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• Math books
  * Intermediate is recommended for students who can expect to pass the AMC 10/12. Before adding any books to this page, please review the [[AoPSWiki:Linking books]] page.
  24 KB (3,177 words) - 12:53, 20 February 2024
• 2005 Alabama ARML TST Problems/Problem 7
  \sum_{n=1}^{\infty} \left(\frac{n^2}{2^n}\right)=z\\ z=1+\frac{3}{2}+\frac{5}{4}+\frac{7}{8}+\frac{9}{16}+\cdots\\
  1 KB (193 words) - 21:13, 18 May 2021
• Mathematics
  ...the development of trustworthy theories and equations which are important to people in most modern technical discplines such as engineering and economic ...aticians then use various methods of formal [[proof]] to extend the axioms to come up with surprising and elegant results. Such methods include [[inducti
  6 KB (902 words) - 12:53, 3 September 2019
• Trivial Inequality
  ...ath>. Since both sides of the inequality are nonnegative, it is equivalent to <math>x+y \ge 2\sqrt{xy}</math>, and thus we have <cmath> \frac{x+y}{2} \ge Another application will be to minimize/maximize quadratics. For example,
  3 KB (560 words) - 22:51, 13 January 2024
• Factoring
  ...or [[polynomial]]) as a product of different terms. This often allows one to find information about an expression that was not otherwise obvious. Using the formula for the sum of a [[geometric sequence]], it's easy to derive the general formula for difference of powers:
  3 KB (532 words) - 22:00, 13 January 2024
• Relatively prime
  Two [[positive]] [[integer]]s <math>m</math> and <math>n</math> are said to be '''relatively prime''' or '''coprime''' if they share no [[common diviso ...ive integers less than any given positive integer that is relatively prime to that number.
  2 KB (245 words) - 15:51, 25 February 2020
• Discriminant
  ...has two [[nonreal]] roots; and if the discriminant is 0, the equation has a real [[double root]]. ...being real. But for polynomials of degree 4 or higher it can be difficult to use it.
  4 KB (734 words) - 19:19, 10 October 2023
• Menelaus' theorem
  ...earity]] of points on each of the three sides (extended when necessary) of a [[triangle]]. pair A=(7,6), B=(0,0), C=(10,0), P=(4,0), Q=(6,8), R;
  5 KB (804 words) - 03:01, 12 June 2023
• Fermat's Little Theorem
  ...a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,658 words) - 16:02, 8 May 2024
• Euler's Totient Theorem
  '''Euler's Totient Theorem''' is a theorem closely related to his [[totient function]]. ...is a positive integer [[relatively prime]] to <math>a</math>, then <math>{a}^{\phi (m)}\equiv 1 \pmod {m}</math>.
  3 KB (542 words) - 17:45, 21 March 2023
• Elementary symmetric sum
  An '''elementary symmetric sum''' is a type of [[summation]]. ...>). 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
• Algebra
  ...the [[variable]], an unknown quantity given a name and usually denoted by a letter or symbol. Many contest problems test one's fluency with [[algebraic Algebra can be used to solve different types of equations, but algebra is also many other things
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  ...ry]] and [[Galois theory]]. Algebraic methods are particularly well-suited to studying properties of individual prime numbers. From an algebraic perspect ...e techniques of [[analysis]] and [[calculus]] are particularly well-suited to studying large-scale properties of prime numbers. The most famous problem i
  5 KB (849 words) - 16:14, 18 May 2021
• Real number
  A '''real number''' is a number that falls on the real number line. It can have any value. Some exam ...tion <math>\mathbb{Z}^{+}</math> means positive integers and <math>\mathbb{Z}^{-}</math> means negative integers. The real numbers can also be divided b
  3 KB (496 words) - 23:22, 5 January 2022
• Euclidean algorithm
  ...the most common of which is the [[nonnegative]] [[integer]]s <math>\mathbb{Z}{\geq 0}</math>, without [[factoring]] them. ...c idea is to repeatedly use the fact that <math>\gcd({a,b}) \equiv \gcd({b,a - b})</math>
  6 KB (924 words) - 21:50, 8 May 2022
• Integer
  ...th>a\div b</math> may or may not be an integer, depending on whether <math>a</math> can be divided by <math>b</math> with no remainder. ...ward definition, an integer is a number that is '''not''' a [[decimal]] or a [[fraction]].
  2 KB (296 words) - 15:04, 5 August 2022
• Natural number
  ...non-negative integers, and <math>\mathbb{Z}_{>0}</math> and <math>\mathbb{Z}^+</math> for the set of positive integers.
  1 KB (162 words) - 21:44, 13 March 2022
• Pi
  ...r]]) of a given [[circle]] to its [[diameter]]. It is approximately equal to 3.141592653. The number pi is one of the most important [[constant]]s in a ...proximates <math>\frac{\pi}{4}</math>. This can simply be multiplied by 4 to approximate <math>\pi</math>.
  8 KB (1,469 words) - 21:11, 16 September 2022
• Injection
  ...t takes distinct values on distinct inputs. Equivalently, an injection is a function for which every value in the [[range]] is the image of exactly one ...: A function <math>f:A\to B</math> is an injection if for all <math>x,y\in A</math>, if <math>f(x)=f(y)</math> then <math>x=y</math>.
  1 KB (228 words) - 01:01, 17 November 2019
• 2011 AMC 10B Problems/Problem 22
  ...ength <math>1</math> and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the py <math> \textbf{(A)}\ 5\sqrt{2} - 7 \qquad\textbf{(B)}\ 7 - 4\sqrt{3} \qquad\textbf{(C)}\ \fra
  4 KB (691 words) - 18:38, 19 September 2021

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