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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]]
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• 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
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• 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.)
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• 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
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• 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>
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• 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.
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• 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>
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• 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
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• 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
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• 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)
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• 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
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• 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
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• 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.
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• 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.
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• 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.
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• 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
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• 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
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• 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>.
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• 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.
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• 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<
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• Zermelo-Fraenkel Axioms
  ...ven property describable by the language of [[set theory]] could be called a [[set]]. As shown by paradoxes such as [[Russell's Paradox]], some restrict The language of set theory consists of a single binary relation <math>\in</math>. As such, all axioms can be written
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• Implicit differentiation
  ...e [[dependent variable]] is treated as a [[function]] of the [[independent variable]] and is differentiated with the [[chain rule]].
  757 bytes (125 words) - 01:04, 26 March 2018
• 2007 AMC 12A Problems/Problem 9
  <math>\mathrm{(A)}\ \frac 23\qquad \mathrm{(B)}\ \frac 34\qquad \mathrm{(C)}\ \frac 45\qquad ...r}</math>. If Yan walks directly to the stadium, then assuming he walks at a rate of <math>1</math>, it will take him <math>x-r</math> units of time. Si
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• Linear equation
  ...]] of the first [[degree]] - i.e. each term does not have any variables to a power other than one. ...onstants, <math>b_i</math> is a series of variables, and <math>c</math> is a [[constant]].
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• 2006 AIME II Problems/Problem 6
  ...d <math> b</math> is not divisible by the square of any prime. Find <math> a+b+c. </math> pair B = (0, 0), C = (1, 0), D = (1, 1), A = (0, 1);
  7 KB (1,067 words) - 12:23, 8 April 2024
• 1989 USAMO Problems/Problem 3
  ...</math> and <math>b</math> such that <math>P(a + bi) = 0</math> and <math>(a^2 + b^2 + 1)^2 < 4 b^2 + 1</math>. ...>\overline{w}</math>, the [[complex conjugate]] of <math>w</math>, is also a root of <math>P</math>.
  2 KB (340 words) - 19:11, 18 July 2016
• 1960 IMO Problems/Problem 2
  For what values of the variable <math>x</math> does the following inequality hold: Set <math>x = -\frac{1}{2} + \frac{a^2}{2}</math>, where <math>a\ge0</math>.
  2 KB (239 words) - 05:51, 7 April 2024
• Excircle
  ...is a [[circle]] [[Tangent line|tangent]] to the extensions of two sides of a [[triangle]] and the third side. pair excenter(pair A, pair B, pair C){
  5 KB (843 words) - 03:02, 1 July 2020
• 2004 AMC 12A Problems/Problem 8
  //Variable Declarations pair A, B, C, D, E;
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• Algebraic manipulation
  ...doing opposite operations (undoing) to any equation to solve for a certain variable (often by isolation). Cross multiplication is a common method of solving [[proportions]]. Essentially, one is multiplying
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• 2002 AMC 12B Problems/Problem 20
  Let <math>\triangle XOY</math> be a [[right triangle|right-angled triangle]] with <math>m\angle XOY = 90^{\circ <math>\mathrm{(A)}\ 24
  3 KB (447 words) - 15:02, 17 August 2023
• 2008 AMC 12B Problems/Problem 21
  ...)</math> to <math>(2,1)</math>. What is the probability that circles <math>A</math> and <math>B</math> intersect? <math>\textbf{(A)} \; \frac {2 + \sqrt {2}}{4} \qquad \textbf{(B)} \; \frac {3\sqrt {3} + 2}
  6 KB (1,008 words) - 11:46, 24 December 2020
• 2008 AIME II Problems/Problem 12
  The well known problem of ordering <math>x</math> elements of a string of <math>y</math> elements such that none of the <math>x</math> elem ...uch that none of them are next to each other. We would also like to choose a place where we can divide this string into two strings such that the left o
  10 KB (1,550 words) - 12:58, 15 July 2023
• 2008 USAMO Problems/Problem 3
  (''Gabriel Carroll'') Let <math>n</math> be a positive integer. Denote by <math>S_n</math> the set of points <math>(x, y) .../math> paths (a partition of <math>S_n</math> into <math>m</math> paths is a set <math>\mathcal{P}</math> of <math>m</math> nonempty paths such that eac
  9 KB (1,585 words) - 01:00, 14 August 2014
• Cauchy Induction
  '''Cauchy Induction''' is a beautiful method of "Proof by [[Induction]]" discovered by [[Augustin Louis For a given statement <math>s</math> over the positive integers greater than or e
  3 KB (658 words) - 16:13, 11 May 2019
• KGS math club
  A group of people on [http://www.gokgs.com Kiseido Go Server] Mathematics roo ...our, or of a less-than-or-equal rank. What is the greatest possible length a parade can reach?
