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  • ...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
  • 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
  • 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
  • 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
  • ...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
  • ....)) 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 (2,324 words) - 16:50, 19 February 2024
  • ...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
  • ...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
  • <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
  • ...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
  • two-digit number such that <math>N = P(N)+S(N)</math>. What is the units digit o A telephone number has the form <math>\text{ABC-DEF-GHIJ}</math>, where each letter represents
    13 KB (1,957 words) - 12:53, 24 January 2024
  • In the expression <math>c\cdot a^b-d</math>, the values of <math>a</math>, <math>b</math>, <math>c</math>, and Minneapolis-St. Paul International Airport is 8 miles southwest of downtown St. Paul and
    13 KB (2,049 words) - 13:03, 19 February 2020
  • {{AMC12 box|year=2006|ab=B|num-b=13|num-a=15}}
    1 KB (227 words) - 17:21, 8 December 2013
  • A sequence <math>a_1,a_2,\dots</math> of non-negative integers is defined by the rule <math>a_{n+2}=|a_{n+1}-a_n|</math> is a 1-1 correspondence between the odd and even numbers less than and relatively p
    5 KB (924 words) - 12:02, 15 June 2022
  • ...ive x- axis, the answer is <math>\dfrac{\tan(BOJ) + \tan(BOJ+60) + \tan(BOJ-60)}{2}</math>. {{AMC12 box|year=2005|ab=B|num-b=23|num-a=25}}
    4 KB (761 words) - 09:10, 1 August 2023
  • ...<math>a</math> in the [[domain]] of the function such that <math>f(a) = (x-a) = 0</math>. Let <cmath>P(x) = c_nx^n + c_{n-1}x^{n-1} + \dots + c_1x + c_0 = \sum_{j=0}^{n} c_jx^j</cmath> with all <math>c_j
    8 KB (1,427 words) - 21:37, 13 March 2022
  • 967 bytes (176 words) - 18:08, 7 April 2012
  • ...0)); pair A=(0,9), B=(9,9), C=(9,0), D=(0,0), E=(2.5-0.5*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), G=(4.5,9), O=(4.5,4.5); draw(A--B--C--D--A);draw(E--O--F);dr ...*sqrt(7),9), F=(6.5-0.5*sqrt(7),9), O=(4.5,4.5), G=O+(E-O)*dir(-90), J=O+(F-O)*dir(-90); draw(A--B--C--D--A);draw(E--O--F);draw(G--O--J);draw(F--G,linet
    13 KB (2,080 words) - 21:20, 11 December 2022
  • ...n range from <math>l = 5</math> to <math>9</math>; notice that <math>d = 10-l</math> gives the number of diagonals. Let <math>R</math> represent a move ...jacent. The <math>D</math>'s split the string into three sections (<math>-D-D-</math>): by the [[Pigeonhole principle]] all of at least one of the two l
    5 KB (897 words) - 00:21, 29 July 2022
  • ...from Pythagorean theorem, <math>SC = \frac{18}{5}</math> and <math>AS = 14-SC = \frac{52}{5}</math>. We can also use the Pythagorean theorem on triangl ...and <math>RE</math> as <math>x</math>. <math>RB</math> then equals <math>13-y</math>. Then, we have two similar triangles.
    13 KB (2,129 words) - 18:56, 1 January 2024

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