\ is\ a\ quadratic\ nonresidue\ modulo\ }\ p. \end{cases}</math>
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• Power of a Point Theorem
  The '''Power of a Point Theorem''' is a relationship that holds between the lengths of the [[line segment]]s form # One of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2
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• Collatz Problem
  ...th>\{n,f(n),f(f(n)),f(f(f(n))),\ldots\}</math> contains 1. This conjecture is still open. Some people have described it as the easiest unsolved problem i ...6m+4\over 2}=3m+2</cmath> we can then observe that; only if <math>m</math> is even will another division by 2 be possible.
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• Algebraic number
  ...[27]{19}}{\sqrt[3]{4}+\sqrt[7]{97}}</math>. A number that is not algebraic is called a [[transcendental number]], such as <math>e</math> or <math>\pi</ma ...mbers is large, there are only [[countable|countably]] many of them. That is, the algebraic numbers have the same [[cardinality]] as the [[natural numbe
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• International Mathematical Olympiad
  The '''International Mathematical Olympiad''' is the pinnacle of all high school [[mathematics competition]]s and the oldest ...eakdown=<u>Problem 1/4</u>: 6.5<br><u>Problem 2/5</u>: 7.5-8<br><u>Problem 3/6</u>: 9.5<br><u>Problem SL1-2</u>: 5.5-7<br><u>Problem SL3-4</u>: 7-8<br><
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• Prime Number Theorem
  The '''Prime Number Theorem''' (PNT) is one of the most celebrated results in [[analytic number theory]]. Indeed, it is
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• Infinite
  ...n]] <math>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element that y
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• Isosceles trapezoid
  ...isosceles trapezoid''' is a geometric figure that lies in a [[plane]]. It is a specific type of [[trapezoid]] in which the legs have the same length. I * the segment joining the midpoints of the bases is perpendicular to the bases
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• Mock AMC
  A '''Mock AMC''' is a contest intended to mimic an actual [[AMC]] (American Mathematics Competi ...popular in the months leading up to the actual [[AMC]] competition. There is no guarantee that community members will make Mock AMCs in any given year,
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• Mock AIME
  A '''Mock AIME''' is a contest that is intended to mimic the [[AIME]] competition. (In more recent years, recurrin ...Y2QwOTc3NWZiYjY0LnBkZg==&rn=TWlsZG9yZiBNb2NrIEFJTUUucGRm Mildorf Mock AIME 3]
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• Permutation
  A '''permutation''' of a [[set]] of <math>r</math> objects is any rearrangement (linear ordering) of the <math>r</math> objects. There a ...of [[infinite]] sets. In this case, a permutation of a set <math>S</math> is simply a [[bijection]] between <math>S</math> and itself.
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• 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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• Mock USAMO
  ...hen a mock USAMO is run on [[AoPS]]/[[MathLinks]], a very wide time window is often allowed to take the mock USAMO. ** [http://www.artofproblemsolving.com/blog/2712 Mock USAMO 3 2006]
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• Modular arithmetic/Introduction
  ...of arithmetic that involves only [[integers]]. This goal of this article is to explain the basics of modular arithmetic while presenting a progression <math>1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 0, \ldots </math>
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• Perfect square
  An [[integer]] <math>n</math> is said to be a '''perfect square''' if there is an integer <math>m</math> so that <math>m^2=n</math>. The first few perfect ...of the first <math>n</math> square numbers (starting with <math>1</math>) is <math>\frac{n(n+1)(2n+1)}{6}</math>
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• Uncountable
  ...or [[countably infinite]]. The most common example of an uncountable set is the set of [[real number]]s <math>\mathbb{R}</math>. == Proof that <math>\mathbb{R}</math> is uncountable ==
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• Modular arithmetic/Intermediate
  ...quiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...<math>n</math>, the relation <math>a \equiv b</math> (mod <math>n</math>) is an [[equivalence relation]] on the set of integers. This relation gives ri
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• Right triangle
  A '''right triangle''' is any [[triangle]] with an angle of 90 degrees (that is, a [[right angle]]). A = (0, 3);
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• 2006 AIME I Problems
  ..., \angle B </math> is a right angle, diagonal <math> \overline{AC} </math> is perpendicular to <math> \overline{CD}, AB=18, BC=21, </math> and <math> CD Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements o
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• 2006 AIME I Problems/Problem 15
  ...n that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possible Suppose <math>b_{i} = \frac {x_{i}}3</math>.
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• 2006 AIME I Problems/Problem 14
  ...notation <math> \lfloor x\rfloor </math> denotes the greatest integer that is less than or equal to <math> x. </math>) currentprojection = perspective(1,-10,3.3);
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• 2006 AIME I Problems/Problem 13
  ...atest integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. ...h>k</math> is odd, then <math>n+1</math> is even, hence <math>k+n-1</math> is odd, and <math>S_n</math> cannot be a perfect square. Hence <math>k</math>
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• 2006 AIME I Problems/Problem 11
  ...h> k </math> for each [[integer]] <math> k, 1 \le k \le 8. </math> A tower is to be built using all 8 cubes according to the rules: ...s than can be constructed. What is the [[remainder]] when <math> T </math> is divided by 1000?
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• 2006 AIME I Problems/Problem 10
  ...d <math> c </math> are positive integers whose [[greatest common divisor]] is 1. Find <math> a^2+b^2+c^2. </math> int[] array={3,3,2};
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• 2006 AIME I Problems/Problem 9
  The [[sequence]] <math> a_1, a_2, \ldots </math> is [[geometric sequence|geometric]] with <math> a_1=a </math> and common [[rat ...<math>a, r</math> [[positive integer]]s. <math>a^{12}r^{66}=8^{2006} = (2^3)^{2006} = (2^6)^{1003}</math> so <math>a^{2}r^{11}=2^{1003}</math>.
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• 2006 AIME I Problems/Problem 8
  ...he area of rhombus <math> \mathcal{T}</math>. Given that <math> K </math> is a [[positive integer]], find the number of possible values for <math> K</ma ...(0,-3.2), F=(-1.65,-1.6), G=(0.45,-1.6), H=(3.75,-1.6), I=(2.1,0), J=(2.1,-3.2), K=(2.1,-1.6);
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• 2006 AIME I Problems/Problem 7
  ...[[region]] <math> C </math> to the area of shaded region <math> B </math> is 11/5. Find the ratio of shaded region <math> D </math> to the area of shade pair A=(1/3,4), B=A+7.5*dir(-17), C=A+7*dir(10);
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• 2006 AIME I Problems/Problem 5
  ...rt{10}+144\sqrt{15}+2006}</math> can be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[p <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2006}</cmath>
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• 2006 AIME I Problems/Problem 4
  ...h> 1!2!3!4!\cdots99!100!. </math> Find the remainder when <math> N </math> is divided by <math>1000</math>. ...ng into our given expression. Since there are clearly more 2s than 5s, it is sufficient to count the number of 5s.
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• 2006 AIME I Problems/Problem 3
  ...]] such that when its leftmost [[digit]] is deleted, the resulting integer is <math>\frac{1}{29}</math> of the original integer. ...7.</math> But <math>a_n</math> is a nonzero digit, so the only possibility is <math>a_n = 7.</math> This gives <cmath>7 \cdot 10^n = 28N_0</cmath> or <cm
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• 2006 AIME I Problems/Problem 2
  ...<math> \mathcal{A} </math> be a 90-[[element]] [[subset]] of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </math> be the sum of the elements o ...995</math> are possible values of S, so the number of possible values of S is <math>4995-4095+1=901</math>.
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• 2022 AIME I Problems/Problem 1
  ...x)</math> and <math>Q(x)</math> cancel, we conclude that <math>R(x)</math> is a linear polynomial. so the slope of <math>R(x)</math> is <math>\frac{106-108}{20-16}=-\frac12.</math>
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• 2021 JMC 10
  What is the value of <cmath>\dfrac{20}{2\cdot1} - \dfrac{2+0}{2/1}?</cmath> <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qquad\te
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• 2006 AMC 12B Problems
  What is <math>( - 1)^1 + ( - 1)^2 + \cdots + ( - 1)^{2006}</math>? .../math>, define <math>x\spadesuit y = (x + y)(x - y)</math>. What is <math>3\spadesuit(4\spadesuit 5)</math>?
  13 KB (2,058 words) - 12:36, 4 July 2023
• 2006 AMC 12A Problems
  Sandwiches at Joe's Fast Food cost <math>3</math> dollars each and sodas cost <math>2</math> dollars each. How many do Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>?
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  ...\%</math> of <math>x</math> and <math>20 \%</math> of <math>y</math>. What is <math>x - y</math>? ...+ 7 = 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
  Alicia earns <math> 20</math> dollars per hour, of which <math>1.45\%</math> is deducted to pay local taxes. How many cents per hour of Alicia's wages are ...ct answer is worth <math>0</math> points, and each problem left unanswered is worth <math>2.5</math> points. If Charlyn leaves <math>8</math> of the <mat
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2003 AMC 12A Problems
  What is the difference between the sum of the first <math>2003</math> even counting ...es and another pair of socks and a shirt 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
  <math>(2x+3)(x-4)+(2x+3)(x-6)=0 </math> ...the result by 9. Instead, she subtracted 9 and then divided the result by 3, giving an answer of 43. What would her answer have been had she worked the
  12 KB (1,792 words) - 13:06, 19 February 2020
• 2000 AMC 12 Problems
  ...ntegers such that the product <math>I \cdot M \cdot O = 2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? == Problem 3 ==
  13 KB (1,948 words) - 12:26, 1 April 2022
• 2001 AMC 12 Problems
  The sum of two 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
  ...ne numbers in the set <math>\{9, 99, 999, 9999, \ldots, 999999999\}</math> is a <math>9</math>-digit number <math>M</math>, all of whose digits are disti 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 <cmath>\frac{2-4+6-8+10-12+14}{3-6+9-12+15-18+21}?</cmath>
  13 KB (1,987 words) - 18:53, 10 December 2022
• 2004 AMC 12B Problems
  <math>(\mathrm {A}) 3\qquad (\mathrm {B}) 6 \qquad (\mathrm {C}) 9 \qquad (\mathrm {D}) 12 \qquad ...>d</math> are 0, 1, 2, and 3, although 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
  ...property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? == Problem 3 ==
  12 KB (1,781 words) - 12:38, 14 July 2022
• 2006 AMC 12B Problems/Problem 2
  .../math>, define <math>x\spadesuit y = (x + y)(x - y)</math>. What is <math>3\spadesuit(4\spadesuit 5)</math>? <math>3\spadesuit -9=-72 \Rightarrow \text{(A)}</math>
  473 bytes (71 words) - 10:44, 4 July 2013
• 2006 AMC 12B Problems/Problem 4
  ...</math>. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the <math>20.00</math> that she will receive i The total price of the items is <math>(8-.01)+(5-.01)+(3-.01)+(2-.01)+(1-.01)=19-.05=18.95</math>
  1 KB (152 words) - 16:11, 8 December 2013
• 2006 AMC 12B Problems/Problem 5
  ...hile Bob is also walking east, but at a speed of 5 miles per hour. If Bob is now 1 mile west of John, how many minutes will it take for Bob to catch up ...Bob is catching up to John is <math>5-3=2</math> miles per hour. Since Bob is one mile behind John, it will take <math>\frac{1}{2} \Rightarrow \text{(A)}
  654 bytes (115 words) - 21:47, 1 August 2020
• 2006 AMC 12B Problems/Problem 7
  The first child can be seated in <math>3</math> spaces. <math>3 \times 2 \times 2 = 12 \Rightarrow \text{(B)}</math>
  1 KB (213 words) - 15:33, 9 April 2024
• 2006 AMC 12B Problems/Problem 8
  ...>y = \frac 14x + b</math> intersect at the point <math>(1,2)</math>. What is <math>a + b</math>?<!-- don't remove the following tag, for PoTW on the Wik <math>\frac{3}{4}(x+y)=a+b</math>
  1 KB (235 words) - 00:46, 6 January 2022
• 2006 AMC 12B Problems/Problem 9
  ...ces for <math>a</math> and <math>b</math>. Thus there are altogether <math>3+10+21=\boxed{34}</math> such integers. If it was 2, there is 1 possibility for the hundreds digit, 3 for the ones digit.
  3 KB (405 words) - 16:17, 4 April 2022
• 2006 AMC 12B Problems/Problem 13
  ...s <math> ABCD</math> is 24, and <math> \angle BAD = 60^\circ</math>. What is the area of rhombus <math> BFDE</math>? ...n, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
  3 KB (447 words) - 03:49, 16 January 2021
• 2006 AMC 12B Problems/Problem 14
  ...> and <math>N</math> are all positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? ...ply that if <math>B=2</math> and <math>J=3</math>, then <math>4B+5J=4(2)+5(3)=23</math>. The problem asks for the total cost of jam, or <math>N(5J)=11(1
  1 KB (227 words) - 17:21, 8 December 2013
• 2006 AMC 12B Problems/Problem 15
  ...h> \overline{BC}</math> are common external tangents to the circles. What is the area of hexagon <math> AOBCPD</math>? ...bf{(B) } 24\sqrt {2} \qquad \textbf{(C) } 36 \qquad \textbf{(D) } 24\sqrt {3} \qquad \textbf{(E) } 32\sqrt {2}</math>
  3 KB (458 words) - 16:40, 6 October 2019
• 2006 AMC 12B Problems/Problem 16
  ...>C</math> at <math>(0,0)</math> and <math>(7,1)</math>, respectively. What is its area? \mathrm{(A)}\ 20\sqrt {3}
  1 KB (203 words) - 16:36, 18 September 2023
• 2006 AMC 12B Problems/Problem 17
  ...<math>6</math> on each die are in the ratio <math>1:2:3:4:5:6</math>. What is the probability of rolling a total of <math>7</math> on the two dice? The probability of getting an <math>x</math> on one of these dice is <math>\frac{x}{21}</math>.
  1 KB (188 words) - 22:10, 9 June 2016
• 2006 AMC 12B Problems/Problem 18
  ...can easily be shown that each location that satisfies these two conditions is indeed reachable. If the object only makes <math>1</math> move, it is obvious that there are only 4 possible points that the object can move to.
