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• Math books
  ...https://artofproblemsolving.com/community/user/243060 cargeek9], currently a junior in high school. It covers the basics of algebra, geometry, combinato ...3265/ref=sr_1_1?ie=UTF8&s=books&qid=1204029652&sr=1-1 Complex Numbers from A to... Z] by [[Titu Andreescu]]
  24 KB (3,177 words) - 12:53, 20 February 2024
• Mathematics
  ..., and the formation of turbulence around an aircraft. Mathematics provides a kind of "quality control" for the development of trustworthy theories and e Modern mathematics is built around a system of [[axiom|axioms]], which is a name given to "the rules of the game." Mathematicians then use various meth
  6 KB (902 words) - 12:53, 3 September 2019
• Inequality
  ...nd relate slightly to [[number theory]]. They deal with [[relations]] of [[variable]]s denoted by four signs: <math>>,<,\ge,\le</math>. For two [[number]]s <math>a</math> and <math>b</math>:
  12 KB (1,798 words) - 16:20, 14 March 2023
• 1955 AHSME Problems/Problem 38
  <math>\textbf{(A)}\ 19 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ ...ns, the system of equation should be constructed. (It doesn't matter which variable is which.)
  1 KB (200 words) - 23:35, 28 August 2020
• Power Mean Inequality
  The '''Power Mean Inequality''' is a generalized form of the multi-variable [[Arithmetic Mean-Geometric Mean]] Inequality. ...th> is flipped, but as <math>k_2<0</math>, <math>x^\frac{1}{k_2}</math> is a decreasing function, the inequality sign is flipped again after applying <m
  3 KB (606 words) - 23:59, 1 July 2022
• Trivial Inequality
  ...and minimizing [[quadratic]] functions. It gives an easy proof of the two-variable case of the [[AMGM | Arithmetic Mean-Geometric Mean]] inequality: <cmath>ax^2+bx+c = a(x^2+\frac{b}{a}x+\frac{b^2}{4a^2})+c-\frac{b^2}{4a} = a(x+\frac{b}{2a})^2+c-\frac{b^2}{4a}.</cmath>
  3 KB (560 words) - 22:51, 13 January 2024
• Polynomial
  ...algebra]] often an arbitrary [[field]]). Note that a [[constant]] is also a polynomial. * <math>4x^2 + 6x - 9</math>, in the variable <math>x</math>
  6 KB (1,100 words) - 01:44, 17 January 2024
• Quadratic equation
  ...s, they do not depend on <math>x</math>) and <math>x</math> is the unknown variable. Quadratic equations are solved using one of three main strategies: [[fact The purpose of factoring is to turn a general quadratic into a product of [[binomial]]s. This is easier to illustrate than to describe.
  2 KB (264 words) - 12:04, 15 July 2021
• Trigonometric substitution
  '''Trigonometric substitution''' is the technique of replacing [[variable]]s in equations with <math>\sin \theta\,</math> or <math>\cos {\theta}\,</m ...n]]s such as <math>\sqrt{a^2+x^2},\sqrt{a^2-x^2}</math> or <math>\sqrt{x^2-a^2}</math>
  1 KB (173 words) - 18:42, 30 May 2021
• 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 ...o add to the confusion, "[[algebra (structure) |algebra]]" is the name for a certain kind of structure in modern algebra.
  3 KB (369 words) - 21:18, 18 June 2021
• Number theory/Advanced
  An elliptic curve is the set of points <math>(x,y)</math> satisfying some two variable third degree equation. Using certain affine transformations it can be shown ...in some subfield (like the reals or the rationals). One also needs to add a limit point, called the point at infinity. As <math>x\to \infty</math>, the
  5 KB (849 words) - 16:14, 18 May 2021
• Jensen's Inequality
  Let <math>{F}</math> be a [[convex function]] of one real variable. Let <math>x_1,\dots,x_n\in\mathbb R</math> and let <math>a_1,\dots, a_n\ge If <math>{F}</math> is a concave function, we have:
  3 KB (623 words) - 13:10, 20 February 2024
• Function
  A '''function''' is a rule that maps one set of values to another set of values, assigning to eac ...is a ''function from <math>A</math> to <math>B</math>'' (written <math>f: A \to B</math>) if and only if
  10 KB (1,761 words) - 03:16, 12 May 2023
• Chain Rule
  Here are some more precise statements for the single-variable and multi-variable cases. ===Single variable Chain Rule===
  12 KB (2,377 words) - 11:48, 22 July 2009
• Complex number
  ...n has no solutions in the real numbers. However, it is possible to define a number, <math> i </math>, such that <math> i = \sqrt{-1} </math>. If we ad ...s the set <math>\mathbb{R}</math> of the [[real number]]s, since <math>a = a + 0i</math>.
