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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

5 KB (886 words) - 21:12, 22 January 2024
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;

2 KB (307 words) - 15:30, 30 March 2024
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

8 KB (1,016 words) - 00:17, 31 December 2023
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

7 KB (792 words) - 10:14, 23 April 2024
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

5 KB (706 words) - 23:49, 29 January 2024
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: 0.5 (School/Chapter), 1 (State/National) Sprint: 1-1.5 (School/Chapter), 2-2.5 (State/National) Target: 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>.

190 bytes (26 words) - 06:13, 16 February 2024
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) - 00:43, 24 April 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=Problem A/B, 1/2: 7 Problem A/B, 3/4: 8 Problem A/B, 5/6: 9}}

4 KB (623 words) - 13:11, 20 February 2024
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 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

4 KB (613 words) - 13:08, 18 July 2023
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.

4 KB (574 words) - 15:28, 22 February 2024
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.)

1 KB (200 words) - 23:35, 28 August 2020
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);

3 KB (415 words) - 18:01, 24 May 2020
2015 IMO Problems
We say that a finite set <math>\mathcal{S}</math> in the plane is balanced ...t points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is

4 KB (692 words) - 22:33, 15 February 2021
United States of America Mathematical Olympiad
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.

3 KB (560 words) - 22:51, 13 January 2024
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_

699 bytes (110 words) - 12:44, 20 September 2015
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

2 KB (422 words) - 16:20, 5 March 2023
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

2 KB (314 words) - 06:45, 1 May 2014
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

3 KB (571 words) - 00:42, 22 October 2021
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>.

10 KB (809 words) - 16:40, 17 March 2024
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

4 KB (734 words) - 19:19, 10 October 2023
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.

3 KB (551 words) - 16:22, 13 September 2023
Euler's Totient Theorem
'''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...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]].

5 KB (849 words) - 16:14, 18 May 2021
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.

1 KB (196 words) - 00:49, 6 January 2021
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

4 KB (580 words) - 15:33, 2 April 2024
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>.

2 KB (277 words) - 16:21, 29 April 2023
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

3 KB (496 words) - 23:22, 5 January 2022
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===

8 KB (1,315 words) - 18:18, 2 March 2024
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

5 KB (768 words) - 20:45, 1 September 2022
Triangle
A '''triangle''' is a type of [[polygon]]. {{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}

4 KB (628 words) - 17:17, 17 May 2018
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''').

2 KB (288 words) - 22:40, 26 January 2021
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:

4 KB (635 words) - 12:19, 2 January 2022
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:

3 KB (623 words) - 13:10, 20 February 2024
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

5 KB (709 words) - 10:28, 19 February 2024
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

6 KB (924 words) - 21:50, 8 May 2022
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>.

4 KB (659 words) - 12:54, 7 March 2022
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

5 KB (935 words) - 13:11, 20 February 2024
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.

2 KB (316 words) - 16:03, 1 January 2024
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

10 KB (1,761 words) - 03:16, 12 May 2023
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.

8 KB (1,192 words) - 17:20, 16 June 2023
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]].

2 KB (296 words) - 15:04, 5 August 2022
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

3 KB (496 words) - 22:14, 5 January 2024
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.

6 KB (350 words) - 12:58, 26 September 2023
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,

1 KB (162 words) - 21:44, 13 March 2022
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]].

9 KB (1,555 words) - 20:05, 2 November 2023
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=Problem 1/4: 6.5 Problem 2/5: 7.5-8 Problem 3/6: 9.5 Problem SL1-2: 5.5-7 Problem SL3-4: 7-8 <

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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>?

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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>?

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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>?

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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

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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 $2366, how many members are in the League?

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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>

1 KB (184 words) - 13:57, 19 January 2021
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 $<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

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