  2 KB (354 words) - 16:57, 28 December 2020
• 2006 AMC 12B Problems/Problem 19
  ..."and the last two 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
  ...th>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 22
  ...are 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
  ...ath>, where <math>a</math> and <math>b</math> are positive integers. What is <math>a+b</math>? MP("90^\circ-\alpha",C,3*dir(30),f);
  7 KB (1,169 words) - 14:04, 10 June 2022
• 2006 AMC 12B Problems/Problem 24
  ...\le \frac{\pi}{2}</math> and <math>0 \le y \le \frac{\pi}{2}</math>. What is the area of the subset of <math>S</math> for which <cmath> \mathrm{(D)}\ \dfrac{3\pi^2}{16}
  3 KB (563 words) - 22:45, 24 October 2021
• 2006 AMC 12B Problems/Problem 25
  A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> for <math>n\geq 1</m ...sequence <math>(a_n)</math> completes at <math>i</math> if <math>i</math> is the minimal positive integer such that <math>a_i = a_{i + 1} = 1</math>. Ot
  5 KB (924 words) - 12:02, 15 June 2022
• 2006 AMC 12A Problems/Problem 10
  For how many real values of <math>x</math> is <math>\sqrt{120-\sqrt{x}}</math> an integer? <math> \textbf{(A) } 3\qquad \textbf{(B) } 6\qquad \textbf{(C) } 9\qquad \textbf{(D) } 10\qquad \t
  1 KB (167 words) - 23:23, 16 December 2021
• 2006 AMC 12A Problems/Problem 13
  ...e centers of three mutually externally tangent [[circle]]s, as shown. What is the sum of the areas of the three circles? <cmath>r_A + r_B = 3</cmath>
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• 2006 AMC 12A Problems/Problem 14
  ...debt could be paid 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? ...mon divisor]]) of <math>a</math> and <math>b</math>. Therefore, the answer is <math>gcd(300,210)=\boxed{\textbf{(C) }30}.</math>
  3 KB (442 words) - 03:13, 8 August 2022
• 2006 AMC 12A Problems/Problem 15
  Suppose <math>\cos x=0</math> and <math>\cos (x+z)=1/2</math>. What is the smallest possible positive value of <math>z</math>? <math> \mathrm{(A) \ } \frac{\pi}{6}\qquad \mathrm{(B) \ } \frac{\pi}{3}\qquad \mathrm{(C) \ } \frac{\pi}{2}\qquad \mathrm{(D) \ } \frac{5\pi}{6} \
  919 bytes (138 words) - 12:45, 4 August 2017
• 2006 AMC 12A Problems/Problem 16
  ...d <math>CD</math> intersect at <math>E</math>, and <math>AE=5</math>. What is <math>CD</math>? dotfactor=3;
  2 KB (286 words) - 10:16, 19 December 2021
• 2006 AMC 12A Problems/Problem 17
  ...th> is tangent to the circle, and <math>AF=\sqrt{9+5\sqrt{2}}</math>. What is <math>r/s</math>? ...rac{5}{9}\qquad \mathrm{(C) \ } \frac{3}{5}\qquad \mathrm{(D) \ } \frac{5}{3}\qquad \mathrm{(E) \ } \frac{9}{5}</math>
  6 KB (958 words) - 23:29, 28 September 2023
• 2006 AMC 12A Problems/Problem 20
  ...s equal probability of being chosen, and all choices are independent. What is the probability that after seven moves the bug will have visited every vert Therefore, starting at <math>A</math>, the bug has a <math>\frac{3}{3}</math> chance of finding a good path to the next vertex, and call it <math
  5 KB (908 words) - 19:23, 22 September 2022
• 2006 AMC 12A Problems/Problem 22
  ...sible from a randomly chosen point on the circle is <math>1/2</math>. What is <math>r</math>? ...quad \rm{(D) \ } 3\sqrt{2}+\sqrt{6}\qquad \mathrm{(E) \ } 6\sqrt{2}-\sqrt{3}</math>
  2 KB (343 words) - 15:39, 14 June 2023
• 2006 AMC 12A Problems/Problem 23
  ...,\ldots ,x^{100})</math>. If <math>A^{100}(S)=(1/2^{50})</math>, then what is <math>x</math>? <cmath>A^2(S)=\left(\frac{1+2x+x^2}{2^2},\frac{x+2x^2+x^3}{2^2},...,\frac{x^{98}+2x^{99}+x^{100}}{2^2}\right)</cmath>
  3 KB (466 words) - 22:40, 29 September 2023
• 2006 AMC 12A Problems/Problem 24
  is simplified by expanding it and combining like terms. How many terms are in if the exponent of <math>y</math> is <math>1</math>, the exponent of <math>z</math> can be all even integers up
  8 KB (1,332 words) - 17:37, 17 September 2023
• 2006 AMC 12A Problems/Problem 25
  How many non-[[empty set | empty]] [[subset]]s <math>S</math> of <math>\{1,2,3,\ldots ,15\}</math> have the following two properties? ...k+1</math>, with no restriction on consecutive numbers. Since this process is easily reversible, we have a [[bijection]].
  8 KB (1,405 words) - 11:52, 27 September 2022
• 2005 AMC 12A Problems/Problem 20
  ...\geq 2</math>. For how many values of <math>x</math> in <math>[0,1]</math> is <math>f^{[2005]}(x) = \frac {1}{2}</math>? ...<math>f(x)=2-2x,\frac{1}{2}\le x\le 1</math>,as long as <math>f(x)</math> is between <math>0</math> and <math>1</math>, <math>x</math> will be in the ri
  3 KB (437 words) - 23:49, 28 September 2022
• 2005 AMC 12A Problems/Problem 25
  ...ly possible side length (red triangle in diagram). Each of these triangles is determined by one vertex of the cube, so in one cube we have 8 equilateral currentprojection=perspective(1/3,-1,1/2);
  4 KB (498 words) - 00:46, 4 August 2023
• 2005 AMC 12B Problems/Problem 2
  ...property that <math>x\%</math> of <math>x</math> is <math>4</math>. What is <math>x</math>? ...h> means <math>0.01x</math>, the statement "<math>x\% \text{ of } x \text{ is 4}</math>" can be rewritten as "<math>0.01x \cdot x = 4</math>":
  1 KB (145 words) - 13:56, 14 December 2021
• 2005 AMC 12B Problems/Problem 4
  ...>A</math> on <math>22</math> of the first <math>30</math> quizzes. If she is to achieve her goal, on at most how many of the remaining quizzes can she e \textbf{(C) }\ 3 \qquad
  1 KB (197 words) - 14:16, 14 December 2021
• 2005 AMC 12B Problems/Problem 6
  ...lies between <math>A</math> and <math>D</math> and <math>CD=8</math>. What is <math>BD</math>? \textbf{(A) }\ 3 \qquad
  2 KB (299 words) - 15:29, 5 July 2022
• 2005 AMC 12B Problems/Problem 7
  What is the area enclosed by the graph of <math>|3x|+|4y|=12</math>? ...equations (using the logic that if <math>|a|=b</math>, then <math>a</math> is either <math>b</math> or <math>-b</math>):
  2 KB (357 words) - 20:15, 27 December 2020
• 2005 AMC 12B Problems/Problem 9
  ...got <math>90</math> points, and the rest got <math>95</math> points. What is the difference between the [[mean]] and the [[median]] score on this exam? ...720}{20}=86</math>. The difference between the mean and median, therefore, is <math>\boxed{\textbf{(B)}\ 1}</math>.
  2 KB (280 words) - 15:35, 16 December 2021
• 2005 AMC 12B Problems/Problem 10
  ...ding term is the sum of the cubes of the digits of the previous term. What is the <math>{2005}^{\text{th}}</math> term of the sequence? ...<math>250</math>. It just so happens that <math>2005\equiv 1\ (\text{mod}\ 3)</math>, which leads us to the answer of <math>\boxed{\textbf{(E) } 250}</m
  1 KB (204 words) - 14:37, 15 December 2021
• 2005 AMC 12B Problems/Problem 11
  ...awn at random without replacement. What is the probability that their sum is &#36;<math>20</math> or more? ...\qquad \textbf{(D) }\ {{{\frac{1}{2}}}} \qquad \textbf{(E) }\ {{{\frac{2}{3}}}}</math>
  4 KB (607 words) - 21:01, 20 May 2023
• 2005 AMC 12B Problems/Problem 13
  ...math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math>. What is <math>x_1x_2...x_{124}</math>? ...)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</mat
  1 KB (203 words) - 19:57, 24 December 2020
• 2005 AMC 12B Problems/Problem 14
  ...o the lines <math>y=x</math>, <math>y=-x</math> and <math>y=6</math>. What is the radius of this circle? ...</math> and the diagonal is <math>k = R+6</math>. The diagonal of a square is <math>\sqrt{2}</math> times the side length. Therefore, <math>R+6 = R\sqrt{
  2 KB (278 words) - 21:12, 24 December 2020
• 2005 AMC 12B Problems/Problem 15
  ...is <math>0</math> and no two of them are the same. Which of the following is '''not''' included among the eight digits? \mathrm{(C)}\ 3 \qquad
  2 KB (411 words) - 21:02, 21 December 2020
• 2005 AMC 12B Problems/Problem 16
  ...radius 1, one per octant, are each tangent to the coordinate planes. What is the radius of the smallest sphere, centered at the origin, that contains th \mathrm {(B)}\ \sqrt{3} \qquad
  2 KB (364 words) - 04:54, 16 January 2023
• 2005 AMC 12B Problems/Problem 17
  <cmath>a\cdot\log_{10}2+b\cdot\log_{10}3+c\cdot\log_{10}5+d\cdot\log_{10}7=2005?</cmath> <cmath>\log_{10}2^{a}+\log_{10}3^{b}+\log_{10}5^{c}+\log_{10}7^{d}=2005</cmath>
  1 KB (159 words) - 21:18, 21 December 2020
• 2005 AMC 12B Problems/Problem 20
  ...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 21
  ...60</math> divisors and <math>7n</math> has <math>80</math> divisors. What is the greatest integer <math>k</math> such that <math>7^k</math> divides <mat ...\mathrm{(B)}\ {{{1}}} \qquad \mathrm{(C)}\ {{{2}}} \qquad \mathrm{(D)}\ {{{3}}} \qquad \mathrm{(E)}\ {{{4}}}</math>
  888 bytes (140 words) - 20:04, 24 December 2020
• 2005 AMC 12B Problems/Problem 22
  A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>.
  4 KB (660 words) - 17:40, 24 January 2021
• 2005 AMC 12B Problems/Problem 23
  ...> we have <math>x^{3}+y^{3}=a \cdot 10^{3z} + b \cdot 10^{2z}.</math> What is the value of <math>a+b?</math> Therefore, <math>x^3 + y^3 = s\cdot\dfrac{3t-s^2}{2} = s(15s-\dfrac{s^2}{2})</math>.
  5 KB (786 words) - 16:49, 31 January 2023
• 2005 AMC 12B Problems/Problem 24
  ...h>m</math> and <math>n</math> are relatively prime positive integers. What is the value of <math>m + n</math>? ...that the slope between 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
  ...o one of the four adjacent vertices, each with equal [[probability]]. What is the probability that no two ants arrive at the same vertex? \qquad\mathrm{(E)}\ \frac {3}{128}</math>
  10 KB (1,840 words) - 21:35, 7 September 2023
• 2006 AMC 10A Problems
  Sandwiches at Joe's Fast Food cost <math> \textdollar 3 </math> each and sodas cost <math> \textdollar 2 </math> each. How many dol Define <math>x\otimes y=x^3-y</math>. What is <math>h\otimes (h\otimes h)</math>?
  13 KB (2,028 words) - 16:32, 22 March 2022
• 1995 AHSME Problems/Problem 6
  ...to the shape of a cube. In the resulting cube, which of the lettered faces is opposite the face marked x? path p=origin--(0,1)--(1,1)--(1,2)--(2,2)--(2,3);
  1 KB (168 words) - 00:49, 14 October 2013
• 2006 AMC 10A Problems/Problem 8
  ...es through the points <math> (2,3) </math> and <math> (4,3) </math>. What is <math>c</math>? Substitute the points <math> (2,3) </math> and <math> (4,3) </math> into the given equation for <math> (x,y) </math>.
  2 KB (348 words) - 23:10, 16 December 2021
• 2006 AMC 12A Problems/Problem 12
  ...ove it. The bottom ring has an outside diameter of <math>3</math> cm. What is the distance, in cm, from the top of the top ring to the bottom of the bott D(CR((0,-39),3));
  2 KB (292 words) - 11:56, 17 December 2021
• 2006 AMC 10A Problems/Problem 15
  .../math> meters in the opposite direction and the circumference of his track is <math>100\pi</math>. ...will meet again in <math>k</math> minutes. So the total amount of meetings is <math>\lfloor\frac{30}{k}\rfloor=\lfloor\frac{150}{\pi}\rfloor=\boxed{\text
  3 KB (532 words) - 17:49, 13 August 2023
• 2006 AMC 10A Problems/Problem 16
  ...h>\overline{AB}</math> and <math>\overline{AC}</math> are congruent. What is the area of <math>\triangle ABC</math>? MP('2', (2*t,3), W); MP('1',(2*t, 5.5), W);</asy>
  5 KB (732 words) - 23:19, 19 September 2023
• 2006 AMC 10A Problems/Problem 17
  ...HE}</math>. In addition, <math>AH=AC=2</math>, and <math>AD=3</math>. What is the area of quadrilateral <math>WXYZ</math> shown in the figure? A=(0,2); B=(1,2); C=(2,2); D=(3,2);
  6 KB (1,066 words) - 00:21, 2 February 2023
• 2006 AMC 10A Problems/Problem 18
  ...quad\textbf{(D) } 10^2\times 26^4\qquad\textbf{(E) } 5\times 10^3\times 26^3\qquad</math> Therefore, the number of distinct license plates is <math> 5\times 10^4\times 26^2 \Longrightarrow \boxed{\mathrm{C}}</math>.
  2 KB (254 words) - 14:39, 5 April 2024
• 2006 AMC 10A Problems/Problem 19
  ...le value for the smallest angle is <math>1</math> and the highest possible is <math>59</math> (since the numbers are distinct), so there are <math>\boxed ==Solution 3 (Quick Summation)==
  2 KB (259 words) - 03:10, 22 June 2023
• 2006 AMC 10A Problems/Problem 20
  ...is the probability that some pair of these integers has a difference that is a multiple of <math>5</math>? ...) } \frac{1}{2}\qquad\textbf{(B) } \frac{3}{5}\qquad\textbf{(C) } \frac{2}{3}\qquad\textbf{(D) } \frac{4}{5}\qquad\textbf{(E) } 1\qquad</math>
  1 KB (187 words) - 08:21, 17 March 2023
• 2006 AMC 10A Problems/Problem 21
  ...itive integers have at least one digit that is a <math>2</math> or a <math>3</math>? ...s and subtracting off those which do not have any <math>2</math>s or <math>3</math>s as digits.