  22 KB (3,358 words) - 15:17, 18 July 2017
• 1988 IMO Problems/Problem 1
  ...r <math>O</math>. Fix <math>P</math> on the small circle and consider the variable chord <math>AP</math> of the small circle. Points <math>B</math> and <math> ...U</math> of <math>AB</math> and <math>V</math> of <math>AC</math> as <math>A</math> varies?</li>
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• 1990 AHSME Problems/Problem 26
  Ten people form a circle. Each picks a number and tells it to the two neighbors adjacent to them in the circle. T <math>\textbf{(A) } 1 \qquad
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• Float
  The '''float''' [[datatype]] is a type of computing [[variable]]. It has three parts: the [[mantissa]], the [[exponent]], and the [[sign]] *Float has a significantly larger range of numbers than fixed. For example, 123450000000
  953 bytes (136 words) - 16:08, 10 March 2011
• 1975 USAMO Problems/Problem 2
  ...denote four points in space and <math>AB</math> the distance between <math>A</math> and <math>B</math>, and so on. Show that pair A=(2,4), B=(0,0), C=(4,0), D=(4,2);
  6 KB (1,096 words) - 23:07, 26 August 2017
• 1976 USAMO Problems/Problem 2
  ...>AX</math> and <math>BY</math>. You may assume that <math>AB</math> is not a diameter. pair A=r*expi(5*pi/6), B=r*expi(pi/6), X=r*expi(pi/3), X1=r*expi(-pi/12), Y=r*expi
  5 KB (848 words) - 23:41, 6 July 2020
• 1976 USAMO Problems
  ...or white. Prove that with <i>any</i> such coloring, the board must contain a rectangle (formed by the horizontal and vertical lines of the board such as * (b) Exhibit a black-white coloring of a <math>4\times 6</math> board in which the four corner squares of every rect
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• 2008 AMC 12B Problems/Problem 20
  ...Michael stops for <math>30</math> seconds at each pail. As Michael passes a pail, he notices the truck ahead of him just leaving the next pail. How man <math>\mathrm{(A)}\ 4\qquad\mathrm{(B)}\ 5\qquad\mathrm{(C)}\ 6\qquad\mathrm{(D)}\ 7\qquad\m
  8 KB (1,241 words) - 17:55, 11 August 2023
• 2002 AMC 10B Problems/Problem 13
  <math>\textbf{(A) } \frac23 \qquad \textbf{(B) } \frac32 \text{ or } -\frac14 \qquad \textbf ...olutions for <math>y</math> and there is no <math>x</math>, we simply find a value of <math>x</math> such that the equation is symmetrical. Therefore,
  1 KB (218 words) - 13:12, 30 November 2019
• 2009 AMC 12B Problems/Problem 20
  ...inside or on <math>Q</math>. The cuts produce <math>n</math> pyramids and a new polyhedron <math>R</math>. How many edges does <math>R</math> have? <math>\mathrm{(A)}\ 200\qquad
  9 KB (1,567 words) - 13:43, 19 August 2023
• Polynomial ring
  ...ring''' <math>R[x]</math> is, informally, "the ring of all polynomials in a [[commutative]] <math>x</math> with coefficients in <math>R</math>." That where <math>N</math> is a [[nonnegative integer]] that varies from sum to sum.
  12 KB (2,010 words) - 00:10, 3 August 2020
• Asymptotic equivalence
  '''Asymptotic equivalence''' is a notion of [[function]]s "eventually" becoming "essentially equal". ...<math>f</math> and <math>g</math> be functions of a [[real number | real]] variable. We say that <math>f</math> and <math>g</math> are '''asymptotically equiv
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• C++
  ...The # include statement includes a file and its functions. Do not include a file if it has the main function. Most include files take the extension ".h
  1,013 bytes (175 words) - 01:45, 28 May 2012
• 2009 AIME II Problems/Problem 9
  For a fixed <math>y'</math>, <math>x</math> can be at most <math>\left\lfloor \df If <math>x=1</math>, we get the equation <math>3y+2z=2005</math>. The variable <math>y</math> must be odd, and it must be between <math>1</math> and <math
  5 KB (845 words) - 15:45, 28 December 2020
• Pascal (language)
  ...ramming. Pascal, which was named after the mathematician Blaise Pascal, is a direct descendent from ALGOL 60, which Wirth helped develop. Pascal also dr Pascal contains some significant language features that allow it to used as a powerful learning tool in introducing structured programming techniques to
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• Substitution
  ...places all instances of a variable (or expression) with another equivalent variable (or expression). ...s. It is generally introduced in a first year high school algebra class. A solution generally exists when the number of equations is exactly equal to
  3 KB (428 words) - 16:53, 10 April 2020
• 2010 AMC 12A Problems/Problem 13
  <math>\textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 2 \qquad \textbf{(D)}\ 4 \ ...which is clearly a circle with radius <math>k</math>, and the red curve is a part of the curve <math>xy=k</math>.
  9 KB (1,622 words) - 20:53, 11 September 2023
• Getting Started With Python Programming
  ...ing language. The only language that AoPS teaches (as of May 16, 2021) in a class is Python. ...''Intermediate Programming with Python'' students should install Python on a regular computer.
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• Python reference
  This page is intended to serve as a brief, beginners reference to the [[Python]] programming language. In Python, there is no need to specify a type when storing information in a variable. It is basic and straightforward to store information.
