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  • In [[algebra]], the '''Cauchy-Schwarz Inequality''', also known as the '''Cauchy–Bunyakovsky–Schwarz Inequality''' or informally as '''Cauchy- ...s in intermediate and olympiad competitions. It is particularly crucial in proof-based contests.
    13 KB (2,048 words) - 14:28, 22 February 2024
  • ...more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to [[Pierre de Fermat]]. ...it is derived by multipling both sides of the theorem by <math>a</math>. The restated form is nice because we no longer need to restrict ourselves to in
    16 KB (2,660 words) - 22:42, 28 August 2024
  • ...te level geometry problems. It also provides the basis for the definition of a [[metric space]] in [[analysis]]. ...ality extends this to [[obtuse triangle| obtuse]] and [[acute triangle]]s. The inequality says:
    7 KB (1,300 words) - 00:11, 28 October 2024
  • [[Image:Incenter_excenter_lemma.png|thumb|right|200px|Diagram of the configuration.]] ...s circumcenter. Then, the theorem states that <math>L</math> is the center of a circle through <math>I</math>, <math>B</math>, <math>I_A</math>, and <mat
    2 KB (292 words) - 17:22, 14 January 2025
  • ...vely. Then, <math>\triangle DEF</math> is called the '''orthic triangle''' of <math>\triangle ABC</math>. ...angle does not exist if <math>\triangle ABC</math> is right. The two cases of when <math>\triangle ABC</math> is either acute or obtuse each carry differ
    8 KB (1,408 words) - 08:39, 10 July 2024
  • ...utive terms is constant. This constant is called the '''common ratio''' of the sequence. ...math> and <math>-3, 1, 5, 9, \ldots</math> are not geometric sequences, as the ratio between consecutive terms varies.
    4 KB (649 words) - 20:09, 19 July 2024
  • ...terms is constant. This constant is called the '''common difference''' of the sequence. ...> and <math>4, 12, 36, 108, \ldots</math> are not arithmetic sequences, as the difference between consecutive terms varies.
    4 KB (736 words) - 01:00, 7 March 2024
  • ...math>a_2<999</math> and <math>a_{2006}=1</math>, how many different values of <math>a_2</math> are possible? ...quence <math>(a_n)</math> completes at <math>i</math> if <math>i</math> is the minimal positive integer such that <math>a_i = a_{i + 1} = 1</math>. Otherw
    6 KB (1,037 words) - 22:32, 9 November 2024
  • ...expressed as a fraction <math>\frac{m}{n}</math> in lowest terms, what is the product <math>mn</math>? ...<math>P</math> are (1) on opposite sides of <math>\ell</math>. and (2) at the same distance from <math>\ell</math>.
    20 KB (3,497 words) - 14:37, 27 May 2024
  • ...[[digit]]s and that its first (leftmost) digit is 9, how many [[element]]s of <math>T_{}^{}</math> have 9 as their leftmost digit? '''Lemma''': For all positive integers n, there's exactly one n-digit power of 9 that does not have a left-most digit 9
    5 KB (762 words) - 00:18, 10 February 2023
  • ...the triangle along the sides of its midpoint triangle. What is the volume of this pyramid? ....S--Q..S--R); draw(S--(16,12,0)); </asy></center><!-- Asymptote renderings of Image:AIME_1999_Solution_15_1.png, Image:AIME_1999_Solution_15_2.png, by Mi
    7 KB (1,175 words) - 12:26, 3 September 2024
  • ...f points that can be reached by the firetruck within six minutes. The area of this region is <math>m/n</math> square miles, where <math>m</math> and <mat ...ighways be the x and y axes. We consider the case where the truck moves in the positive x direction.
    6 KB (1,042 words) - 14:08, 1 January 2025
  • ...>. Find all polynomials <math>f</math> with integer coefficients such that the sequence <math>\{ p(f(n^2))-2n) \}_{n \in \mathbb{Z} \ge 0}</math> is bound Let <math>f(x)</math> be a non-constant polynomial in <math>x</math> of degree <math>d</math> with
    9 KB (1,699 words) - 12:48, 11 April 2020
  • ...ath>S</math> and <math>T</math> respectively. Prove that the circumcircles of triangles <math>SAE</math>, <math>SBF</math>, <math>TCF</math>, and <math>T ...BF</math> be <math>X</math>, and let the intersection of the circumcircles of <math>TCF</math> and <math>TDE</math> be <math>Y</math>.
    5 KB (986 words) - 21:46, 18 May 2015
  • ...number|prime]], and let <math>a</math> be any integer. Then we can define the [[Legendre symbol]] ...{}{}{a}{p}</math> as the unique nontrivial multiplicative [[homomorphism]] of <math>\mathbb{F}_p^\times</math> into <math>\mathbb{R}^\times</math>, exten
    7 KB (1,182 words) - 15:46, 28 April 2016
  • A '''Dedekind domain''' is a [[integral domain]] <math>R</math> satisfying the following properties: * Every [[prime ideal]] of <math>R</math> is a [[maximal ideal]].
    9 KB (1,648 words) - 15:36, 14 October 2017
  • ...If <math>f(m,49)</math> is an integer, find the sum of all possible values of <math>m</math>. Lemma: <math>\cot{B}-\cot{A}=2\cot{\angle{AMP}}</math>
    3 KB (541 words) - 16:32, 22 November 2023
  • ...e known as [[corollary|corollaries]], if they are an immediate consequence of some other theorem. [[Mathematics]] is in large part the process of proving new theorems.
    535 bytes (83 words) - 14:52, 13 January 2025
  • Prove that the fraction <math>\frac{21n+4}{14n+3}</math> is irreducible for every natural ...eatest common divisor of <math>a, b </math> as <math>(a,b) </math>, we use the [[Euclidean algorithm]]:
    5 KB (767 words) - 09:59, 23 July 2023
  • ...of which are circumscriptible quadrilaterals. Prove that all the faces of the prism are circumscriptible quadrilaterals. ...P such that P is equidistant from any point on one circle and any point on the other circle.
    3 KB (509 words) - 22:22, 15 August 2012

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