  4 KB (525 words) - 21:38, 7 February 2024
• 2006 AMC 10A Problems/Problem 24
  ...ent faces of a unit cube are joined to form a regular [[octahedron]]. What is the volume of this octahedron? ...) } \frac{1}{6}\qquad\textbf{(C) } \frac{1}{4}\qquad\textbf{(D) } \frac{1}{3}\qquad\textbf{(E) } \frac{1}{2}\qquad</math>
  2 KB (292 words) - 10:19, 19 December 2021
• Set
  ...ames really do not define the meaning of the word ''set''; all they can do is replace it in various sentences. So, instead of defining what sets are, one ...uch as the following: <math>\{1,4,5,3,24,4,4,5,6,2\}</math> Such an entity is actually called a multiset.
  11 KB (2,021 words) - 00:00, 17 July 2011
• Newman's Tauberian Theorem
  '''Newman's Tauberian Theorem''' is a [[tauberian theorem]] (which is well-defined by this formula for <math>\Re s>0</math>) admits an
  6 KB (1,034 words) - 07:55, 12 August 2019
• 2006 USAMO Problems
  if and only if <math>s</math> is not a divisor of <math>p-1</math>. ...rms of <math>k</math>, the minimum value of <math>N</math> for which there is a set of <math>2k+1</math> distinct positive integers that has sum greater
  3 KB (520 words) - 09:24, 14 May 2021
• 1991 AJHSME Problems
  == Problem 3 == [[1991 AJHSME Problems/Problem 3|Solution]]
  17 KB (2,246 words) - 13:37, 19 February 2020
• 1990 AJHSME Problems/Problem 1
  What is the smallest sum of two <math>3</math>-digit numbers that can be obtained by placing each of the six digits draw((1,1)--(3,1)--(3,3)--(1,3)--cycle); draw((1,4)--(3,4)--(3,6)--(1,6)--cycle);
  1 KB (191 words) - 17:12, 29 October 2016
• 2022 USAJMO Problems/Problem 1
  <math>\bullet</math> <math>a_n-g_n</math> is divisible by <math>m</math> for all integers <math>n>1</math>; <math>\bullet</math> <math>a_2-a_1</math> is not divisible by <math>m</math>.
  4 KB (792 words) - 00:29, 13 April 2024
• 2006 AIME II Problems/Problem 2
  ...th>\log_{10} 75</math>, and <math>\log_{10} n</math>, where <math>n</math> is a positive integer. Find the number of possible values for <math>n</math>. ...number of positive integer <math>n</math> which satisfies this requirement is <math>\boxed{893}</math>.
  1 KB (164 words) - 14:58, 14 April 2020
• 2005 AIME II Problems/Problem 1
  ...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> ...ial coefficient]] <math>{n \choose 6} = \frac{n\cdot(n-1)\cdot(n-2)\cdot(n-3)\cdot(n-4)\cdot(n-5)}{6\cdot5\cdot4\cdot3\cdot2\cdot1}</math>.
  1 KB (239 words) - 11:54, 31 July 2023
• 2005 AIME II Problems/Problem 2
  ...uests. Given that the [[probability]] each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are [[ *Person 1: <math>\frac{9 \cdot 6 \cdot 3}{9 \cdot 8 \cdot 7} = \frac{9}{28}</math>
  4 KB (628 words) - 11:28, 14 April 2024
• 2005 AIME II Problems/Problem 4
  <math>15^7 = 3^7\cdot5^7</math> so <math>15^7</math> has <math>8\cdot8 = 64</math> divisor <math>\gcd(15^7, 18^{11}) = 3^7 </math> which has 8 divisors.
  3 KB (377 words) - 18:36, 1 January 2024
• 2005 AIME II Problems/Problem 13
  ...th> P(17)=10 </math> and <math> P(24)=17. </math> Given that <math> P(n)=n+3 </math> has two distinct integer solutions <math> n_1 </math> and <math> n_ ...h>(x-17)(x-24)</math> to be a factor of <math>10</math>. Hence the answer is <math>19\cdot 22=\boxed{418}</math>.
  4 KB (642 words) - 14:55, 12 August 2019
• 2005 AIME II Problems/Problem 5
  ...og b=3\log a </math> or <math>\log b=2\log a </math>, so either <math> b=a^3 </math> or <math> b=a^2 </math>. ...e <math> b=a^3 </math>, note that <math> 12^3=1728 </math> while <math> 13^3=2197 </math>. Therefore, for this case, all values of <math>a</math> from <
  3 KB (547 words) - 19:15, 4 April 2024
• 2005 AIME II Problems/Problem 6
  ...agical. For example, eight cards form a magical stack because cards number 3 and number 6 retain their original positions. Find the number of cards in t ...s 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
  ...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> ...he guests. Given that the probability each guest got one roll of each type is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are re
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 7
  It follows that <math>(x + 1)^{48} = (\sqrt[16]5)^{48} = 5^3 = \boxed{125}</math>. ...+1) = (y^{15}+y^{14}+y^{13}+y^{12}+y^{11}+y^{10}+y^9+y^8+y^7+y^6+y^5+y^4+y^3+y^2+y+1)=\frac{y^{16}-1}{y-1}</cmath>
  2 KB (279 words) - 12:33, 27 October 2019
• 2005 AIME II Problems/Problem 8
  .../math> and <math> p </math> are [[relatively prime]], and <math> n </math> is not divisible by the square of any [[prime]], find <math> m+n+p. </math> ...= (-10,0), C2 = (4,0), C3 = (0,0), H = (-10-28/3,0), T = 58/7*expi(pi-acos(3/7));
  4 KB (693 words) - 13:03, 28 December 2021
• 2005 AIME II Problems/Problem 9
  ...positive integers <math> n </math> less than or equal to <math>1000</math> is <math> (\sin t + i \cos t)^n = \sin nt + i \cos nt </math> true for all rea ...t 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 \leq 1000</
  6 KB (1,154 words) - 03:30, 11 January 2024
• 2005 AIME II Problems/Problem 12
  ...h> and <math> r </math> are [[positive]] [[integer]]s and <math> r </math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math ...- y</math> again, we know have <math>xy = (400 - y)y = 150^2</math>. This is a quadratic with roots <math>200 \pm 50\sqrt{7}</math>. Since <math>y < x</
  13 KB (2,080 words) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  ...at the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math> m </math> and <math> n </math> are re ...,0,-3)--(0,-3,0)--(3,0,0)--(0,0,-3)--(0,3,0)--(0,0,3)--(3,0,0)--(0,3,0)--(-3,0,0));
  3 KB (436 words) - 03:10, 23 September 2020
• 2005 AIME II Problems/Problem 11
  ...>a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k = 1,2,\ldots, m-1. </math> Find <math>m. </math> <math>a_{k}a_{k+1} = a_{k-1}a_{k} - 3 </math>.
  3 KB (499 words) - 18:52, 21 November 2022
• Euler's identity
  ...er's Formula''' is <math>e^{i\theta}=\cos \theta+ i\sin\theta</math>. It is named after the 18th-century mathematician [[Leonhard Euler]]. ...umbers]] and/or [[trigonometry]]. Euler's formula replaces "[[cis]]", and is a superior notation, as it encapsulates several nice properties:
  3 KB (452 words) - 23:17, 4 January 2021
• 2006 AMC 12A Problems/Problem 4
  A digital watch displays hours and minutes with AM and PM. What is the largest possible sum of the digits in the display? 1+2&9&6&3\\
  2 KB (257 words) - 11:20, 2 January 2022
• 2005 AIME I Problems
  ...ose common difference is <math> k. </math> For example, <math> S_3 </math> is the sequence <math> 1,4,7,10,\ldots. </math> For how many values of <math> == Problem 3 ==
  6 KB (983 words) - 05:06, 20 February 2019
• 2005 AIME I Problems/Problem 2
  ...ose common difference is <math> k</math>. For example, <math> S_3 </math> is the [[sequence]] <math> 1,4,7,10,\ldots. </math> For how many values of <ma ...h>. Thus the requested number of values is <math>12</math>, and the answer is <math>\boxed{012}</math>.
  2 KB (303 words) - 01:31, 5 December 2022
• 2005 AIME I Problems/Problem 3
  ...ivisor]]s (positive integral [[divisor]]s excluding itself), each of which is less than 50? ...so <math>n</math> must be in the form <math>n=p\cdot q</math> or <math>n=p^3</math> for distinct [[prime number]]s <math>p</math> and <math>q</math>.
  2 KB (249 words) - 09:37, 23 January 2024
• 2005 AIME I Problems/Problem 4
  ...5</math>, so this number works and no larger number can. Thus, the answer is <math>\boxed{294}</math>. ...factors of <math>69</math> are <math>(1,69), (3,23)</math>; <math>x</math> is maximized for the first case. Thus, <math>x = \frac{69 + 1}{2} = 35</math>,
  8 KB (1,248 words) - 11:43, 16 August 2022
• 2005 AIME I Problems/Problem 5
  ...te parts to this problem: one is the color (gold vs silver), and the other is the orientation. ...t occur at all, for <math>9</math> total configurations. Thus, the answer is <math>70 \cdot 9 = \boxed{630}</math>.
  5 KB (830 words) - 01:51, 1 March 2023
• 2005 AIME I Problems/Problem 6
  Let <math> P </math> be the product of the nonreal roots of <math> x^4-4x^3+6x^2-4x=2005. </math> Find <math> \lfloor P\rfloor. </math> The left-hand side of that [[equation]] is nearly equal to <math>(x - 1)^4</math>. Thus, we add 1 to each side in ord
  4 KB (686 words) - 01:55, 5 December 2022
• 2005 AIME I Problems/Problem 7
  ...DE</math> is concurrent with line <math>BC</math>. Then, <math>ABED</math> is an isosceles trapezoid so <math>AD=BE=10</math>, and <math>BC=8</math> and ...</math>. The [[Pythagorean Theorem]] yields that <math>GC^2 = 12^2 - \sqrt{3}^2 = 141</math>, so <math>EF = GC = \sqrt{141}</math>. Therefore, <math>AB
  4 KB (567 words) - 20:20, 3 March 2020
• 2005 AIME I Problems/Problem 8
  ...2^{222x+1} + 1 </math> has three [[real]] [[root]]s. Given that their sum is <math>m/n</math> where <math> m </math> and <math> n </math> are [[relative ...</math> and <math>x_1 + x_2 + x_3 = \frac{2}{111}</math>. Thus the answer is <math>111 + 2 = \boxed{113}</math>.
  1 KB (161 words) - 19:50, 2 January 2022
• 2005 AIME I Problems/Problem 9
  ...[probability]] of the entire [[surface area]] of the larger cube is orange is <math> \frac{p^a}{q^br^c}, </math> where <math> p,q, </math> and <math> r < ...orientations, so from these cubes we gain a factor of <math>\left(\frac{2}{3}\right)^6</math>.
  4 KB (600 words) - 21:44, 20 November 2023
• 2005 AIME I Problems/Problem 10
  ...[[midpoint]] <math>M</math> of [[line segment]] <math>\overline{BC}</math> is <math>\left(\frac{35}{2}, \frac{39}{2}\right)</math>. The equation of the m ...tion for the triangle will give a smaller value of <math>p+q</math>, which is provable by following these steps over again) (alternatively, we could use
  5 KB (852 words) - 21:23, 4 October 2023
• 2005 AIME I Problems/Problem 11
  ...e]] whose sides have length 8. Given the maximum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> ...n it touches both other sides of the square. This can happen only when it is arranged so that the center of the semicircle lies on one diagonal of the s
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME I Problems/Problem 12
  ...squares less than <math>n</math>. So <math>S(1), S(2)</math> and <math>S(3)</math> are odd, while <math>S(4), S(5), \ldots, S(8)</math> are even, and ...t the numbers between <math>1^2</math> and <math>2^2</math>, between <math>3^2</math> and <math>4^2</math>, and so on, all the way up to the numbers bet
  4 KB (647 words) - 02:29, 4 May 2021
• 2005 AIME I Problems/Problem 13
  ...th>U</math> represent a move upwards, and <math>D</math> to be a move that is diagonal. [[Casework]] upon the number of diagonal moves: *'''Case ''' <math>d = 1</math>: It is easy to see only <math>2</math> cases.
  5 KB (897 words) - 00:21, 29 July 2022
• 2005 AIME I Problems/Problem 14
  ...e the area of <math> S. </math> Find the remainder when <math> 10K </math> is divided by <math>1000</math>. Consider a point <math>E</math> such that <math>AE</math> is [[perpendicular]] to <math>BD</math>, <math>AE</math> intersects <math>BD</
  3 KB (561 words) - 14:11, 18 February 2018
• 2005 AIME I Problems/Problem 15
  ...re <math> m </math> and <math> n </math> are integers and <math> n </math> is not [[divisor | divisible]] by the [[perfect square | square]] of a prime, ...thout loss of generality, let <math>AC < AB</math>, so that <math>E</math> is between <math>D</math> and <math>C</math>. Let the length of the median be
  5 KB (906 words) - 23:15, 6 January 2024
• 2005 AIME II Problems/Problem 15
  ...or which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </math> and interna ...centers is <math>r_1 + r_2</math>, and if they are internally tangent, it is <math>|r_1 - r_2|</math>. So we have
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2005 AIME II Problems/Problem 14
  ...th> \overline{BC} </math> with <math> CD=6. </math> Point <math> E </math> is on <math> \overline{BC} </math> such that <math> \angle BAE\cong \angle CAD ...{BE} - 1 \Longrightarrow BE = \frac{13^2 \cdot 15}{463}</math>. The answer is <math>q = \boxed{463}</math>.