  28 KB (4,762 words) - 21:20, 12 June 2023
• 1969 IMO Problems
  ...ers <math>a</math> with the following property: the number <math>z = n^4 + a</math> is not prime for any natural number <math>n</math>. ...math>a_1, a_2, \cdots, a_n</math> be real constants, <math>x</math> a real variable, and
  3 KB (428 words) - 13:34, 29 January 2021
• 1973 IMO Problems
  ...nit vectors such that points <math>P_1, P_2, \ldots, P_n</math> all lie in a plane containing <math>g</math> and on one side of <math>g</math>. Prove th ...s in space not lying in the same plane such that, for any two points <math>A</math> and <math>B</math> of <math>M</math>, one can select two other point
  3 KB (602 words) - 15:44, 29 January 2021
• Irreducible polynomial
  It's a nonconstant polynomial (meaning it has a variable) that can't be factored into at least 2 nonconstant polynomials. Examples i
  185 bytes (28 words) - 14:44, 31 May 2020
• 1993 AHSME Problems
  ...<math>c</math>, define <math>\boxed{a,b,c}</math> to mean <math>a^b-b^c+c^a</math>. Then <math>\boxed{1,-1,2}</math> equals <math>\text{(A)} \ -4 \qquad \text{(B)} \ -2 \qquad \text{(C)} \ 0 \qquad \text{(D)} \ 2 \
  20 KB (2,814 words) - 08:15, 27 June 2021
• Introduction to Programming Week 2
  Computers work with data, which might be a number, a piece of text, or a true/false value. ...e quotes (<tt>'this is a string'</tt>) or double quotes (<tt>"this is also a string"</tt>).
  4 KB (569 words) - 22:05, 15 August 2020
• 2001 AMC 10 Problems/Problem 10
  <math>\textbf{(A) }18\qquad\textbf{(B) }19\qquad\textbf{(C) }20\qquad\textbf{(D) }22\qquad\t Since we know every variable's value, we can substitute them in to find <math> x+y+z = 4+6+12 = \boxed{\
  2 KB (257 words) - 23:42, 9 April 2024
• 2001 AMC 10 Problems/Problem 20
  ...formed by cutting an isosceles right triangle from each of the corners of a square with sides of length <math> 2000 </math>. What is the length of each <math> \textbf{(A) } \frac{1}{3}(2000) \qquad \textbf{(B) } {2000(\sqrt{2}-1)} \qquad \textbf
  3 KB (460 words) - 09:59, 1 March 2024
• 2001 AMC 10 Problems/Problem 22
  <math> \textbf{(A)}\ 43 \qquad \textbf{(B)}\ 44 \qquad \textbf{(C)}\ 45 \qquad \textbf{(D)}\ We know that <math> y+z=2v </math>, so we could find one variable rather than two.
  5 KB (721 words) - 16:44, 9 August 2022
• Introduction to Programming Week 3
  ====Part 1: What is a Function?==== A '''function''' in [[Python]] is a block of code that performs a task. In programming terminology, functions aren't used, they're '''called
  4 KB (683 words) - 00:20, 30 October 2012
• Introduction to Programming Week 4
  ...statements''', which will execute the block of code following them only if a specified condition is met. If the condition isn't met the code will be sk The condition in an if-statement must be an expression that evaluates to a [[boolean]] (that is, either true or false). For this we can use [[operato
  5 KB (787 words) - 13:28, 30 March 2011
• Range function
  ...It's most frequently used in for-loops, to define the values the iterator variable should take on. ...han to generate them all beforehand and store them in memory. If you need a real [[list]] for some reason you can call the [[type conversion functions|
  3 KB (420 words) - 15:40, 10 April 2011
• Introduction to Programming Week 8
  This week we focused on a very popular and very powerful paradigm called '''object-oriented programmi ...of object is called a [[class (Computer Science)|class]], and an object of a particular class is also called an '''instance''' of that class.
  5 KB (755 words) - 23:04, 27 May 2013
• Introduction to Programming Week 9
  ...s supplying. For example, we could give the user the option of specifying a list of values in our Die init with Then if the user specifies a list of values init will use that, but if the user doesn't it'll will use [
  2 KB (403 words) - 19:23, 5 May 2011
• Isinstance function
  ...ul because Python is dynamically typed, so the class of object stored in a variable may not be known until runtime. It has the following syntax: It will return True if object is an instance of Class or a subclass of Class, and False otherwise.
  612 bytes (94 words) - 17:53, 2 September 2011
• Olympiad books
  Here is a list of '''Olympiad Books''' that have Olympiad-level problems used to trai *''Inequalities: A Mathematical Olympiad Approach'' - '''Radmila Bulajich Manfrino, Jose Anton
  17 KB (2,261 words) - 00:30, 22 April 2024
• 2007 AMC 10B Problems/Problem 23
  A pyramid with a square base is cut by a plane that is parallel to its base and <math>2</math> units from the base. <math>\textbf{(A) } 2 \qquad\textbf{(B) } 2+\sqrt{2} \qquad\textbf{(C) } 1+2\sqrt{2} \qquad\
  4 KB (616 words) - 13:52, 4 April 2024

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