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• Overcounting
  ...otal count via subtraction or division. The idea of strategic overcounting is fundamental to [[combinatorics]] and plays a role in incredibly important c An example of a classic problem is as follows:
  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 (1,996 words) - 12:01, 18 May 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 (1,581 words) - 18:59, 9 May 2024
• 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
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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>?
  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
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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
  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
• 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]].
  6 KB (957 words) - 23:49, 7 March 2024
• 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
  8 KB (1,346 words) - 12:53, 8 October 2023
• 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>\# (
  2 KB (263 words) - 00:54, 17 November 2019
• 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
  11 KB (2,082 words) - 15:23, 2 January 2022
• 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]].
  2 KB (372 words) - 19:04, 30 May 2015
• 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>
  4 KB (691 words) - 18:38, 19 September 2021
• 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]].
  7 KB (1,265 words) - 13:22, 14 July 2021
• 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
  4 KB (680 words) - 12:54, 16 October 2023
• 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
  6 KB (1,003 words) - 09:11, 7 June 2023
• 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>.
  2 KB (398 words) - 16:57, 29 December 2021
• 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>
  8 KB (1,397 words) - 21:55, 20 January 2024
• 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
  3 KB (368 words) - 19:26, 6 June 2015
• 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
  5 KB (860 words) - 15:36, 10 December 2023
• 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);
  3 KB (575 words) - 15:27, 19 March 2023
• 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
  2 KB (280 words) - 15:30, 22 February 2024
• 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:
  4 KB (658 words) - 16:19, 28 April 2024
• 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
  4 KB (644 words) - 12:55, 7 March 2022
• 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
  4 KB (736 words) - 02:00, 7 March 2024
• 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.''"
  3 KB (453 words) - 11:13, 9 June 2023

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