  13 KB (2,129 words) - 18:56, 1 January 2024
• 2004 AIME I Problems/Problem 15
  f(x)=\begin{cases}1 & \text{if }x = 1\\ \frac x{10} & \text{if }x\text{ is divisible by 10}\\ x+1 & \text{otherwise}\end{cases} ...st <math>n</math> such that <math>x_n=1</math>. (For example, <math>d(100)=3</math> and <math>d(87)=7</math>.) Let <math>m</math> be the number of posit
  9 KB (1,491 words) - 01:23, 26 December 2022
• 2004 AIME I Problems/Problem 14
  ...> and <math> c </math> are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> real x = 20 - ((750)^.5)/3, CE = 8*(6^.5) - 4*(5^.5), CD = 8*(6^.5), h = 4*CE/CD;
  4 KB (729 words) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 13
  ...oots of the form <math> z_k = r_k[\cos(2\pi a_k)+i\sin(2\pi a_k)], k=1, 2, 3,\ldots, 34, </math> with <math> 0 < a_1 \le a_2 \le a_3 \le \cdots \le a_{3 ...nomial]] <math>P</math> is very difficult to 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
  ...on <math> [z] </math> denotes the [[floor function|greatest integer]] that is less than or equal to <math> z. </math> <math>\left\lfloor\log_2\left(\frac{1}{x}\right)\right\rfloor</math> is even when
  2 KB (303 words) - 22:28, 11 September 2020
• 2004 AIME I Problems/Problem 11
  ...s a 3-inch radius. The entire [[surface]] of the cone, including its base, is painted. A [[plane]] [[parallel]] to the base of the cone divides the cone ...face area]] <math>A = \pi r^2 + \pi r \ell</math>, where <math>\ell</math> is the [[slant height]] of the cone. Using the [[Pythagorean Theorem]], we ge
  5 KB (839 words) - 22:12, 16 December 2015
• 2004 AIME I Problems/Problem 10
  ...[[probability]] that the circle will not touch diagonal <math> AC </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ ...nter of the circle must be in the <math>34 \times 13</math> rectangle that is one unit away from the sides of rectangle <math>ABCD</math>. We want to fin
  5 KB (836 words) - 07:53, 15 October 2023
• 2004 AIME I Problems/Problem 9
  ...<math> U_1 </math> is similar to <math> U_2 </math> and <math> V_1 </math> is similar to <math> V_2. </math> The minimum value of the area of <math> U_1 ...h>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 1
  ...to right. What is the sum of the possible remainders when <math> n </math> is divided by <math>37</math>? ...2) + 10(n + 1) + n = 3210 + 1111n</math>, for <math>n \in \lbrace0, 1, 2, 3, 4, 5, 6\rbrace</math>.
  2 KB (374 words) - 14:53, 27 December 2019
• 2004 AIME I Problems/Problem 2
  ...atest element of <math>A</math> and the greatest element of <math>B</math> is <math>99</math>. Find <math>m.</math> ...must be <math>2</math>. Therefore, the largest element in <math>A</math> is <math>2 + \frac{m-1}{2}</math>.
  8 KB (1,437 words) - 21:53, 19 May 2023
• 2004 AIME I Problems/Problem 4
  ...<math> S </math> enclose a region whose [[area]] to the nearest hundredth is <math>k</math>. Find <math> 100k</math>. ...e at each corner of the square. The area enclosed by all of the midpoints is <math>4-4\cdot \left(\frac{\pi}{4}\right)=4-\pi \approx .86</math> to the n
  3 KB (532 words) - 09:22, 11 July 2023
• 2004 AIME I Problems/Problem 5
  ...</math> and <math> n </math> are relatively prime positive integers. What is <math> m+n </math>? From here, we see the largest possible value of <math>a+b</math> is <math>349</math>.
  3 KB (436 words) - 18:31, 9 January 2024
• 2004 AIME I Problems/Problem 6
  ...s [[odd integer | odd]] and <math> a_i>a_{i+1} </math> if <math> i </math> is [[even integer | even]]. How many snakelike integers between 1000 and 9999 ...into two cases: 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
  ...ath>Q(x)</math> is some polynomial [[divisibility | divisible]] by <math>x^3</math>. ...x)</math>, where <math>R(x)</math> is some polynomial divisible by <math>x^3</math>.
  5 KB (833 words) - 19:43, 1 October 2023
• 2004 AIME I Problems/Problem 8
  There are no regular 3-pointed, 4-pointed, or 6-pointed stars. All regular 5-pointed stars are sim ...of this <math>n</math>-gon in a counterclockwise direction: <math>0, 1, 2, 3, \ldots, n-1.</math>
  4 KB (620 words) - 21:26, 5 June 2021
• 2004 AIME I Problems
  ...to right. What is the sum of the possible remainders when <math> n </math> is divided by 37? ...t 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
  ...th>256</math> by <math>1</math> strip of quadruple thickness. This process is repeated <math>8</math> more times. After the last fold, the strip has beco Number the squares <math>0, 1, 2, 3, ... 2^{k} - 1</math>. In this case <math>k = 10</math>, but we will consi
  6 KB (899 words) - 20:58, 12 May 2022
• 2004 AIME II Problems/Problem 14
  ...ht <math> 7 </math>'s in this way. For how many values of <math> n </math> is it possible to insert <math> + </math> signs so that the resulting expressi ...g by <math>7</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 13
  ...of triangle <math> ABC </math> and the area of triangle <math> EBD </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ ...B \parallel CE, BC \parallel AD, </math> it follows that <math>ABCF</math> is a [[parallelogram]], and so <math>\triangle ABC \cong \triangle CFA</math>.
  3 KB (486 words) - 22:15, 7 April 2023
• 2004 AIME II Problems/Problem 12
  ..., </math> and <math> p </math> are [[positive integer]]s, <math> n </math> is not [[divisibility | divisible]] by the [[perfect square | square]] of any real r = (-60 + 48 * 3^.5)/23;
  3 KB (431 words) - 23:21, 4 July 2013
• 2004 AIME II Problems/Problem 10
  ...ath> S, </math> the [[probability]] that it is divisible by <math>9</math> is <math> p/q, </math> where <math> p </math> and <math> q </math> are relativ ...{40}{2}</math> because we’re choosing 2 1s to go in 40 digit slots. This is equal to 780; we have found <math>q</math>, our denominator.
  7 KB (1,091 words) - 18:41, 4 January 2024
• 2004 AIME II Problems/Problem 9
  ...ression. Let <math> a_n </math> be the greatest term in this sequence that is less than <math>1000</math>. Find <math> n+a_n. </math> ...th>. This happens with <math>f(7)f(8) = 29 \cdot 33 = 957</math>, and this is the <math>2(8) = 16</math>th term of the sequence.
  3 KB (538 words) - 21:33, 30 December 2023
• 2004 AIME II Problems/Problem 8
  ...the prime factorization of <math>2004^{2004}</math> is <math>2^{4008}\cdot 3^{2004}\cdot 167^{2004}</math>. ...ample, the number of divisors of <math>2004=2^2\cdot 3^1\cdot 167^1</math> is <math>(2+1)(1+1)(1+1)=12</math>.
  2 KB (353 words) - 18:08, 25 November 2023
• 2004 AIME II Problems/Problem 7
  ...th> CF=3 </math> are given. The perimeter of rectangle <math> ABCD </math> is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ pair A=origin, B=(25,0), C=(25,70/3), D=(0,70/3), E=(8,0), F=(22,70/3), Bp=reflect(E,F)*B, Cp=reflect(E,F)*C;
  9 KB (1,501 words) - 05:34, 30 October 2023
• 2004 AIME II Problems/Problem 6
  ...t the end of the process are in the [[ratio]] <math> 3: 2: 1, </math>what is the least possible total for the number of bananas? ...c{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</math>.
  6 KB (950 words) - 14:18, 15 January 2024
• 2004 AIME II Problems/Problem 5
  ...urther behind schedule. Given that all workers work at the same rate, what is the minimum number of additional workers, beyond the <math>800</math> worke ...0}{800}(60)=\frac{150}{8}</math>. The train then has <math>60-15-\frac{50}{3}-\frac{150}{8}=230/24</math> minutes left to travel 250 miles, and doing th
  4 KB (592 words) - 19:02, 26 September 2020
• 2004 AIME II Problems/Problem 4
  ...ath>n</math>-digit number, for a total of <math>(2^1 - 2) + (2^2 - 2) + (2^3 -2) + (2^4 - 2) = 22</math> such numbers (or we can list them: <math>AB, BA ...s we can form, for a total of <math>(2^0 - 1) + (2^1 - 1) + (2^2 - 1) + (2^3 - 1) = 11</math> such numbers (or we can list them: <math>A0, A00, A0A, AA0
  3 KB (508 words) - 01:16, 19 January 2024
• 2004 AIME II Problems/Problem 3
  ...gruent]] 1-cm [[cube (geometry) | cube]]s [[face]] to face. When the block is viewed so that three of its faces are visible, exactly <math>231</math> of ...s close together as possible, which occurs when the smaller block is <math>3 \times 7 \times 11</math>. Then the extra layer makes the entire block <ma
  2 KB (377 words) - 11:53, 10 March 2014
• 2004 AIME II Problems/Problem 2
  ...ability]] that they get the same color combination, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are [[relat ...c{28}{153}</math>. So the probability that they both pick two red candies is <math>\frac{9}{38} \cdot \frac{28}{153} = \frac{14}{323}</math>. The same
  2 KB (330 words) - 13:42, 1 January 2015
• 2004 AIME II Problems/Problem 1
  ...y [[prime]]. Find the [[remainder]] when the product <math> abcdef </math> is divided by 1000. .../math>; the rest of the area of the circle is then equal to <math>\frac{2}{3}r^2\pi</math>.
  2 KB (329 words) - 23:20, 4 July 2013
• 2004 AIME II Problems
  ...re of any prime. Find the remainder when the product <math> abcdef </math> is divided by 1000. ...obability that they get the same color combination, irrespective of order, is <math> m/n, </math> where <math> m </math> and <math> n </math> are relativ
  9 KB (1,410 words) - 05:05, 20 February 2019
• 1983 AIME Problems
  == Problem 3 == What is the product of the real roots of the equation <math>x^2 + 18x + 30 = 2 \sqr
  7 KB (1,104 words) - 12:53, 6 July 2022
• 2019 AIME I Problems/Problem 2
  ...9, 20</math> distinct from <math>J</math>. The value of <math>B - J</math> is at least <math>2</math> with a probability that can be expressed in the for ...because <math>B \ne J</math>, so the probability that <math>B-J < 0</math> is <math>\frac{1}{2}</math> by symmetry.
  5 KB (830 words) - 22:15, 28 December 2023
• 1984 AIME Problems
  ..._{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an arithmetic progression with common difference 1, and <math>a_1+a_2+a_3+\ ...sitive multiple of <math>15</math> such that every digit of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>.
  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 - == Problem 3 ==
  5 KB (847 words) - 15:48, 21 August 2023
• 1987 AIME Problems
  An ordered pair <math>(m,n)</math> of non-negative integers is called "simple" if the addition <math>m+n</math> in base <math>10</math> re What is the largest possible distance between two points, one on the sphere of radi
  6 KB (869 words) - 15:34, 22 August 2023
• 1988 AIME Problems
  ...rder -- the correct five buttons. The sample shown below has <math>\{1, 2, 3, 6, 9\}</math> as its combination. Suppose that these locks are redesigned == Problem 3 ==
  6 KB (902 words) - 08:57, 19 June 2021
• 1989 AIME Problems
  == Problem 3 == Suppose <math>n_{}^{}</math> is a positive integer and <math>d_{}^{}</math> is a single digit in base 10. Find <math>n_{}^{}</math> if
  7 KB (1,045 words) - 20:47, 14 December 2023
• 1990 AIME Problems
  The [[increasing sequence]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are ne 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
• 1991 AIME Problems
  ...overline {AB}</math> of length 4 and <math>\overline {CB}</math> of length 3. Divide <math>\overline {AB}</math> into 168 congruent segments with points == Problem 3 ==
  7 KB (1,106 words) - 22:05, 7 June 2021
• 1992 AIME Problems
  ...n its decimal representation, there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are == Problem 3 ==
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1993 AIME Problems
  == Problem 3 == <center><math>\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\
  8 KB (1,275 words) - 06:55, 2 September 2021
• 1994 AIME Problems
  ...erfect square. What is the remainder when the 1994th term of the sequence is divided by 1000? ...<math>P^{}_{}</math> to a circle of radius 20. Square <math>ABCD\,</math> is constructed with <math>A\,</math> and <math>B\,</math> on the larger circle
  7 KB (1,141 words) - 07:37, 7 September 2018
• 1995 AIME Problems
  ...</math> The total area enclosed by at least one of <math>S_{1}, S_{2}, S_{3}, S_{4}, S_{5}</math> can be written in the form <math>m/n,</math> where <m == Problem 3 ==
  6 KB (1,000 words) - 00:25, 27 March 2024
• 1996 AIME Problems
  ...magic square, the sum of the three entries in any row, column, or diagonal is the same value. The figure shows four of the entries of a magic square. Fin ...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
  == Problem 3 == ...number is exactly nine times the product Sarah should have obtained. What is the sum of the two-digit number and the three-digit number?
  7 KB (1,098 words) - 17:08, 25 June 2020
• 1998 AIME Problems
  For how many values of <math>k</math> is <math>12^{12}</math> the [[least common multiple]] of the positive integers == Problem 3 ==
  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 ter ...rigin cuts this figure into two congruent polygons. The slope of the line is <math>m/n,</math> where <math>m_{}</math> and <math>n_{}</math> are relativ
  7 KB (1,094 words) - 13:39, 16 August 2020
• 2000 AIME I Problems
  ...ositive integer <math>n</math> such that no matter how <math>10^{n}</math> is expressed as the product of any two positive integers, at least one of thes ...<math>D</math> across the y-axis. The area of pentagon <math>ABCDE</math> is <math>451</math>. Find <math>u + v</math>.