  5 KB (860 words) - 15:36, 10 December 2023
• 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
• Limit
  ...of mathematics. It took several centuries to articulate the definition of a limit and to make it rigorous. ...ue to which a function grows close when its argument is near (but not at!) a particular value. For example,
  7 KB (1,325 words) - 13:51, 1 June 2015
• 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
• 2004 AMC 12A Problems
  {{AMC12 Problems|year=2004|ab=A}} <math>\text{(A) } 0.0029 \qquad \text{(B) } 0.029 \qquad \text{(C) } 0.29 \qquad \text{(D)
  13 KB (1,953 words) - 00:31, 26 January 2023
• 2004 AIME II Problems/Problem 6
  ...es the rest equally between the other two. Given that each monkey receives a [[whole number]] of bananas whenever the bananas are divided, and the numbe ...> is divisible by <math>72</math> (however, since the denominator contains a <math>27</math>, the factors of <math>3</math> cancel, and it only really n
  6 KB (950 words) - 14:18, 15 January 2024
• 1984 AIME Problems/Problem 4
  Let <math>S</math> be a list of positive integers--not necessarily distinct--in which the number <m == Solution 2 (One Variable) ==
  2 KB (319 words) - 03:38, 16 January 2023
• 1985 AIME Problems/Problem 12
  ...= \frac{n}{729}</math> be the probability that the bug is at vertex <math>A</math> when it has crawled exactly <math>7</math> meters. Find the value of == Solution 1 (Single Variable Recursion) ==
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1986 AIME Problems/Problem 9
  pair C=(0,0),A=(510,0),B=IP(circle(C,450),circle(A,425)); pair Da=IP(Circle(A,289),A--B),E=IP(Circle(C,324),B--C),Ea=IP(Circle(B,270),B--C);
  11 KB (1,850 words) - 18:07, 11 October 2023
• 1988 AIME Problems/Problem 6
  ...od will be to use the given numbers to set up equations to solve for <math>a</math> and <math>b</math>, and then calculate <math>(*)</math>. \hline a & & 103 & & \\
  5 KB (878 words) - 23:06, 20 November 2023
• 1995 AIME Problems/Problem 7
  ...+ \cos t) = \frac{3}{2}</math>. Completing the square on the left in the variable <math>(\sin t + \cos t)</math> gives <math>\sin t + \cos t = -1 \pm \sqrt{\ {{AIME box|year=1995|num-b=6|num-a=8}}
  3 KB (427 words) - 09:23, 13 December 2023
• 1997 AIME Problems/Problem 4
  NOTE: It can be seen that there is no apparent need to use the variable x as a 5,12,13 right triangle has been formed. ...2+k_2k_3+k_3k_1}</math> where each of <math>k_i</math> is the curvature of a circle with radius <math>r_i</math>, and the curvature is defined as <math>
  2 KB (354 words) - 22:33, 2 February 2021
• 2003 AIME II Problems/Problem 13
  ...ne of the two vertices where it is not currently located, and crawls along a side of the triangle to that vertex. Given that the probability that the bu ...o return to the original vertex, the net number of clockwise steps must be a multiple of 3, i.e., <math>\#CW - \#CCW \equiv 0 \pmod{3}</math>. Since <m
  15 KB (2,406 words) - 23:56, 23 November 2023
• Equation
  An '''equation''' is a [[relation]] which states that two [[expression]]s are equal, identical, or A unique aspect to equations is the ability to modify an original equation by
  5 KB (932 words) - 12:57, 26 July 2023
• Identity
  ...x^2+2x+1</math> is an identity, since it holds regardless of the choice of variable. Therefore, it is sometimes written <math>(x+1)^2\equiv x^2+2x+1</math>. ...</math>. If we instead took <math>G</math> to be addition (<math>G(a, b) = a + b</math>), <math>0</math> would be the identity.
  1 KB (238 words) - 13:38, 14 July 2021
• Mock AIME 1 2006-2007 Problems/Problem 9
  Let <math>a_{n}</math> be a [[geometric sequence]] of [[complex number]]s with <math>a_{0}=1024</math> Let <math>a_{n}</math> be a geometric sequence for <math>n\in\mathbb{Z}</math> with <math>a_{0}=1024</m
  5 KB (744 words) - 19:46, 20 October 2020
• Mock AIME 1 2006-2007 Problems/Problem 14
  ...ath>n</math> are positive integers. Construct circle <math>O</math> with a variable radius that is tangent to <math>AB</math> at <math>T</math>. Let <math>P</ ...ee below an attempted solution to understand why this problem doesn't have a solution:
  3 KB (541 words) - 17:32, 22 November 2023
• Mock AIME 1 2006-2007/Problems
  Let <math>\star (x)</math> be the sum of the digits of a positive integer <math>x</math>. <math>\mathcal{S}</math> is the set of pos ...\sqrt{7}</math>, <math>CA=1</math>, and <math>AB=3</math>. If <math>\angle A=\frac{\pi}{n}</math> where <math>n</math> is an integer, find the remainder
  8 KB (1,355 words) - 14:54, 21 August 2020
• Expected value
  Given an event with a variety of different possible outcomes, the '''expected value''' is what on A video that goes over the type of Expected value, practical examples, and pr
  5 KB (789 words) - 20:56, 10 May 2024
• 2006 Romanian NMO Problems
  ...<math>2 \cdot \frac{CN}{BC} = \frac{AM}{AB}</math>. Let <math>P</math> be a point on the line <math>AC</math>. Prove that the lines <math>MN</math> and ...nimal <math>n</math>, such that for each coloring, there exists a line and a column with at least 3 unit squares of the same color (on the same line or
  11 KB (1,779 words) - 14:57, 7 May 2012
• Derivative
  ...ith respect to one of the [[variable]]s. Note that not every function has a derivative.
