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• 2015 IMO Problems
  We say that a finite set <math>\mathcal{S}</math> in the plane is <i> balanced </i> if, for any two different points <math>A</math>, <math>B</math> in <math>\mathcal{S}</math>, there is
  4 KB (709 words) - 14:00, 1 June 2024
• Fermat's Little Theorem
  ...] (which students should study more at the introductory level if they have a hard time following the rest of this article). This theorem is credited to ...}</math> is not [[divisibility|divisible]] by <math>{p}</math>, then <math>a^{p-1}\equiv 1 \pmod {p}</math>.
  16 KB (2,660 words) - 22:42, 28 August 2024
• Number theory/Advanced
  ...vertically upward as the point at infinity. We denote it by <math>\mathcal{O}</math>. ...points. This means that given 2 points on the curve, they can be added in a way that satisfies the normal laws of addition, like associativity, commuta
  5 KB (849 words) - 15:14, 18 May 2021
• Law of Sines
  The '''Law of Sines''' is a useful identity in a [[triangle]], which, along with the [[law of cosines]] and the [[law of tan ...<math>B</math>, <math>c</math> opposite to <math>C</math>, and where <math>R</math> is the circumradius:
  4 KB (658 words) - 15:19, 28 April 2024
• Trigonometry
  ...s and angles of triangles through the '''trigonometric functions'''. It is a fundamental branch of mathematics, and its discovery paved the way towards ...definition of trigonometry is preferred in order to extend trigonometry to a complex domain.
  8 KB (1,217 words) - 19:15, 7 September 2023
• Root-Mean Square-Arithmetic Mean-Geometric Mean-Harmonic mean Inequality
  ...power]], [[arithmetic mean]], [[geometric mean]], and [[harmonic mean]] of a set of [[positive]] [[real number]]s <math>x_1,\ldots,x_n</math> that says ...x_n} \ge \sqrt[n_3]{\frac{x_1^{n_3}+\cdots+x_{n_3}^{n_3}}{n_3}} \ge \frac{n}{\frac{1}{x_1}+\cdots+\frac{1}{x_n}} \ge \sqrt[n_4]{\frac{x_1^{n_4}+\cdots+
  5 KB (913 words) - 14:44, 14 August 2024
• Power of a Point Theorem
  ...e lengths of the [[line segment]]s formed when two [[line]]s [[intersect]] a [[circle]] and each other. ...of the lines is [[tangent line|tangent]] to the circle while the other is a [[secant line|secant]] (middle figure). In this case, we have <math> AB^2 =
  5 KB (854 words) - 13:30, 30 June 2024
• Algebraic geometry
  ...> be a [[prime ideal]] of <math>R</math>. Then <math>V(I)=\{p\in\mathbb{A}^n\mid f(p)=0\mathrm{\ for\ all\ } f\in I\}</math> is called an '''affine alge ...s are algebraic varieties. A projective space <math>\mathbb{P}^n</math> is a quotient set with an equivalence class satisfying
  2 KB (361 words) - 00:59, 24 January 2020
• Mock AMC
  ...an actual [[AMC]] (American Mathematics Competitions 8, 10, or 12) exam. A number of '''Mock AMC''' competitions have been hosted on the [[Art of Prob == Tips for Writing a Mock AMC ==
  51 KB (6,135 words) - 05:44, 2 November 2024
• 2006 AIME I Problems/Problem 13
  ...st integer <math> n </math> less than 1000 such that <math> S_n </math> is a [[perfect square]]. .../math> but not by <math>2^n, \ldots,</math> and <math>2^{n-1}-2^{n-2} = 2^{n-2}</math> elements of <math>S</math> that are divisible by <math>2^1</math>
  10 KB (1,702 words) - 21:23, 25 July 2024
• LaTeX:Symbols
  This article provides a short list of commonly used LaTeX symbols. .../math> on the web, (technically an AJAX library simulating it.)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list o
  16 KB (2,315 words) - 19:35, 4 November 2024
• 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 (1,965 words) - 21:18, 7 September 2024
• 2000 AMC 12 Problems
  ...2001 </math>. What is the largest possible value of the sum <math>I + M + O</math>? <math>\textbf{(A)}\ 23 \qquad \textbf{(B)}\ 55 \qquad \textbf{(C)}\ 99 \qquad \textbf{(D)}\
  13 KB (1,948 words) - 09:35, 16 June 2024
• 2003 AMC 12B Problems
  \text {(A) } -1 \qquad \text {(B) } -\frac{2}{3} \qquad \text {(C) } \frac{2}{3} \qqu ...ks. A green pill costs 1 dollar more than a pink pill, and Al's pills cost a total of 546 dollars for the two weeks. How much does one green pill cost?
  13 KB (1,987 words) - 17:53, 10 December 2022
• 2005 AIME II Problems
  {{AIME Problems|year=2005|n=II}} ...imes the number of possible sets of 3 cards that can be drawn. Find <math> n. </math>
  7 KB (1,119 words) - 20:12, 28 February 2020
• 2005 AIME II Problems/Problem 12
  ...math> is not divisible by the [[square]] of any [[prime]], find <math> p+q+r. </math> ...abel("\(x\)",E/2+G/2,(0,1));label("\(y\)",G/2+F/2,(0,1)); label("\(450\)",(O+G)/2,(-1,1));
  13 KB (2,080 words) - 12:14, 23 July 2024
• 2005 AIME I Problems/Problem 11
  ...mum value of <math> d </math> is <math> m - \sqrt{n},</math> find <math> m+n. </math> We note that aligning the base of the semicircle with a side of the square is certainly non-optimal. If the semicircle is tangent
  4 KB (707 words) - 10:11, 16 September 2021
• 2005 AIME II Problems/Problem 15
  ...h> a </math> for which the line <math> y=ax </math> contains the center of a circle that is externally [[tangent (geometry)|tangent]] to <math> w_2 </ma Let <math>w_3</math> have center <math>(x,y)</math> and radius <math>r</math>. Now, if two circles with radii <math>r_1</math> and <math>r_2</mat
  12 KB (2,001 words) - 19:26, 23 July 2024
• 2004 AIME I Problems/Problem 14
  ...are [[positive]] [[integer]]s, and <math> c </math> is prime. Find <math> a+b+c. </math> ...-8,4), B=(0,-8,h), C=(Cxy.x,Cxy.y,0), D=(A.x,A.y,0), E=(B.x,B.y,0), O=(O.x,O.y,h);
  4 KB (729 words) - 00:00, 27 November 2022
• 2004 AIME II Problems/Problem 12
  ...> k </math> and <math> p </math> are [[relatively prime]]. Find <math> k+m+n+p. </math> ...of the center circle be <math>r</math> and its center be denoted as <math>O</math>.
  3 KB (431 words) - 22:21, 4 July 2013

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