  7 KB (1,204 words) - 03:40, 4 January 2023
• 2001 AIME I Problems
  ...h>\mathcal{S}</math>, and the mean of <math>\mathcal{S}\cup\{2001\}</math> is <math>27</math> more than the mean of <math>\mathcal{S}</math>. Find the me == Problem 3 ==
  7 KB (1,212 words) - 22:16, 17 December 2023
• 2002 AIME I Problems
  ...it arrangement that reads the same left-to-right as it does right-to-left) is <math>m/n</math>, where <math>m</math> and <math>n</math> are relatively pr size(250);real x=sqrt(3);
  8 KB (1,374 words) - 21:09, 27 July 2023
• 2003 AIME I Problems
  <center><math> \frac{((3!)!)!}{3!} = k \cdot n!, </math></center> ...k </math> and <math> n </math> are positive integers and <math> n </math> is as large as possible, find <math> k + n. </math>
  6 KB (965 words) - 16:36, 8 September 2019
• 2000 AIME II Problems
  <center><math>\frac 2{\log_4{2000^6}} + \frac 3{\log_5{2000^6}}</math></center> A point whose coordinates are both integers is called a lattice point. How many lattice points lie on the hyperbola <math
  6 KB (947 words) - 21:11, 19 February 2019
• 2001 AIME II Problems
  ...t and 85 percent of the school population, and the number who study French is between 30 percent and 40 percent. Let <math>m</math> be the smallest numbe == Problem 3 ==
  8 KB (1,282 words) - 21:12, 19 February 2019
• 2002 AIME II Problems
  ...s between <math>100</math> and <math>999</math>, inclusive; <math>y</math> is the number formed by reversing the digits of <math>x</math>; and <math>z=|x ...7,12,10)</math>, <math>Q=(8,8,1)</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
  ...> of three positive integers is 6 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</m ...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
  <math>x^{120}=w^5</math>, <math>y^{120}=w^3</math>, and <math>(xyz)^{120}=w^{10}</math>. ...=w</math>. It now becomes clear that one way to find <math>\log_z w</math> is to find what <math>x^{12}</math> and <math>y^{12}</math> are in terms of <m
  4 KB (642 words) - 03:14, 17 August 2022
• 1983 AIME Problems/Problem 2
  It is best to get rid of the [[absolute value]]s first. Adding these together, we find that the sum is equal to <math>30-x</math>, which attains its minimum value (on the given i
  1 KB (184 words) - 20:16, 14 January 2023
• 1983 AIME Problems/Problem 3
  What is the product of the [[real]] [[root]]s of the [[equation]] <math>x^2 + 18x + ...d moreover, plugging in <math>y=-6</math>, we get <math>-6=6</math>, which is obviously false). Hence we have <math>y=10</math> as the only solution for
  3 KB (532 words) - 05:18, 21 July 2022
• 1983 AIME Problems/Problem 4
  ...d that of <math>BC</math> is <math>2</math> cm. The angle <math>ABC</math> is a right angle. Find the square of the distance (in centimeters) from <math> ...tem to get <math>x = 1</math> and <math>y = 5</math>, such that the answer is <math>1^2 + 5^2 = \boxed{026}</math>.
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 5
  .../math> 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 6
  After some quick division, our answer is <math>\boxed{035}</math>. === Solution 3 (cheap and quick) ===
  3 KB (361 words) - 20:20, 14 January 2023
• 1983 AIME Problems/Problem 7
  ...ch other. If <math>P</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...24}=1-\frac{420}{552}=1-\frac{35}{46}=\frac{11}{46}</math>, and the answer is <math>11+46=\boxed{057}</math>.
  9 KB (1,392 words) - 20:37, 19 January 2024
• 1983 AIME Problems/Problem 8
  What is the largest <math>2</math>-digit [[prime]] factor of the integer <math>n = ...h>3p<200</math>. The largest such prime is <math>\boxed{061}</math>, which is our answer.
  2 KB (243 words) - 20:23, 14 January 2023
• 1983 AIME Problems/Problem 9
  ...e x\sin x \le \frac{\pi}{2}</math>, this value of <math>\frac{2}{3}</math> is attainable by the [[Intermediate Value Theorem]]). ...We show this possible with the same methods in Solution 1; thus the answer is <math>\boxed{012}</math>.
  4 KB (722 words) - 20:25, 14 January 2023
• 1983 AIME Problems/Problem 10
  ...h>, <math>1005</math> and <math>1231</math> have something in common: each is a <math>4</math>-digit number beginning with <math>1</math> that has exactl ...ath>, <math>x\neq1</math>, and <math>y\neq1</math>. Hence, there are <math>3\cdot9\cdot8=216</math> numbers of this form.
  5 KB (855 words) - 20:26, 14 January 2023
• 1983 AIME Problems/Problem 11
  ...dges have length <math>s</math>. Given that <math>s=6\sqrt{2}</math>, what is the volume of the solid? triple A=(0,0,0),B=(s,0,0),C=(s,s,0),D=(0,s,0),E=(-s/2,s/2,6),F=(3*s/2,s/2,6);
  5 KB (865 words) - 21:11, 6 February 2023
• 1983 AIME Problems/Problem 12
  ...from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</math>. draw((-2,-2*sqrt(3))--(-2,2*sqrt(3)));
  2 KB (412 words) - 18:23, 1 January 2024
• 1983 AIME Problems/Problem 13
  ...3, 6,9\}</math> is <math>9-6+3-2+1=5</math> and for <math>\{5\}</math> it is simply <math>5</math>. Find the sum of all such alternating sums for <math> Let <math>S</math> be a non-[[empty set | empty]] [[subset]] of <math>\{1,2,3,4,5,6\}</math>.
  5 KB (894 words) - 22:02, 5 April 2024
• 1983 AIME Problems/Problem 14
  ...units apart. At <math>P</math>, one of the points of intersection, a line is drawn in such a way that the chords <math>QP</math> and <math>PR</math> hav <asy>size(160); defaultpen(linewidth(.8pt)+fontsize(11pt)); dotfactor=3; pair O1=(0,0), O2=(12,0); path C1=Circle(O1,8), C2=Circle(O2,6); pair P=in
  13 KB (2,149 words) - 18:44, 5 February 2024
• 1983 AIME Problems/Problem 15
  ...is expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? add(pathticks(A--F,1,0.5,0,3.5));
  19 KB (3,221 words) - 01:05, 7 February 2023
• 1984 AIME Problems/Problem 1
  ..._{98}</math> if <math>a_1</math>, <math>a_2</math>, <math>a_3\ldots</math> is an [[arithmetic progression]] with common difference 1, and <math>a_1+a_2+a One approach to this problem is to apply the formula for the sum of an [[arithmetic series]] in order to fi
  4 KB (576 words) - 21:03, 23 December 2023
• 1984 AIME Problems/Problem 2
  ...[multiple]] of <math>15</math> such that every [[digit]] of <math>n</math> is either <math>8</math> or <math>0</math>. Compute <math>\frac{n}{15}</math>. Any multiple of 15 is a multiple of 5 and a multiple of 3.
  1 KB (187 words) - 20:05, 29 May 2021
• 1984 AIME Problems/Problem 3
  ...smaller [[triangle]]s <math>t_{1}</math>, <math>t_{2}</math>, and <math>t_{3}</math> in the figure, have [[area]]s <math>4</math>, <math>9</math>, and < D(A--B--C--cycle); D(A+(B-A)*3/4--A+(C-A)*3/4); D(B+(C-B)*5/6--B+(A-B)*5/6);D(C+(B-C)*5/12--C+(A-C)*5/12);
  4 KB (726 words) - 13:39, 13 August 2023
• 1984 AIME Problems/Problem 4
  ...moved, the average of the remaining numbers drops to <math>55</math>. What is the largest number that can appear in <math>S</math>? ...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 5
  ...b^3 = 2^{15}</math> and that <math>\log a^3 b = 21\log 2 \Longrightarrow a^3 b = 2^{21}</math>. If we multiply the two equations together, we get that < ...2}{2 \ln 2}} = \frac{12 \ln 2}{\frac{1}{3} + 1} = \frac{12 \ln 2}{\frac{4}{3}} = 9 \ln 2</math>. This means that <math>\frac{\ln ab}{\ln 2} = 9</math>.
  5 KB (782 words) - 14:49, 1 August 2023
• 1984 AIME Problems/Problem 6
  ...l [[area]] of the parts of the three circles to the other side of it. What is the [[absolute value]] of the [[slope]] of this line? The line passes through the center of the bottom circle; hence it is the circle's [[diameter]] and splits the circle into two equal areas. For t
  6 KB (1,022 words) - 19:29, 22 January 2024
• 1984 AIME Problems/Problem 7
  The [[function]] f is defined on the [[set]] of [[integer]]s and satisfies <math>f(n)=\begin{case n-3&\mbox{if}\ n\ge 1000\\
  4 KB (617 words) - 18:01, 9 March 2022
• 1984 AIME Problems/Problem 8
  The equation <math>z^6+z^3+1=0</math> has complex roots with argument <math>\theta</math> between <mat ...ithin the desired range that satisfies our original equation <math>x^6 + x^3 + 1 = 0</math>.
  3 KB (430 words) - 19:05, 7 February 2023
• 1984 AIME Problems/Problem 9
  ...rc</math> angle. Find the [[volume]] of the tetrahedron in <math>\mbox{cm}^3</math>. triple A=(0,0,0),B=(3,0,0),C=(1.8,10,0),D=(1.5,4,4),Da=(D.x,D.y,0),Db=(D.x,0,0);
  6 KB (947 words) - 20:44, 26 November 2021
• 1984 AIME Problems/Problem 10
  ..., where <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 ...ightarrow 4c-w=89</math> so <math>w\equiv 3\pmod{4}</math>. But if <math>w=3</math>, then <math>c=23</math>, which was the result given; otherwise <math
  7 KB (1,163 words) - 23:53, 28 March 2022
• 1984 AIME Problems/Problem 11
  ...not change the probability of the birch trees being near each other. That is, in the end, you multiply the numerator by the number of ways to arrange th ...5} = 792</math> total ways to arrange the twelve trees, so the probability is <math>\frac{56}{792} = \frac{7}{99}</math>.
  7 KB (1,115 words) - 00:52, 7 September 2023
• 1984 AIME Problems/Problem 12
  ...math>x</math>. If <math>x=0</math> is a root for <math>f(x)=0</math>, what is the least number of roots <math>f(x)=0</math> must have in the interval <ma Since <math>0</math> is a root, all multiples of <math>10</math> are roots, and anything congruent
  3 KB (588 words) - 14:37, 22 July 2020
• 1984 AIME Problems/Problem 14
  What is the largest even integer that cannot be written as the sum of two odd compo ...+ 6n</math> for nonnegative <math>n</math> are odd composites. We now have 3 cases:
  8 KB (1,346 words) - 01:16, 9 January 2024
• 1984 AIME Problems/Problem 15
  ...\frac{w^2}{6^2-7^2}=1 </math><br /><math> \frac{x^2}{8^2-1}+\frac{y^2}{8^2-3^2}+\frac{z^2}{8^2-5^2}+\frac{w^2}{8^2-7^2}=1 </math></div> Rewrite the system of equations as <cmath>\frac{x^{2}}{t-1}+\frac{y^{2}}{t-3^{2}}+\frac{z^{2}}{t-5^{2}}+\frac{w^{2}}{t-7^{2}}=1.</cmath>
  6 KB (1,050 words) - 18:07, 16 January 2024
• 1984 AIME Problems/Problem 13
  Find the value of <math>10\cot(\cot^{-1}3+\cot^{-1}7+\cot^{-1}13+\cot^{-1}21).</math> ...= \frac{\tan(x)+\tan(y)}{1-\tan(x)\tan(y)}</math>. Let <math>a = \cot^{-1}(3)</math>, <math>b=\cot^{-1}(7)</math>, <math>c=\cot^{-1}(13)</math>, and <ma
  3 KB (473 words) - 12:06, 18 December 2018
• 1995 IMO Problems/Problem 2
  ...h> \frac{1}{a^3(b+c)} + \frac{1}{b^3(c+a)} + \frac{1}{c^3(a+b)} \geq \frac{3}{2}. </cmath> ...c^3(a+b)} &= \frac{x^3}{xyz(1/y+1/z)} + \frac{y^3}{xyz(1/z+1/x)} + \frac{z^3}{xyz(1/x+1/z)} \\
  6 KB (1,122 words) - 12:23, 6 January 2022
• 1985 AIME Problems/Problem 15
  ...o fold into a [[polyhedron]]. What is the [[volume]] (in <math>\mathrm{cm}^3</math>) of this polyhedron? ...e]], so the volume is <math>\frac12 \cdot 12^3 = 864</math>, so our answer is <math>\boxed{864}</math>.
  2 KB (245 words) - 22:44, 4 March 2024
• 1985 AIME Problems/Problem 14
  ...us we must have <math>n > 10</math>, so <math>n = 15</math> and the answer is <math>15 + 10 = \boxed{25}</math>. ...e weakest <math>10</math> who gained <math>45</math> points vs them, which is a contradiction since it must be larger. Thus, <math>n=\boxed{25}</math>.
  5 KB (772 words) - 22:14, 18 June 2020
• 1985 AIME Problems/Problem 13
  ...>\ldots</math> are of the form <math>a_n=100+n^2</math>, where <math>n=1,2,3,\ldots</math> For each <math>n</math>, let <math>d_n</math> be the greatest ...r | divides]] <math>100+n^2</math>, it must divide <math>2n+1</math> if it is going to divide the entire [[expression]] <math>100+n^2+2n+1</math>.
  4 KB (671 words) - 20:04, 6 March 2024
• 1985 AIME Problems/Problem 12
  ...te end. Let <math>p = \frac{n}{729}</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. ...gers <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> meters.
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 11
  ...>-plane and is [[tangent line | tangent]] to the <math>x</math>-axis. What is the length of its [[major axis]]? ...ath>F_2 Y \leq F’_2 Y</math> with equality if and only if <math>Y</math> is on the <math>x</math>-axis. Now, we have
  5 KB (932 words) - 17:00, 1 September 2020
• 1985 AIME Problems/Problem 10
  where <math>x</math> is a [[real number]], and <math>\lfloor z \rfloor</math> denotes the greatest ...or \le 3</math>. But according to this the maximum we can get is <math>1+2+3 = 6</math>, so we only need to try the first 6 numbers.
  12 KB (1,859 words) - 18:16, 28 March 2022
• 1985 AIME Problems/Problem 9
  ...[[rational number]], is expressed as a [[fraction]] in lowest terms, what is the sum of its numerator and denominator? pair O = (0,0), A = r*expi(pi/3);
  5 KB (763 words) - 16:20, 28 September 2019
• 1985 AIME Problems/Problem 8
  ...difference, is as small as possible. For this minimum <math>M</math>, what is <math>100M</math>? Then <math>M</math> is the greatest of the <math>7</math> absolute values. So basically you are as
  2 KB (377 words) - 02:17, 16 February 2021
• 1985 AIME Problems/Problem 7
  .../math> are [[positive integer]]s such that <math>a^5 = b^4</math>, <math>c^3 = d^2</math>, and <math>c - a = 19</math>. Determine <math>d - b</math>. ...th>s - t^2 = 1</math>. Then <math>s = 10, t = 3</math> and so <math>d = s^3 = 1000</math>, <math>b = t^5 = 243</math> and <math>d-b=\boxed{757}</math>.
