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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
• Vector
  ...y, but also most abstractly, a vector is any object which is an element of a given vector space. ...(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
• Complex plane
  ...point two units to the right and one unit down from the origin corresponds to the complex number <math>2 - i</math>.
  892 bytes (134 words) - 16:52, 3 September 2017
• Complex number
  The '''complex numbers''' arise when we try to solve [[equation]]s such as <math> x^2 = -1 </math>. ...solutions of <math> x^2 = -1 </math> but we can now find ''all'' solutions to ''every'' polynomial. (See the [[Fundamental Theorem of Algebra]] for more
  5 KB (860 words) - 15:36, 10 December 2023
• Cis
  ...plex number]]s. For all complex numbers <math>z</math>, we can write <math>z=r\mathrm{cis }(\theta)=r\cos \theta + ir\sin \theta</math>. Notice that <ma Once one gets used to the notation, it is almost always preferred to write <math>re^{i\theta}</math> rather than <math>r\mathrm{cis }(\theta)</m
  1 KB (171 words) - 20:59, 11 July 2023
• Polar form
  ...ows us to graph complex numbers given an [[angle]] <math>\theta</math> and a [[radius]] or [[magnitude]] <math>r</math>. ...u do not understand this notation.) This represents a complex number <math>z</math> that is <math>r</math> units away from the origin, and <math>\theta<
  633 bytes (105 words) - 13:35, 1 April 2022
• Roots of unity
  The '''Roots of unity''' are a topic closely related to [[trigonometry]]. Roots of unity come up when we examine the [[complex numb ...making <math>r^n=1\Rightarrow r=1</math> (magnitude is always expressed as a positive real number). This leaves us with <math>e^{ni\theta} = e^{2\pi ik
  3 KB (558 words) - 21:36, 11 December 2011
• Angle bisector
  ...at the angle between this line and <math>\overline{AB}</math> is congruent to the angle between this line and <math>\overline{AC}</math>: pair A,B,C,D,E,F;
  3 KB (575 words) - 15:27, 19 March 2023
• Fermat's Last Theorem
  ...> with <math>n \geq 3</math>, there are no solutions to the equation <math>a^n + b^n = c^n</math>. ...marvelous demonstration of this proposition that this margin is too narrow to contain.''"
  3 KB (453 words) - 11:13, 9 June 2023
• 2005 USAMO Problems/Problem 4
  ...math> from the end of leg <math>L_i \; (i = 1,2,3,4)</math> and still have a stable table? ...can be placed so that 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
• Diophantine equation
  A '''Diophantine equation''' is an [[equation]] relating [[integer]] (or some ...antine equation has infinitely many solutions, [[parametric form]] is used to express the relation between the variables of the equation.
  9 KB (1,434 words) - 13:10, 20 February 2024
• Area
  ...]al figure occupies. The size of a region in higher dimensions is referred to as [[volume]]. It is often possible to find the area of a region bounded by parts of [[circle]]s and [[line segment]]s through elemen
  6 KB (1,181 words) - 22:37, 22 January 2023
• Exponential form
  ...> as <math>z=re^{i\theta}</math>, which is the general exponential form of a complex number. So <math>z=re^{i\theta}</math> looks like:
  1 KB (238 words) - 22:51, 20 February 2022
• Floor function
  ...]</math>. The action of this function is the same as "rounding down." On a [[positive]] argument, this function is the same as "dropping everything af ...+b\rfloor\ge \lfloor a\rfloor+\lfloor b \rfloor</math> for all real <math>(a,b)</math>.
  3 KB (508 words) - 21:05, 26 February 2024
• Rational approximation
  ...and larger <math>p</math> and <math>q</math>. So, the reasonable question to ask here is how well can we approximate <math>x</math> by rationals with no ...th> can be approximated by a rational number <math>\frac{p}{q}</math> with a given denominator <math>q\ge 1</math> with an error not exceeding <math>\fr
  7 KB (1,290 words) - 12:18, 30 May 2019
• Complex analysis
  ...ctions of a real variable that cannot occur in complex variables. Here are a few spectacular results in complex analysis. ...ly connected]] [[domain]] ''D'', and let <math>\Gamma\subseteq D</math> be a [[simple closed Jordan curve]]. Then for any <math>z_0</math> in the interi
  2 KB (271 words) - 22:06, 12 April 2022
• Fundamental Theorem of Algebra
  ...tant]] [[polynomial]] with [[complex number|complex]] [[coefficient]]s has a complex [[root]]. In fact, every known proof of this theorem involves some ...is [[monic]]. Then <math>1/P(z)</math> is an [[entire]] function; we wish to show that it is bounded. It is clearly bounded when <math>n=0</math>; we n
  5 KB (832 words) - 14:22, 11 January 2024
• Surjection
  ..., the range is a subset of the codomain.) In adjectival form, we say that a function is ''surjective'' or ''onto''. ...defined by <math>f(x) = x+1</math> is not surjective because there exists a [[natural number]] which is not one more than any other natural number.
