Search results

  • This is the '''AMC historical results''' page. This page should include results for *Mean: 68.3
    17 KB (1,921 words) - 13:00, 28 April 2024
  • ...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
  • 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
  • ...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
  • 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
  • ...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
  • 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
  • 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
  • We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> ...t points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
    4 KB (692 words) - 22:33, 15 February 2021
  • 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
  • ...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
  • ...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
  • ...)! + 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
  • ...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
  • 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
  • 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''' (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
  • ...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
  • ...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
  • ...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
  • ...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
  • ...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
  • ...+ 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
  • ...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
  • ...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
  • ...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
  • ...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
  • ...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
  • 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
  • ...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
  • 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
  • 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
  • ...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
  • 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''' 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
  • 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''' 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
  • ...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
  • 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
  • ...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
  • 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
  • ...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
  • ...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
  • 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
  • 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
  • ...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
  • ..., 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
  • ...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
  • 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
  • ...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
  • ...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
  • 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''' 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
  • ...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
  • ...(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
  • ...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
  • ...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
  • ...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
  • 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
  • ...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
  • ...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
  • ...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
  • ...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
  • ...<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
  • ...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
  • ...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
  • ...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
  • 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
  • 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
  • ...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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  • ...], 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
  • ...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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  • ...<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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  • ...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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  • ...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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  • ...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''' 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''' 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
  • 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''' 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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  • ...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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  • 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
  • ..., 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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  • 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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  • 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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  • ...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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  • ...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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  • ...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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  • ...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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  • 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
  • ...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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  • ...<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
  • 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
  • ...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
  • ...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
  • ...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
  • 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
  • ...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
  • ...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
  • ...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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