  1 KB (222 words) - 11:04, 4 November 2022
• 1985 AIME Problems/Problem 6
  As shown in the figure, [[triangle]] <math>ABC</math> is divided into six smaller triangles by [[line]]s drawn from the [[vertex | v ...> share the same altitude from <math>C</math>, so the ratio of their areas is the same as the ratio of their bases. Moreover, the two pairs of bases are
  5 KB (789 words) - 03:09, 23 January 2023
• 1985 AIME Problems/Problem 5
  ...e sum of the first 1492 terms is 1985, and the sum of the first 1985 terms is 1492? ...-b) = a</math> and <math>a_8 = a - (a - b) = b</math>. Since the sequence is recursively defined by the first 2 terms, after this point it must continue
  2 KB (410 words) - 13:37, 1 May 2022
• 1985 AIME Problems/Problem 4
  .... Find the value of <math>n</math> if the the [[area]] of the small square is exactly <math>\frac1{1985}</math>. ...frac{1}{\sqrt{2n^2 - 2n + 1}}</math>. But the height of the parallelogram is the side of the little square, so <math>2n^2 - 2n + 1 = 1985</math>. Solvi
  3 KB (484 words) - 21:40, 2 March 2020
• 1985 AIME Problems/Problem 3
  ...nd <math>c</math> are [[positive integer]]s which satisfy <math>c=(a + bi)^3 - 107i</math>, where <math>i^2 = -1</math>. ...6</math> (since we know <math>a</math> is positive). Thus <math>c = 6^3 - 3\cdot 6 = \boxed{198}</math>.
  1 KB (205 words) - 18:58, 10 March 2024
• 1986 AIME Problems/Problem 15
  ...]s through <math>A</math> and <math>B</math> lie along the lines <math>y=x+3</math> and <math>y=2x+4</math> respectively, find the area of triangle <mat ...}{1 + \tan \theta_1 \tan \theta_2} = \frac{2-1}{1 + 2 \cdot 1 } = \frac{1}{3}. </cmath>
  11 KB (1,722 words) - 09:49, 13 September 2023
• 1986 AIME Problems/Problem 14
  ...find the shortest distance between <math>AC</math> and that corner, which is <math>\frac {wl}{\sqrt {w^2 + l^2}}</math>. ...ctively. (This would give us the guess that the sides are of the ratio 1:2:3, but let's provide the complete solution.)
  2 KB (346 words) - 13:13, 22 July 2020
• 1986 AIME Problems/Problem 13
  ...ord of instances in which a tail is immediately followed by a head, a head is immediately followed by a head, and etc. We denote these by <tt>TH</tt>, <t ...mine what happens to the last coin toss. Adding <tt>HH</tt> or <tt>TT</tt> is simply an [[identity]] for the last coin toss, so we will ignore them for n
  3 KB (445 words) - 19:44, 8 January 2023
• 1986 AIME Problems/Problem 12
  ...Suppose 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? ...at least <math>\dbinom{6}{0} + \dbinom{6}{1} + \dbinom{6}{2} + \dbinom{6}{3} + \dbinom{6}{4}=57</math> of its subsets have at most four elements (the n
  2 KB (364 words) - 19:41, 1 September 2020
• 1986 AIME Problems/Problem 11
  The [[polynomial]] <math>1-x+x^2-x^3+\cdots+x^{16}-x^{17}</math> may be written in the form <math>a_0+a_1y+a_2y^ ...By the [[Binomial Theorem]], this is <math>(-1) \cdot (-1)^{15}{18 \choose 3} = \boxed{816}</math>.
  6 KB (872 words) - 16:51, 9 June 2023
• 1986 AIME Problems/Problem 10
  ...h>m \equiv 358</math> mod <math>666</math>. We see that there are no other 3-digit integers that are <math>358</math> mod <math>666</math>, so <math>m = === Solution 3 ===
  3 KB (565 words) - 16:51, 1 October 2023
• 1986 AIME Problems/Problem 9
  ...=450</math>, and <math>AC=510</math>. An interior [[point]] <math>P</math> is then drawn, and [[segment]]s are drawn through <math>P</math> [[parallel]] ...}{18}\right)+\frac{1}{102}}=\frac{10}{\frac{1}{5}\cdot\frac{35}{306}+\frac{3}{306}}=\frac{10}{\frac{10}{306}} = \boxed{306}</math></center>
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1986 AIME Problems/Problem 8
  ...l [[divisor]]s of a number excluding itself) of <math>1000000</math>. What is the integer nearest to <math>S</math>? ...math>48</math> are proper. The sum of multiple logarithms of the same base is equal to the logarithm of the products of the numbers.
  3 KB (487 words) - 20:52, 16 September 2020
• 1986 AIME Problems/Problem 7
  ...nteger]]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. ...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 5
  ...largest [[positive integer]] <math>n</math> for which <math>n^3+100</math> is [[divisible]] by <math>n+10</math>? ...h>; we can double-check manually and we find that indeed <math>900\mid 890^3+100</math>.
  2 KB (338 words) - 19:56, 15 October 2023
• 1986 AIME Problems/Problem 4
  ...e fifth given equation gives <math>x_5 = 65</math>, so our answer is <math>3\cdot17 + 2\cdot65 = \boxed{181}</math>. <cmath>3x_4+2x_5=3(x_1+42)+2(x_1+90)=\boxed{181}</cmath>
  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
• 1986 AIME Problems/Problem 2
  == Solution 3 (Geometry) == ...rt5,2\sqrt6,</math> and <math>2\sqrt7,</math> by Heron's Formula, the area is the square root of the original expression.
  3 KB (460 words) - 00:44, 5 February 2022
• 1987 AIME Problems/Problem 15
  ...{440}{441}T_2</math>. Additionally, the area of triangle <math>ABC</math> is equal to both <math>T_1 + T_2 + 441</math> and <math>T_3 + T_4 + T_5 + 440. ...and <math>ABC</math> is <math>441</math>, and the ratio between the sides is <math>\sqrt {441} = 21</math>. As a result, <math>AB = 21\sqrt {440} = \sqr
  5 KB (838 words) - 18:05, 19 February 2022
• 1987 AIME Problems/Problem 14
  ...+ 2b^2 - 2ab\right)\left(a^2 + 2b^2 + 2ab\right).</math> Each of the terms is in the form of <math>x^4 + 324.</math> Using Sophie Germain, we get that x^4 + 324 &= x^4 + 4\cdot 3^4 \\
  7 KB (965 words) - 10:42, 12 April 2024
• 1987 AIME Problems/Problem 12
  ...</math>, where <math>n</math> is a [[positive integer]] and <math>r</math> is a [[positive]] [[real number]] less than <math>1/1000</math>. Find <math>n< ...<math>r</math>, <math>3n^2 + 3nr + r^2 > 3\cdot 19^2 > 1000</math>, so it is possible for <math>r</math> to be less than <math>\frac{1}{1000}</math>. H
  4 KB (673 words) - 19:48, 28 December 2023
• 1987 AIME Problems/Problem 11
  ...the largest possible value of <math>k</math> for which <math>3^{11}</math> is expressible as the sum of <math>k</math> consecutive [[positive integer]]s. <math>3^{11} = (n + 1) + (n + 2) + \ldots + (n + m) = \frac{1}{2} m(2n + m + 1)</ma
  3 KB (418 words) - 18:30, 20 January 2024
• 1987 AIME Problems/Problem 10
  ...teps are visible on the escalator at a given time? (Assume that this value is constant.) ...er the time it took Bob to climb, the [[ratio]] of their distances covered is the same as the ratio of their speeds, so <math>\frac{e}{b} = \frac{x - 75}
  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 <math>k</math> such that <math>\frac{8}{15} < \frac{n}{n + .../math>. Thus, <math>48n < 56k < 49n</math>. <math>k</math> is unique if it is within a maximum [[range]] of <math>112</math>, so <math>n = 112</math>.
  2 KB (393 words) - 16:59, 16 December 2020
• 1987 AIME Problems/Problem 6
  ...re <math>XY = YB + BC + CZ = ZW = WD + DA + AX</math>, and <math>PQ</math> is [[parallel]] to <math>AB</math>. Find the [[length]] of <math>AB</math> (i ...umber is also equal to one quarter the area of the entire rectangle, which is <math>\frac{19\cdot AB}{4}</math>, so we have <math>AB = XY + 87</math>.
  3 KB (530 words) - 07:46, 1 June 2018
• 1987 AIME Problems/Problem 5
  If we move the <math>x^2</math> term to the left side, it is factorable with [[SFFT|Simon's Favorite Factoring Trick]]: ...ath>y^2 - 10 = 39</math>, so <math>y^2 = 49</math>. Thus, <math>3x^2 y^2 = 3 \times 4 \times 49 = \boxed{588}</math>.
  1 KB (160 words) - 04:44, 21 January 2023
• 1987 AIME Problems/Problem 4
  ...h is when <math>x = 60</math> and <math>y = \pm 15</math>. Since the graph is [[symmetry|symmetric]] about the y-axis, we just need [[casework]] upon <ma *<math>x - 60 > 0</math>. Then <math>y = -\frac{3}{4}x+60</math>.
  2 KB (371 words) - 17:25, 13 February 2024
• 1987 AIME Problems/Problem 3
  ...nice'' if it is equal to the product of its distinct proper divisors. What is the sum of the first ten nice numbers? ...of the distinct 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
  What is the largest possible [[distance]] between two [[point]]s, one on the [[sphe {{AIME box|year=1987|num-b=1|num-a=3}}
  697 bytes (99 words) - 18:46, 14 February 2014
• 1987 AIME Problems/Problem 1
  An [[ordered pair]] <math>(m,n)</math> of [[non-negative]] [[integer]]s is called "simple" if the [[addition]] <math>m+n</math> in base <math>10</math ...pair]]s will be <math>(1 + 1)(4 + 1)(9 + 1)(2 + 1) = 2\cdot 5\cdot 10\cdot 3 = \boxed{300}</math>.
  1 KB (191 words) - 14:42, 17 September 2016
• 1988 AIME Problems/Problem 15
  ...typed during the day, and the boss delivers them in the order <math>1, 2, 3, 4, 5, 6, 7, 8, 9</math>. ...ch typing 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
  ...icular]] to <math>y=2x</math>, so the slope of <math>\overline{PP'}</math> is <math>\frac{-1}{2}</math>. Thus <math>\frac{y' - y}{x' - x} = \frac{-1}{2} ...math>, which is unchanged by the reflection, into the expression. But this is not necessary. We see that <math>b=-7</math>, <math>c=-12</math>, so <math>
  4 KB (700 words) - 17:21, 3 May 2021
• 1988 AIME Problems/Problem 13
  ...</math> and <math>b</math> are integers such that <math>x^2 - x - 1</math> is a factor of <math>ax^{17} + bx^{16} + 1</math>. ...Fibonacci sequence. Carrying out this pattern, we find that the remainder is <cmath>(F_{16}b + F_{17}a)x + F_{15}b + F_{16}a + 1 = 0.</cmath> Since the
  10 KB (1,585 words) - 03:58, 1 May 2023
• 1988 AIME Problems/Problem 12
  ...the product <math>abc</math> if <math>a + b + c = 43</math> and <math>d = 3</math>. ...]] <math>\frac {d}{a + d} + \frac {d}{b + d} + \frac {d}{c + d} = 1</math> is a form of [[Ceva's Theorem]].
  4 KB (727 words) - 23:37, 7 March 2024
• 1988 AIME Problems/Problem 11
  ...h> be [[complex number]]s. A 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> ...14 + 43i</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
  The number of segments joining the vertices of the polyhedron is <math>{48\choose2} = 1128</math>. We must now subtract out those segments t ...t each of its endpoints, the number of edges <math>E</math> is <math>\frac{3}{2}V = 72</math>.
  5 KB (811 words) - 19:10, 25 January 2021
• 1988 AIME Problems/Problem 9
  ...<math>20k + 8 \equiv 88 \pmod{100}</math>. This is true if the tens digit is either <math>4</math> or <math>9</math>. Casework: ...le value for the hundreds digit is <math>4</math>, and so <math>442</math> is a valid solution.
  6 KB (893 words) - 08:15, 2 February 2023
• 1988 AIME Problems/Problem 8
  & = \frac{91}{3}\cdot f(4,6) \\ ==Solution 3 (Number Theory)==
  4 KB (538 words) - 13:24, 12 October 2021
• 1988 AIME Problems/Problem 7
  .../math> divides <math>BC</math> into [[segment]]s of length 3 and 17. What is the area of triangle <math>ABC</math>? ...So, <math>\tan \alpha = \frac {17}{h}</math> and <math>\tan \beta = \frac {3}{h}</math>. Using the tangent addition formula <math>\tan (\alpha + \beta)
  1 KB (178 words) - 23:25, 20 November 2023
• 1988 AIME Problems/Problem 6
  It is possible to place positive integers into the vacant twenty-one squares of t ...ce is <math>103 - 2b</math>, so that square also has a value of <math>2b + 3(103 - 2b) = 309 - 4b</math>. Equating, we get <math>148 - 3a = 309 - 4b \Lo
  5 KB (878 words) - 23:06, 20 November 2023
• 1988 AIME Problems/Problem 4
  What is the smallest possible value of <math>n</math>? ...,1</math> and find that the <math>LHS</math> is <math>3</math> and the RHS is <math>1.</math> Similarly testing <math>1,-1,-1,1</math> yields <math>4</ma
  2 KB (394 words) - 10:21, 27 January 2024
• 1988 AIME Problems/Problem 3
  2^{3 \log_2(\log_8x)} &= \log_2x\\ (\log_8x)^3 &= \log_2x\\
  3 KB (481 words) - 21:52, 18 November 2020
• 1988 AIME Problems/Problem 2
  Note that this revolves between the two numbers. Since <math>1988</math> is even, we thus have <math>f_{1988}(11) = f_{4}(11) = \boxed{169}</math>. {{AIME box|year=1988|num-b=1|num-a=3}}
  696 bytes (103 words) - 19:16, 27 February 2018
• 1988 AIME Problems/Problem 1
  ...order -- the correct five buttons. The sample shown below has <math>\{1,2,3,6,9\}</math> as its [[combination]]. Suppose that these locks are redesigne ...ath>9</math>, the number of ways to choose a set of <math>x</math> buttons is <math>\sum^{9}_{k=1}{10 \choose k}</math>.
