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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 (777 words) - 11:32, 15 March 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) - 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=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 (3,477 words) - 16:43, 1 January 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-

11 KB (1,980 words) - 12:52, 20 February 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]].

5 KB (892 words) - 21:52, 1 May 2021
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.

4 KB (547 words) - 17:23, 30 December 2020
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

2 KB (351 words) - 10:39, 1 October 2015
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

788 bytes (120 words) - 10:32, 8 November 2021
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>?

1,007 bytes (165 words) - 00:28, 30 December 2023
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.

6 KB (936 words) - 15:38, 22 February 2024
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

1 KB (153 words) - 13:11, 14 May 2019
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

2 KB (383 words) - 10:49, 4 September 2022
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.

1 KB (197 words) - 10:59, 14 April 2024
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>

3 KB (602 words) - 09:01, 7 June 2023
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>.

1 KB (185 words) - 20:24, 6 March 2024
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>.

8 KB (1,469 words) - 21:11, 16 September 2022

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