  794 bytes (131 words) - 22:39, 13 May 2020
• Prime Number Theorem
  of primes less than or equal to <math>x</math>. In other words, it states <cmath> \lim_{x\to \infty} \frac{\pi(x) \log x}{x} = 1 . </cmath>
  10 KB (1,729 words) - 19:52, 21 October 2023
• Infinite
  ...h>f:S\to\mathbb{Z}</math>. If this is not the case, <math>S</math> is said to be [[finite]]. In simplified language, a set is infinite if it doesn't end, i.e. you can always find another element
  1 KB (186 words) - 23:19, 16 August 2013
• Mock AIME
  A '''Mock AIME''' is a contest that is intended to mimic the [[AIME]] competition. (In more recent years, recurring competitio ** [https://artofproblemsolving.com/community/c5h2987334 Or(z)IME]
  8 KB (933 words) - 22:41, 23 May 2024
• Commutative property
  ...a [[binary operation]]) is said to have the '''commutative property''' or to ''be commutative'' if the order of its arguments does not affect the value. ...eal number]]s, [[integer]]s, etc.) because <math>\displaystyle a + b = b + a</math>. However, the operation of [[division]] is not commutative over the
  2 KB (301 words) - 17:46, 16 March 2012
• Algebraic topology
  ...of [[topology]] using methods from [[abstract algebra]]. In general, given a [[topological space]], we can associate various algebraic objects, such as ...a,1)=q(a)</math>, and <math>g(0,b)=g(1,b)=x</math>. We call <math>g</math> a [[homotopy]]. Now define <math>\pi_1(X)=L/\sim</math>. That is, we equate a
  3 KB (479 words) - 15:35, 1 December 2015
• Modular arithmetic/Intermediate
  ...or <math>a \equiv b</math> (mod <math>n</math>), if the difference <math>{a - b}</math> is divisible by <math>n</math>. ...ing a wide range of number-theoretic problems, including finding solutions to [[Diophantine equation|Diophantine equations]], testing whether certain lar
  14 KB (2,317 words) - 19:01, 29 October 2021
• Subset
  ...r than <math>B</math> itself. In the latter case, <math>A</math> is called a ''proper subset''. The following is a true statement:
  1 KB (217 words) - 09:32, 13 August 2011
• 2006 AIME I Problems/Problem 5
  ...n be written as <math> a\sqrt{2}+b\sqrt{3}+c\sqrt{5}, </math> where <math> a, b, </math> and <math> c </math> are [[positive]] [[integer]]s. Find <math> <cmath> a\sqrt{2}+b\sqrt{3}+c\sqrt{5} = \sqrt{104\sqrt{6}+468\sqrt{10}+144\sqrt{15}+2
  3 KB (439 words) - 18:24, 10 March 2015
• 2021 JMC 10
  <math>\textbf{(A) } 3 \qquad\textbf{(B) } 7 \qquad\textbf{(C) } 8 \qquad\textbf{(D) } 9 \qqu <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12A Problems
  {{AMC12 Problems|year=2006|ab=A}} ...and sodas cost <math>2</math> dollars each. How many dollars will it cost to purchase <math>5</math> sandwiches and <math>8</math> sodas?
  15 KB (2,223 words) - 13:43, 28 December 2020
• 2005 AMC 12A Problems
  {{AMC12 Problems|year=2005|ab=A}} (\mathrm {A}) \ 1 \qquad (\mathrm {B}) \ 2 \qquad (\mathrm {C})\ 5 \qquad (\mathrm {D})
  13 KB (1,971 words) - 13:03, 19 February 2020
• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} ...ed to pay local taxes. How many cents per hour of Alicia's wages are used to pay local taxes?
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2000 AMC 12 Problems
  <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\ <math>\textbf{(A)}\ 2000^{2001} \qquad \textbf{(B)}\ 4000^{2000} \qquad \textbf{(C)}\ 2000^{
  13 KB (1,948 words) - 12:26, 1 April 2022

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