  1 KB (181 words) - 18:23, 26 August 2019
• 1989 AIME Problems/Problem 15
  ...ven that <math>AP=6</math>, <math>BP=9</math>, <math>PD=6</math>, <math>PE=3</math>, and <math>CF=20</math>, find the area of <math>\triangle ABC</math> ...<math>RST</math>. We'll make use of the following fact: if <math>P</math> is a point in the interior of triangle <math>XYZ</math>, and line <math>XP</ma
  13 KB (2,091 words) - 00:20, 26 October 2023
• 1989 AIME Problems/Problem 14
  ...+i</math> using the integers <math>0,1,2,\ldots,n^2</math> as digits. That is, the equation is true for a unique choice of non-negative integer <math>m</math> and digits
  2 KB (408 words) - 17:28, 16 September 2023
• 1989 AIME Problems/Problem 13
  ...members 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? ...we can take at most one from each of the pairs: <math>[2,9]</math>, <math>[3,7]</math>, <math>[4,11]</math>, <math>[6,10]</math>. Now, <math>1989 = 180\
  2 KB (274 words) - 04:07, 17 December 2023
• 1989 AIME Problems/Problem 11
  ...r D\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 ...> (in other words, <math>x \in [1,1000]</math>). Indeed, <math>D(x)</math> is symmetric about <math>x = 500.5</math>; consider replacing all of numbers <
  5 KB (851 words) - 18:01, 28 December 2022
• 1989 AIME Problems/Problem 10
  pair A = (0,0), B = (3, 0), C = (1, 4); ...wo angles in the triangle. So, the cotangent of any angle in the triangle is directly proportional to the sum of the squares of the two adjacent sides,
  8 KB (1,401 words) - 21:41, 20 January 2024
• 1989 AIME Problems/Problem 9
  Taking the given equation modulo <math>2,3,</math> and <math>5,</math> respectively, we have n^5&\equiv0\pmod{3}, \\
  6 KB (874 words) - 15:50, 20 January 2024
• 1989 AIME Problems/Problem 8
  ...+3)^2x_4+(k+4)^2x_5+(k+5)^2x_6+(k+6)^2x_7</cmath> for some <math>k\in\{1,2,3\}.</math> f(3)&=9a+3b+c&&=123,
  8 KB (1,146 words) - 04:15, 20 November 2023
• 1989 AIME Problems/Problem 6
  ...t, frozen lake. The [[distance]] between <math>A</math> and <math>B</math> is <math>100</math> meters. Allie leaves <math>A</math> and skates at a [[spee pair A=(0,0),B=(10,0),C=6*expi(pi/3);
  5 KB (864 words) - 19:55, 2 July 2023
• 1989 AIME Problems/Problem 5
  ...st terms, be the probability that the coin comes up heads in exactly <math>3</math> out of <math>5</math> flips. Find <math>i+j</math>. ...<math>{5\choose3}(h)^3(1-h)^2 = 10\left(\frac{1}{3}\right)^3\left(\frac{2}{3}\right)^2 = \frac{40}{243}</math>, so <math>i+j=40+243=\boxed{283}</math>.
  2 KB (258 words) - 00:07, 25 June 2023
• 1989 AIME Problems/Problem 4
  ...[[perfect square]] and <math>a+b+c+d+e</math> is a [[perfect cube]], what is the smallest possible value of <math>c</math>? ...dot y^3</math>, <math>3^3</math> must be a factor of <math>c</math>. <math>3^35^2 = \boxed{675}</math>, which works as the solution.
  3 KB (552 words) - 12:41, 3 March 2024
• 1989 AIME Problems/Problem 3
  Suppose <math>n</math> is a [[positive integer]] and <math>d</math> is a single [[digit]] in [[base 10]]. Find <math>n</math> if ...must be [[divisible]] by 37, and the only digit for which this is possible is <math>d = 9</math>. Thus <math>4d + 1 = 37</math> and <math>n = \boxed{750
  3 KB (499 words) - 22:17, 29 March 2024
• 1989 AIME Problems/Problem 2
  ...mber and <math>{10 \choose 2} = 45</math> have 2 members. Thus the answer is <math>1024 - 1 - 10 - 45 = \boxed{968}</math>. ...{n \choose 0}+{n \choose 1} + {n \choose 2} + \dots + {n \choose n}</math> is equivalent to <math>2^n</math>
  911 bytes (135 words) - 08:30, 27 October 2018
• 1989 AIME Problems/Problem 1
  ...+1} = \sqrt{(870)(868) +1} = \sqrt{(868 +1)^2} = \boxed{869}</math>. This is because we have that <math>a=868</math> as per the equation <math>(a+1)^2 = == Solution 3 (Symmetry with Generalization) ==
  4 KB (523 words) - 00:12, 8 October 2021
• 1990 AIME Problems/Problem 14
  ...rline{AP}</math> and <math>\overline{BP}</math> are joined, and the figure is then creased along segments <math>\overline{CP}</math> and <math>\overline{ label("$13\sqrt{3}$", A--D, S);
  7 KB (1,086 words) - 08:16, 29 July 2023
• 1990 AIME Problems/Problem 13
  ...9^{4000}_{}</math> has 3817 [[digit]]s and that its first (leftmost) digit is 9, how many [[element]]s of <math>T_{}^{}</math> have 9 as their leftmost d If there is exactly 1 n-digit power of 9, then such a number <math>m</math> cannot begi
  5 KB (762 words) - 01:18, 10 February 2023
• 1990 AIME Problems/Problem 12
  ...]s of the 12-gon can be written in the form <math>a + b \sqrt{2} + c \sqrt{3} + d \sqrt{6},</math> where <math>a^{}_{}</math>, <math>b^{}_{}</math>, <ma *The length of each of the 12 sides is <math>2 \cdot 12\sin 15</math>. <math>24\sin 15 = 24\sin (45 - 30) = 24\fra
  6 KB (906 words) - 13:23, 5 September 2021
• 1990 AIME Problems/Problem 11
  ...h>, which decreases as <math>a</math> increases. Thus, <math>n = 23</math> is the greatest possible value to satisfy the given conditions. ...annot be less than or equal to <math>n</math>, else the product of <math>n-3</math> consecutive positive integers will be less than <math>n!</math>.
  3 KB (519 words) - 09:28, 28 June 2022
• 1990 AIME Problems/Problem 10
  ...unity]]. The set <math>C = \{zw : z \in A ~ \mbox{and} ~ w \in B\}</math> is also a set of complex roots of unity. How many distinct elements are in <m ...^8, n^{16}, \ldots n^{144}\}</math> and of set <math>B</math> as <math>\{n^3, n^6, \ldots n^{144}\}</math>. <math>n^x</math> can yield at most <math>144
  3 KB (564 words) - 04:47, 4 August 2023
• 1990 AIME Problems/Problem 9
  A [[fair]] coin is to be tossed <math>10_{}^{}</math> times. Let <math>\frac{i}{j}^{}_{}</math ...obability <math>\frac{144}{1024} = \frac{9}{64}</math>. Thus, our solution is <math>9 + 64 = \boxed{073}</math>.
  3 KB (425 words) - 19:31, 30 July 2021
• 1990 AIME Problems/Problem 8
  ...ng columns of three targets each and one 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 7
  ...s lengths of side <math>15,\ 20,\ 25</math>, indicating that it is a <math>3-4-5</math> [[right triangle]]. At this point, we just need to find another ...2},\ \frac{15}{2}</math>. It follows that <math>\frac{QP'}{RP'} = \frac{5}{3}</math>, and so <math>P' = \left(\frac{5x_R + 3x_Q}{8},\frac{5y_R + 3y_Q}{8
  8 KB (1,319 words) - 11:34, 22 November 2023
• 1990 AIME Problems/Problem 6
  ...them. On September 1 she catches a random sample of 70 fish and finds that 3 of them are tagged. To calculate the number of fish in the lake on May 1, s ...ember is proportional to the percentage of tagged fish in May, <math>\frac{3}{42} = \frac{60}{x} \Longrightarrow \boxed{x = 840}</math>.
  2 KB (325 words) - 13:16, 26 June 2022
• 1990 AIME Problems/Problem 5
  Let <math>n^{}_{}</math> be the smallest positive [[integer]] that is a multiple of <math>75_{}^{}</math> and has exactly <math>75_{}^{}</math> p ...fore, <math>n = 2^43^45^2</math> and <math>\frac{n}{75} = \frac{2^43^45^2}{3 \cdot 5^2} = 16 \cdot 27 = \boxed{432}</math>.
  1 KB (175 words) - 03:45, 21 January 2023
• 1990 AIME Problems/Problem 4
  ...- 29 \Longleftrightarrow 0 = (x - 13)(x + 3)</math>. The positive [[root]] is <math>\boxed{013}</math>. {{AIME box|year=1990|num-b=3|num-a=5}}
  1 KB (156 words) - 07:35, 4 November 2022
• 1990 AIME Problems/Problem 3
  ...q s\geq 3)</math> such that each [[interior angle]] of <math>P_1^{}</math> is <math>\frac{59}{58}</math> as large as each interior angle of <math>P_2^{}< The formula for the interior angle of a regular sided [[polygon]] is <math>\frac{(n-2)180}{n}</math>.
  3 KB (516 words) - 19:18, 16 April 2024
• 1990 AIME Problems/Problem 2
  Find the value of <math>(52+6\sqrt{43})^{3/2}-(52-6\sqrt{43})^{3/2}</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
• 1990 AIME Problems/Problem 1
  The [[increasing sequence]] <math>2,3,5,6,7,10,11,\ldots</math> consists of all [[positive integer]]s that are ne ...r is the biggest non-square and non-cube less than <math>529</math>, which is <math>\boxed{528}</math>.
  2 KB (283 words) - 23:11, 25 June 2023
• 2006 AMC 10B Problems/Problem 2
  ...</math>, define <math> x \spadesuit y = (x+y)(x-y) </math>. What is <math> 3 \spadesuit (4 \spadesuit 5) </math>? ...4 \spadesuit 5) = 3 \spadesuit((4+5)(4-5)) = 3 \spadesuit (-9) = (3+(-9))(3-(-9)) = \boxed{\textbf{(A)}-72}</math>
  633 bytes (85 words) - 10:33, 19 December 2021
• 2006 AMC 10B Problems/Problem 4
  ...side the smaller circle and inside the larger circle is painted blue. What is the ratio of the blue-painted area to the red-painted area? <math> \textbf{(A) } 2\qquad \textbf{(B) } 3\qquad \textbf{(C) } 6\qquad \textbf{(D) } 8\qquad \textbf{(E) } 9 </math>
  1 KB (172 words) - 10:47, 19 December 2021
• 2006 AMC 10B Problems/Problem 5
  ...of the square are parallel to the sides of the two given rectangles. What is the smallest possible area of the square? ...tely enclosed in a square with a side length of 5. The area of this square is <math>5^2 = 25</math>.
  1 KB (242 words) - 18:35, 15 August 2023
• 2006 AMC 10B Problems/Problem 7
  Which of the following is equivalent to <math> \sqrt{\frac{x}{1-\frac{x-1}{x}}} </math> when <math> x ...math>. As no other option choice fits, <math>\boxed{\textbf{(A)}-x}</math> is the correct solution.
  1 KB (179 words) - 10:33, 19 August 2022
• 2006 AMC 10B Problems/Problem 11
  What is the tens digit in the sum <math> 7!+8!+9!+...+2006!</math> <math> \textbf{(A) } 1\qquad \textbf{(B) } 3\qquad \textbf{(C) } 4\qquad \textbf{(D) } 6\qquad \textbf{(E) } 9 </math>
  1 KB (170 words) - 14:00, 26 January 2022
• 2006 AMC 10B Problems/Problem 12
  ...y=\frac{1}{4}x+b </math> intersect at the point <math> (1,2) </math>. What is <math> a+b </math>? <math> \textbf{(A) } 0\qquad \textbf{(B) } \frac{3}{4}\qquad \textbf{(C) } 1\qquad \textbf{(D) } 2\qquad \textbf{(E) } \frac{9
  1 KB (220 words) - 20:07, 27 November 2023
• 2006 AMC 10B Problems/Problem 14
  ...ac1a </math> are the roots of the equation <math> x^2-px+q=0 </math>. What is <math>q</math>? ...of the form <math> x^2 + bx + c = 0 </math>, the product of the [[root]]s is <math>c</math> ([[Vieta's Formulas]]).
  2 KB (264 words) - 21:10, 19 September 2023
• 1991 AIME Problems/Problem 15
  ...unique positive integer <math>n^{}_{}</math> for which <math>S_n^{}</math> is also an integer. Find this <math>n^{}_{}</math>. ...he value of <math>S_n</math>. The minimum value of <math>S_n</math>, then, is the length of the straight line connecting the bottom vertex of the first r
  4 KB (658 words) - 16:58, 10 November 2023
• 1991 AIME Problems/Problem 14
  A [[hexagon]] is inscribed in a [[circle]]. Five of the sides have length <math>81</math> an ...18)), D=expi(-pi/2+acos(475/486)+2*acos(7/18)), E=expi(-pi/2+acos(475/486)+3*acos(7/18)), F=expi(-pi/2-acos(475/486)-acos(7/18));
  2 KB (284 words) - 03:56, 23 January 2023
• 1991 AIME Problems/Problem 13
  ...blue. What is the largest possible number of red socks in the drawer that is consistent with this data? ...cks are drawn randomly, without replacement, both are red or both are blue is given by
  7 KB (1,328 words) - 20:24, 5 February 2024
• 2006 AMC 10B Problems/Problem 15
  ...D</math> is <math>24</math> and <math> \angle BAD = 60^\circ </math>. What is the area of rhombus <math>BFDE</math>? ...n, B=(2,0), C=(3, sqrt(3)), D=(1, sqrt(3)), E=(1, 1/sqrt(3)), F=(2, 2/sqrt(3));
  3 KB (445 words) - 22:01, 20 August 2022
• 1991 AIME Problems/Problem 12
  ...ath>\overline{CD}</math>, and <math>\overline{DA}</math>, respectively. It is given that <math>PB^{}_{}=15</math>, <math>BQ^{}_{}=20</math>, <math>PR^{}_ <center><asy>defaultpen(fontsize(12)+linewidth(1.3)); pair A=(0,28.8), B=(38.4,28.8), C=(38.4,0), D=(0,0), O, P=(23.4,28.8), Q
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...re <math>a,b,c^{}_{}</math> are positive integers and <math>c^{}_{}</math> is not divisible by the square of any prime. Find <math>a+b+c^{}_{}</math>. _Diagram by 1-1 is 3_
  4 KB (740 words) - 19:33, 28 December 2022
• 1991 AIME Problems/Problem 10
  ...s written as a [[fraction]] in [[irreducible fraction|lowest terms]], what is its [[numerator]]? aab & 4 & 2 & 3 \\
  5 KB (813 words) - 06:10, 25 February 2024
• 2006 AMC 10B Problems/Problem 17
  ...randomly selects one ball from his bag and puts it into Alice's bag. What is the probability that after this process the contents of the two bags are th ...} \frac{1}{6}\qquad \textbf{(C) } \frac{1}{5}\qquad \textbf{(D) } \frac{1}{3}\qquad \textbf{(E) } \frac{1}{2} </math>
  1 KB (211 words) - 04:32, 4 November 2022
• 1991 AIME Problems/Problem 9
  ...and that <math>\csc x+\cot x=\frac mn,</math> where <math>\frac mn</math> is in lowest terms. Find <math>m+n^{}_{}.</math> ...problem is much easier computed if we consider what <math>\sec (x)</math> is, then find the relationship between <math>\sin( x)</math> and <math>cos (x)
  10 KB (1,590 words) - 14:04, 20 January 2023
• 2006 AMC 10B Problems/Problem 18
  ...}}{a_{n-2}} </math> for each positive integer <math> n \ge 3 </math>. What is <math> a_{2006} </math>? ...\mathrm{(C) \ } \frac{3}{2}\qquad \mathrm{(D) \ } 2\qquad \mathrm{(E) \ } 3 </math>
  1 KB (158 words) - 01:33, 29 May 2023
• 1991 AIME Problems/Problem 8
  Since <math>-a</math> is an integer, we need <math>\sqrt{a^2-24a}</math> to be an integer (let this Which implies that <math>b^2 + 144</math> is a [[perfect square]] also (let this be <math>c^2</math>). Then
  2 KB (310 words) - 11:25, 13 June 2023
• 1991 AIME Problems/Problem 5
  ...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
  The [[range]] of the [[sine]] function is <math>-1 \le y \le 1</math>. It is [[periodic function|periodic]] (in this problem) with a period of <math>\fr ...ath>x < 1</math>, we can count <math>4</math> more solutions. The solution is <math>154 + 1 + 4 = \boxed{159}</math>.
