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• Brute forcing
  1 KB (190 words) - 13:22, 5 May 2023
• Real number
  Let <math>\alpha\subset\mathbb{Q}</math> be non-empty
  3 KB (496 words) - 23:22, 5 January 2022
• Pi
  ...imes(0,1)}\frac {x^ky^\ell}{1-xy}\,dx\,dy</math>. Expanding <math>\frac 1{1-xy}=\sum_{j=0}^\infty x^jy^j</math> and integrating term by term, we get <ma <math>I(k,\ell)=\frac 1{\ell-k}\left[\frac 1{k+1}+\frac 1{k+2}+\dots+\frac 1\ell\right]</math>
  8 KB (1,469 words) - 21:11, 16 September 2022
• Chain Rule
  Here are some more precise statements for the single-variable and multi-variable cases. ===Multi-dimensional Chain Rule===
  12 KB (2,377 words) - 11:48, 22 July 2009
• Muirhead's Inequality
  ...an inequality ''can'' be proved with AM-GM before demonstrating the full AM-GM proof. ...</math> from the [[AM-GM Inequality#Weighted AM-GM Inequality|(weighted) AM-GM inequality]].
  8 KB (1,346 words) - 12:53, 8 October 2023
• Vector
  ...ected line segment. In many situations, a vector is best considered as an n-tuple of numbers (often real or complex). Most generally, but also most abst ...h (or magnitude) and the angle it makes with some fixed line (usually the x-axis) or by describing it as an arrow beginning at the origin and ending at
  7 KB (1,265 words) - 13:22, 14 July 2021
• Area
  ...math>c</math> are the lengths of the sides and <math>s</math> is the [[semi-perimeter]] <math>s=\frac{a+b+c}{2}</math>. ...>, where <math>r</math> is the radius of the [[incircle]] and s is the semi-perimeter.
  6 KB (1,181 words) - 22:37, 22 January 2023
• Improper fractional base
  787 bytes (118 words) - 19:20, 23 October 2010
• Newton's Sums
  <center><math> P(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0</math></center> <cmath>a_nP_1 + a_{n-1} = 0</cmath>
  4 KB (690 words) - 13:11, 20 February 2024
• Fundamental Theorem of Algebra
  <cmath> \lvert P(z) \rvert \ge \lvert z^n \rvert - (R-1) \lvert z^{n-1} \rvert = \lvert z^{n-1} \rvert \cdot \bigl[ \lvert z \rvert - (R-1) \bigr]
  5 KB (832 words) - 14:22, 11 January 2024
• ...bda_a = \lambda_b = 1/2</math>, this is the elementary form of the [[Cauchy-Schwarz Inequality]]. ...he left-hand side of the desired inequality but may contribute to the right-hand side.
  4 KB (774 words) - 12:12, 29 October 2016
• Mock AMC
  Listed below are the [higher-quality] Mock AMCs that have been hosted over AoPS in the past. Feel free to {| class="wikitable" style="text-align:center;width:100%"
  51 KB (6,175 words) - 20:58, 6 December 2023
• 2006 AIME I Problems
  Let set <math> \mathcal{A} </math> be a 90-element subset of <math> \{1,2,3,\ldots,100\}, </math> and let <math> S </ma A collection of 8 cubes consists of one cube with edge-length <math> k </math> for each integer <math> k, 1 \le k \le 8. </math> A
  7 KB (1,173 words) - 03:31, 4 January 2023
• 2006 AIME I Problems/Problem 15
  Given that a sequence satisfies <math> x_0=0 </math> and <math> |x_k|=|x_{k-1}+3| </math> for all integers <math> k\ge 1, </math> find the minimum possi {{AIME box|year=2006|n=I|num-b=14|after=Last Question}}
  6 KB (910 words) - 19:31, 24 October 2023
• 2006 AIME I Problems/Problem 13
  ...For each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 ...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) - 00:45, 16 November 2023
• LaTeX:Symbols
  ....)) maintains a [http://docs.mathjax.org/en/latest/tex.html#supported-latex-commands list of supported commands]. [http://mirrors.ctan.org/info/symbols/comprehensive/symbols-a4.pdf The Comprehensive LaTeX Symbol List].
  16 KB (2,307 words) - 15:24, 8 June 2024
• 2006 AIME I Problems/Problem 10
  ...will not work in general). Denote the intersection of the line and the [[x-axis]] as <math>(x, 0)</math>. ...ezoid]] and a [[rectangle]], and the areas are <math>\frac{1}{2}((1-x) + (2-x))(3)</math> and <math>2 \cdot 1 = 2</math>, totaling <math>\frac{13}{2} -
  4 KB (731 words) - 17:59, 4 January 2022
• 2006 AIME I Problems/Problem 8
  ...ath>44.5^2</math> is also less than <math>2006</math>, so we have numbers 1-44, times two because 0.5 can be added to each of time, plus 1, because 0.5 ...have <math>[GIHJ]=0,</math> implying the minimum possible positive-integer-valued area is <math>1.</math>
  5 KB (730 words) - 15:05, 15 January 2024
• 2021 JMC 10
  <math>\textbf{(A) }\pi-e \qquad\textbf{(B) }2\pi-2e\qquad\textbf{(C) }2e\qquad\textbf{(D) }2\pi \qquad\textbf{(E) }2\pi +e</m ...h>74</math> and <math>83</math> are pretentious. How many pretentious three-digit numbers are odd?
  12 KB (1,784 words) - 16:49, 1 April 2021
• 2006 AMC 12B Problems
  ...1.99</math>, and <math>\textdollar0.99</math>. Mary will pay with a twenty-dollar bill. Which of the following is closest to the percentage of the <ma How many even three-digit integers have the property that their digits, read left to right, are
  13 KB (2,058 words) - 12:36, 4 July 2023

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