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• 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 (2,658 words) - 16:02, 8 May 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:
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• 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
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• Harmonic mean
  ...math>n</math> [[positive]] [[real number]]s <math> x_1, x_2... x_n </math> is defined to be: <math> \frac{n} {\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_ ...ate <math>\frac 3{\frac 13 + \frac 16 - \frac 12} = \frac 30</math>, which is obviously problematic.
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• LaTeX
  ...d [[math|mathematical]] and scientific writing. <math>\text{\LaTeX}</math> is very handy for producing equations such as <cmath>1+2+3+4+5+\sin \pi = \frac{5\cdot 6}{2}+0=15.</cmath>
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• North Carolina MathCounts
  In the North Carolina MathCounts State Competition, the Countdown Round is unofficial in that it doesn't affect individual results. * 1987 - Ashley Reiter (3), Stephen London (41), Tim Ross (37), Ghene Faulcon, Coach: Caroline Wolfe
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• Divisibility
  In [[number theory]], '''divisibility''' is the ability of a number to evenly divide another number. The study of divis ...th>a</math> is a '''multiple''' of <math>b</math>, and that <math>a</math> is '''divisible''' or '''evenly divisible''' by <math>b</math>.
  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
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• Divisibility rules
  ..., in particular, a number is divisible by 2 if and only if its units digit is divisible by 2, i.e. if the number ends in 0, 2, 4, 6 or 8. === Divisibility Rule for 3 and 9===
  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
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• Triangle
  A '''triangle''' is a type of [[polygon]]. {{asy image|<asy>draw((0,1)--(2,0)--(3,2)--cycle);</asy>|right|A triangle.}}
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• Greatest common divisor
  ...common factor''')) of two or more [[integer]]s is the largest integer that is a [[divisor]] of all the given numbers. The GCD is sometimes called the '''greatest common factor''' ('''GCF''').
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• Overcounting
  ...otal count via subtraction or division. The idea of strategic overcounting is fundamental to [[combinatorics]] and plays a role in incredibly important c An example of a classic problem is as follows:
  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.
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• 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
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• Euclidean algorithm
  ...(GCD) of two elements of a [[Euclidean domain]], the most common of which is the [[nonnegative]] [[integer]]s <math>\mathbb{Z}{\geq 0}</math>, without [ The basic idea is to repeatedly use the fact that <math>\gcd({a,b}) \equiv \gcd({b,a - b})</m
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• Generating function
  ...function) is <math>c_0 + c_1 x + c_2 x^2 + \cdots </math> and the sequence is <math>c_0, c_1, c_2,\ldots</math>. ...n}=2^n</math>(let <math>{x}=1</math>), also <math>{n \choose 1}+{n \choose 3}+\cdots={n \choose 0}+{n \choose 2}+\cdots</math>.
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• Counting number
  ...at we count numbers of objects using positive integers (for example, <math>3</math> pencils). These are just the numbers in the set of {1,2,3,4,..}
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• Binomial Theorem
  ...efficient]]. In other words, the coefficients when <math>(a + b)^n</math> is expanded and like terms are collected are the same as the entries in the <m For example, <math>(a + b)^5 = a^5 + 5 a^4 b + 10 a^3 b^2 + 10 a^2 b^3 + 5 a b^4 + b^5</math>, with coefficients <math>1 = \binom{5}{0}</math>, <m
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• Prime number
  A '''prime number''' (or simply '''prime''') is a [[positive integer]] <math>p>1</math> whose only positive [[divisor | div ...fined as being neither prime nor [[composite number|composite]] because it is its only factor among the [[natural number|natural numbers]].
  6 KB (985 words) - 12:38, 25 February 2024
• Recursion
  ...f the sequence in terms of previous values: <math>F_0=1, F_1=1, F_2=2, F_3=3, F_4=5, F_5=8</math>, and so on. Often, it is convenient to convert a recursive definition into a closed-form definition.
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• Function
  ...e value in the second. For instance, one function may map 1 to 1, 2 to 4, 3 to 9, 4 to 16, and so on. This function has the rule that it takes its inp ...]] between <math>A</math> and <math>B</math>.) We say that <math>f</math> is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \t
  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.
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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>.
  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]].
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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
  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
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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
  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
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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>
  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
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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
  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);
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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.)
  7 KB (1,276 words) - 20:51, 6 January 2024
• 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>
  3 KB (508 words) - 21:05, 26 February 2024
• 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>.
  5 KB (838 words) - 17:20, 3 January 2023
• 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.
  5 KB (803 words) - 16:25, 10 August 2020
• 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
  8 KB (1,217 words) - 20:15, 7 September 2023
• 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
  6 KB (936 words) - 10:37, 27 November 2023
• 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.
  5 KB (912 words) - 20:06, 14 March 2023
• 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
  2 KB (334 words) - 20:52, 13 March 2022
• 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
  7 KB (1,201 words) - 16:59, 19 February 2024
• 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
  2 KB (308 words) - 02:27, 1 May 2024
• 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>
  5 KB (778 words) - 13:10, 29 November 2017
• 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
  5 KB (827 words) - 17:30, 21 February 2024
• 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.
  1 KB (231 words) - 19:45, 24 February 2020
• 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
  1,006 bytes (151 words) - 21:56, 22 April 2022
• International Mathematical Olympiad
  The '''International Mathematical Olympiad''' is the pinnacle of all high school [[mathematics competition]]s and the oldest ...eakdown=<u>Problem 1/4</u>: 6.5<br><u>Problem 2/5</u>: 7.5-8<br><u>Problem 3/6</u>: 9.5<br><u>Problem SL1-2</u>: 5.5-7<br><u>Problem SL3-4</u>: 7-8<br><
  3 KB (490 words) - 03:32, 23 July 2023

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