  2 KB (300 words) - 16:01, 26 November 2019
• 1991 AIME Problems/Problem 3
  ...th> for <math>k = 0,1,2,\ldots,1000</math>. For which <math>k_{}^{}</math> is <math>A_k^{}</math> the largest? ...lues of <math>k>165.8</math>, the largest possible value of <math>k</math> is <math>\boxed{166}</math>.
  5 KB (865 words) - 12:13, 21 May 2020
• 1991 AIME Problems/Problem 2
  ...line {AB}</math> of [[length]] 4 and <math>\overline {CB}</math> of length 3. Divide <math>\overline {AB}</math> into 168 [[congruent]] [[segment]]s wit pair A=(0,0),B=(4,0),C=(4,3),D=(0,3);
  4 KB (595 words) - 12:51, 17 June 2021
• 1991 AIME Problems/Problem 1
  ...nd so there are no integral solutions for <math>(x,y)</math>. The solution is <math>5^2 + 11^2 = \boxed{146}</math>. ...ctor of <math>11</math> is <math>(5,11)</math>, and checking shows that it is correct.
  4 KB (628 words) - 22:05, 7 June 2021
• 1992 AIME Problems/Problem 15
  ...ne a positive integer <math>n^{}_{}</math> to be a factorial tail if there is some positive integer <math>m^{}_{}</math> such that the decimal representa .../math> is a multiple of <math>5</math>, <math>f(m) = f(m+1) = f(m+2) = f(m+3) = f(m+4)</math>.
  2 KB (358 words) - 01:54, 2 October 2020
• 1992 AIME Problems/Problem 14
  == Solution 3 == A consequence of Ceva's theorem sometimes attributed to Gergonne is that <math>\frac{AO}{OA'}=\frac{AC'}{C'B}+\frac{AB'}{B'C}</math>, and simil
  4 KB (667 words) - 01:26, 16 August 2023
• 1992 AIME Problems/Problem 13
  ...ath>. So if we can write: <math>b^2=-(a+n)^2+m</math>, then <math>m</math> is the maximum value of <math>b^2</math> (this follows directly from the [[tri Then the area is <math>9\cdot\frac{1}{2} \cdot \frac{40\cdot 41}{9} = \boxed{820}</math>.
  4 KB (703 words) - 02:40, 29 December 2023
• 1992 AIME Problems/Problem 9
  .../math>. A circle with center <math>P^{}_{}</math> on <math>AB^{}_{}</math> is drawn tangent to <math>BC^{}_{}</math> and <math>AD^{}_{}</math>. Given tha ...<math>AP=x=\frac{161}{3}</math>. This gives us a final answer of <math>161+3=\boxed{164}</math>
  5 KB (874 words) - 10:27, 22 August 2021
• 1992 AIME Problems/Problem 8
  ...1,a_3-a_2,a_4-a_3,\ldots)</math>, whose <math>n^{\mbox{th}}_{}</math> term is <math>a_{n+1}-a_n^{}</math>. Suppose that all of the terms of the sequence ...binom{n-1}{1}\Delta a_n + \binom{n-1}{2}\Delta^2 a_n +\binom{n-1}{3}\Delta^3 a_n + ...</math>
  5 KB (778 words) - 21:36, 3 December 2022
• 1992 AIME Problems/Problem 7
  ...</math> is <math>120^{}_{}</math>, the area of face <math>BCD^{}_{}</math> is <math>80^{}_{}</math>, and <math>BC=10^{}_{}</math>. Find the volume of the ...ot \sin 30^\circ=8</math>. Therefore, the volume is <math>\frac{8\cdot120}{3}=\boxed{320}</math>.
  800 bytes (114 words) - 17:40, 14 March 2017
• 1992 AIME Problems/Problem 6
  ...irs of consecutive integers in <math>\{1000,1001,1002,\ldots,2000\}</math> is no carrying required when the two integers are added? 0\leq C\leq 8 & 1 & A & B & C+1 & 0\leq A,B,C\leq 4 & 5^3 \\
  3 KB (455 words) - 02:03, 10 July 2021
• 1992 AIME Problems/Problem 4
  In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are \text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\vspace{4pt}
  3 KB (476 words) - 14:13, 20 April 2024
• 1992 AIME Problems/Problem 3
  ...es, winning three and losing one. At the end of the weekend, her win ratio is greater than <math>.503</math>. What's the largest number of matches she co ...ches 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
  ...s [[decimal representation]], there are at least two digits and each digit is less than any digit to its right. How many ascending positive integers are ...ition for 0 is at the leftmost end of the number, i.e. a leading 0), there is exactly one ascending number with those digits.
  2 KB (336 words) - 05:18, 4 November 2022
• 1992 AIME Problems/Problem 1
  ...ac{49}{30}.</math> Following this pattern, our answer is <math>4(10)+8(1+2+3+\cdots+9)=\boxed{400}.</math> ...us the sum of the smallest <math>8</math> rational numbers satisfying this is <math>\frac12\cdot8\cdot1=4</math>. Now refer to solution 1.
  1 KB (190 words) - 20:02, 23 February 2022
• 2006 AMC 10B Problems/Problem 19
  ..., respectively. What is the area of the shaded region in the figure, which is bounded by <math>BD</math>, <math>BE</math>, and the minor arc connecting < pair O=origin, A=(1,0), C=(0,1), B=(1,1), D=(1, sqrt(3)), E=(sqrt(3), 1), point=B;
  5 KB (873 words) - 15:39, 29 May 2023
• 2006 AMC 10B Problems/Problem 20
  ...6,178)</math>, <math>D=(8,y)</math>, for some integer <math>y</math>. What is the area of rectangle <math>ABCD</math>? Therefore the area of rectangle <math>ABCD</math> is <math> 200\sqrt{101}\cdot2\sqrt{101} = 40,400 \Rightarrow E </math>
  4 KB (594 words) - 15:45, 30 July 2023
• 2006 AMC 10B Problems/Problem 21
  ...math>6</math>, on each die are in the ratio <math>1:2:3:4:5:6</math>. What is the probability of rolling a total of <math>7</math> on the two dice? <math>2</math>, <math>3</math>, <math>4</math>, <math>5</math>, and <math>6</math> are <math>2x</ma
  3 KB (484 words) - 19:09, 15 October 2023
• 2006 AMC 10B Problems/Problem 22
  ...ath>, and <math>N</math> are positive integers with <math>N>1</math>. What is the cost of the jam Elmo uses to make the sandwiches? ...nly possible positive integer pairs <math>(N , 4B+5J)</math> whose product is <math>253</math> are: <math> (1,253) ; (11,23) ; (23,11) ; (253,1) </math>
  2 KB (394 words) - 00:51, 25 November 2023
• 2006 AMC 10B Problems/Problem 23
  ...te sides. The areas of the three triangles are 3, 7, and 7, as shown. What is the area of the shaded quadrilateral? pair A = (0,0), B = (3,0), C = (1.4, 2), D = B + 0.4*(C-B), Ep = A + 0.3*(C-A);
  5 KB (861 words) - 00:53, 25 November 2023
• 2006 AMC 10B Problems/Problem 24
  ...ath> and <math>BC</math> are common external tangents to the circles. What is the area of the [[concave]] [[hexagon]] <math>AOBCPD</math>? fill((-3,7)--(-3,-7)--(-7,-7)--(-7,7)--cycle, white);</asy>
  4 KB (558 words) - 14:38, 6 April 2024
• 2006 AMC 10B Problems/Problem 25
  ..."and the last two digits just happen to be my age." Which of the following is <b><i>not</i></b> the age of one of Mr. Jones's children? ...the license plate. Since at least one of <math>4</math> or <math>8</math> is contained in <math>S</math>, we have <math>4 | m</math>.
  5 KB (878 words) - 14:39, 3 December 2023
• 1993 AIME Problems/Problem 15
  A=(8,0); B=origin; C=(3,4); H=(3,0); draw(A--B--C--cycle); draw(C--H); ...s of the two right triangles, the distance between the two tangency points is simply <math>\frac{n-2}{2n+2}=\frac{n-2}{2(n+1)}</math>.
  3 KB (449 words) - 21:39, 21 September 2023
• 1993 AIME Problems/Problem 14
  ...a larger rectangle (with one vertex on each side) is called unstuck if it is possible to rotate (however slightly) the smaller rectangle about its cente ...mbinations of positive 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 13
  ...again. If <math>t\,</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? ...3}.</cmath> Finally, the sum of the numerator and denominator is <math>160+3=\boxed{163}.</math>
  8 KB (1,231 words) - 20:06, 26 November 2023
• 1993 AIME Problems/Problem 12
  ...th>\overline{P_n L}</math>. Given that <math>P_7 = (14,92)\,</math>, what is <math>k + m\,</math>? ...>(r,s)</math> and we want to find <math>(u,v)</math> so <math>(r,s)</math> is the midpoint of <math>(u,v)</math> and <math>(p,q)</math>, then <math>u=2r-
  4 KB (611 words) - 13:59, 15 July 2023
• 1993 AIME Problems/Problem 11
  ...st in the first game, and that the probability that he wins the sixth game is <math>m/n\,</math>, where <math>m\,</math> and <math>n\,</math> are relativ .../math>, and the probability of the second person winning is <math>\frac{1}{3}</math>.
  7 KB (1,058 words) - 20:57, 22 December 2020
• 1993 AIME Problems/Problem 10
  ...h>T</math> triangular faces and <math>P</math> pentagonal faces meet. What is the value of <math>100P+10T+V</math>? ...=\frac{3t+5p}{2}</math>, (the factor <math>2</math> in the [[denominator]] is because we are counting twice each edge, since two adjacent faces share one
  4 KB (716 words) - 20:50, 17 April 2022
• 1993 AIME Problems/Problem 9
  ...will have more than one label and some points will not have a label. What is the smallest integer that labels the same point as <math>1993</math>? ...}</math>. Therefore, one of <math>1993 - n</math> or <math>1994 + n</math> is odd, and each of them must be a multiple of <math>125</math> or <math>16</m
  3 KB (488 words) - 02:06, 22 September 2023
• 1993 AIME Problems/Problem 8
  ...distinct subsets of <math>S\,</math> so that the union of the two subsets is <math>S\,</math>? The order of selection does not matter; for example, the ...math> elements of <math>S.</math> So our final answer is then <math>\frac {3^6 - 1}{2} + 1 = \boxed{365}.</math>
  9 KB (1,400 words) - 14:09, 12 January 2024
• 1993 AIME Problems/Problem 7
  ...the box. If <math>p</math> is written as a fraction in lowest terms, what is the sum of the numerator and denominator? There is a total of <math>P(1000,6)</math> possible ordered <math>6</math>-tuples <m
  5 KB (772 words) - 09:04, 7 January 2022
• 1993 AIME Problems/Problem 6
  What is the smallest [[positive]] [[integer]] that can be expressed as the sum of n ...e find that the least possible value of <math>b = 45</math>, so the answer is <math>10(45) + 45 = 495</math>.
  3 KB (524 words) - 18:06, 9 December 2023
• 1993 AIME Problems/Problem 5
  ...h>n \ge 1\,</math>, define <math>P_n(x) = P_{n - 1}(x - n)\,</math>. What is the [[coefficient]] of <math>x\,</math> in <math>P_{20}(x)\,</math>?<!-- do ...0</math> into the function definition, we get <math>P_0(x-210) = (x - 210)^3 + 313(x - 210)^2 - 77(x - 210) - 8</math>. We only need the coefficients of
  2 KB (355 words) - 13:25, 31 December 2018
• 1993 AIME Problems/Problem 4
  ...k) = (c-a)(d-c) = 93</math>. Hence <math>(c - a,d - c) = (1,93),(3,31),(31,3),(93,1)</math>. ...1,c - 28,c,c + 3),</math> <math>(c - 1,c + 92,c,c + 93),</math> <math>(c - 3,c + 28,c,c + 31)</math>. The last two solutions don't follow <math>a < b <
  8 KB (1,343 words) - 16:27, 19 December 2023
• 1993 AIME Problems/Problem 3
  <center><math>\begin{array}{|c|c|c|c|c|c|c|c|c|} \hline n & 0 & 1 & 2 & 3 & \dots & 13 & 14 & 15 \\ :(b) those who caught <math>3</math> or more fish averaged <math>6</math> fish each;
  2 KB (364 words) - 00:05, 9 July 2022
• 1993 AIME Problems/Problem 2
  ...^2 - 39^2 = \left|\sum_{i=0}^9 \frac{(4i+1)^2}{2} - \sum_{i=0}^9 \frac{(4i+3)^2}{2}\right|</math>. Applying [[difference of squares]], we see that ...)^2 - (4i+3)^2}{2}\right| &= \left|\sum_{i=0}^9 \frac{(4i+1+4i+3)(4i+1-(4i+3))}{2}\right|\\ &= \left|\sum_{i=0}^9 -(8i+4) \right|.
  2 KB (241 words) - 11:56, 13 March 2015

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