  509 bytes (64 words) - 19:26, 3 March 2010
• Navier-Stokes Equation
  ...ate how sticky (viscous) a fluid is. Thus, the Navier-Stokes equations are a dynamical statement of the balance of forces acting at any given region of ...odel weather, ocean currents, water flow in a pipe, motion of stars inside a galaxy, and flow around an airfoil (wing). They are also used in the design
  3 KB (553 words) - 22:08, 2 May 2022
• System of equations
  ...[[equation]]s which share the same [[variable]]s. Below is an example of a system of equations. <cmath>\left\{\begin{array}{l}a^2 + b^2 = 25\\a+b=7\end{array}\right.</cmath>
  5 KB (784 words) - 23:27, 30 July 2020
• Linear congruence
  A '''Linear Congruence''' is a [[congruence]] [[modular arithmetic|mod]] p of the form ...c</math>, and <math>p</math> are [[constant]]s and <math>x</math> is the [[variable]] to be solved for.
  1 KB (225 words) - 15:52, 3 August 2022
• Asymptote: Basics
  ...nded with the "[/asy]" tag. Each command in Asymptote must be separated by a semicolon (;), similar to programming languages like C and Java. This conve To write comments (Lines that are not interpreted by Asymptote), start a line with two forward slashes.
  5 KB (801 words) - 20:44, 6 May 2024
• Coefficient
  ...]] of a specified object. The object can be a [[variable]], a [[vector]], a [[function]], or anything else that might be subject to multiplication. ...ficients of: 5 also has a coefficient in <math>5a + b</math>, namely <math>a</math>.
  1 KB (227 words) - 22:38, 6 October 2020
• Expression
  ...ally consist of [[number]]s, [[variable]]s, and [[operator]]s, arranged in a sensible way. Thus, <math>3 - \frac x4</math> is an arithmetic expression, ...t an arithmetic expression, this equivalence of logical expressions is not a logical expression. This is amazing. Again please help us by expanding it
  2 KB (239 words) - 14:42, 11 March 2023
• ELMO
  ...OP]]. Its initials have stood for many things over time, originating from a backronym originally called the "Experimental Lincoln Math Olympiad". The b ..., with an additional 'S' "*, having the same problems as the ELMO but with a new appearance to make it hard to read.
  5 KB (773 words) - 19:16, 17 June 2022
• Asymptote: Macros and Packages
  ...+r,b+r)</math>. In addition, you want it to simply return the pair <math>(a,b)</math> if no value of <math>r</math> is specified, so you want <math>r</ real a,b;
  3 KB (515 words) - 00:43, 24 April 2019
• Periodic function
  ...] <math>f</math> is '''periodic''' if for all <math>x</math>, there exists a <math>p</math> such that <math>f(x + p) = f(x)</math>. The smallest positiv
  515 bytes (80 words) - 15:47, 21 August 2011
• 1960 IMO Problems
  For what values of the variable <math>x</math> does the following inequality hold: ...nteger). Let <math>\alpha</math> be the acute angle subtending, from <math>A</math>, that segment which contains the midpoint of the hypotenuse. Let <ma
  3 KB (511 words) - 21:21, 20 August 2020
• 2007 AIME I Problems/Problem 8
  We can see that <math>Q_1</math> and <math>Q_2</math> must have a [[root]] in common for them to both be [[factor]]s of the same cubic. Let this root be <math>a</math>.
  4 KB (728 words) - 00:11, 29 November 2023
• 2007 AIME I Problems/Problem 14
  A [[sequence]] is defined over [[non-negative]] integral indexes in the follo Define a function <math>f(n)</math> on the non-negative integers, as <cmath>f(n) = \
  13 KB (2,185 words) - 23:30, 5 December 2022
• 1989 USAMO Problems
  ...tegers <math>0 < a,\ b,\ c,\ d < 1000000</math> such that <math>T_{1988} = a S_{1989} - b</math> and <math>U_{1988} = c S_{1989} - d</math>. ...laying in at least one game. Prove that within this schedule there must be a set of 6 games with 12 distinct players.
  2 KB (326 words) - 18:52, 18 July 2016
• LaTeX:Commands
  | <math>a^{i+1}_3</math>||a^{i+1}_3||<math>x^{3^2}</math>||x^{3^2} Notice that we can apply both a subscript and a superscript at the same time. For subscripts or superscripts with more than
  12 KB (1,898 words) - 15:31, 22 February 2024
• Imaginary number
  An '''imaginary number''' is a [[complex number]] whose [[real part]] is equal to 0. In the [[complex pla ...math>, which is <math>\sqrt{-1}</math>. <math>i</math> can be treated as a variable with the distinctive property that <math>i^2= -1</math>. Powers of <math>i<
  2 KB (283 words) - 00:18, 8 March 2012

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