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• 2006 AMC 12B Problems/Problem 15
  ...s <math> A</math> and <math> B</math> are on the circle centered at <math> O</math>, and points <math> C</math> and <math> D</math> are on the circle ce unitsize(0.4 cm); defaultpen(linewidth(0.7) + fontsize(11));
  3 KB (458 words) - 16:40, 6 October 2019
• 2005 AIME II Problems
  ...the faces of <math> O, </math> and that the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math ...</math> be a sequence of reals such that <math> a_0 = 37, a_1 = 72, a_m = 0, </math> and <math> a_{k+1} = a_{k-1} - \frac 3{a_k} </math> for <math> k =
  7 KB (1,119 words) - 21:12, 28 February 2020
• 2005 AIME II Problems/Problem 12
  [[Square]] <math>ABCD </math> has [[center]] <math> O,\ AB=900,\ E </math> and <math> F </math> are on <math> AB </math> with <ma ...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) - 21:20, 11 December 2022
• 2005 AIME II Problems/Problem 10
  ...the faces of <math> O, </math> and that the ratio of the volume of <math> O </math> to that of <math> C </math> is <math> \frac mn, </math> where <math import three; currentprojection = perspective(4,-15,4); defaultpen(linewidth(0.7));
  3 KB (436 words) - 03:10, 23 September 2020
• 2005 AIME I Problems/Problem 11
  ...oints at which the "corners" of the semicircle touch the square. Let <math>O</math> be the center of the semicircle. ...of the semicircle as <math>r</math>. Draw the [[perpendicular]] from <math>O</math> to <math>AB</math>, which forms a <math>45-45-90</math> triangle. Th
  4 KB (707 words) - 11:11, 16 September 2021
• 2005 AIME II Problems/Problem 15
  ...cle]]s <math> x^2+y^2+10x-24y-87=0 </math> and <math> x^2 +y^2-10x-24y+153=0, </math> respectively. Let <math> m </math> be the smallest positive value size(220); pointpen = black; pen d = linewidth(0.7); pathpen = d;
  12 KB (2,000 words) - 13:17, 28 December 2020
• 2004 AIME I Problems/Problem 14
  import three; defaultpen(fontsize(10)+linewidth(0.62)); ...-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) - 01:00, 27 November 2022
• 2004 AIME I Problems/Problem 10
  ...= 36/5</math>. Since <math>\triangle AOR \sim \triangle AED</math> (<math>O</math> is the center of the circle), we find that <math>AR = 5</math> since ...s <math>y = \frac{5}{12}x + c</math>. Manipulating, <math>5x - 12y + 12c = 0</math>. We need to find the value of <math>c</math>, which can be determine
  5 KB (836 words) - 07:53, 15 October 2023
• 2004 AIME II Problems/Problem 12
  ...of the center circle be <math>r</math> and its center be denoted as <math>O</math>. pointpen = black; pathpen = black+linewidth(0.7); pen d = linewidth(0.7) + linetype("4 4"); pen f = fontsize(8);
  3 KB (431 words) - 23:21, 4 July 2013
• 1983 AIME Problems
  Let <math>f(x)=|x-p|+|x-15|+|x-p-15|</math>, where <math>0 < p < 15</math>. Determine the [[minimum]] value taken by <math>f(x)</math> defaultpen(linewidth(0.6)+fontsize(11));
  7 KB (1,104 words) - 03:13, 27 May 2024
• 1992 AIME Problems
  ...she has played. At the start of a weekend, her win ratio is exactly <math>0.500</math>. During the weekend, she plays four matches, winning three and l \text{Row 0: } & & & & & & & 1 & & & & & & \\\vsp
  8 KB (1,117 words) - 05:32, 11 November 2023
• 1996 AIME Problems
  ...th>b</math>, and <math>c</math>, and that the roots of <math>x^3+rx^2+sx+t=0</math> are <math>a+b</math>, <math>b+c</math>, and <math>c+a</math>. Find < ...a^{\circ}}+i\sin{\theta^{\circ}})</math>, where <math>0<r</math> and <math>0\leq \theta <360</math>. Find <math>\theta</math>.
  6 KB (931 words) - 17:49, 21 December 2018
• 2000 AIME II Problems
  ..., <math>BC=14</math>, <math>CA=15</math>, and that the distance from <math>O</math> to triangle <math>ABC</math> is <math>\frac{m\sqrt{n}}k</math>, wher ...egers, <math>m</math> and <math>r</math> are relatively prime, and <math>r>0</math>. Find <math>m+n+r</math>.
  6 KB (947 words) - 21:11, 19 February 2019
• 2002 AIME II Problems
  ...th> is a right angle. A circle of radius <math>19</math> with center <math>O</math> on <math>\overline{AP}</math> is drawn so that it is tangent to <mat ...th>68</math>. The x-axis and the line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be
  7 KB (1,177 words) - 15:42, 11 August 2023
• 1983 AIME Problems/Problem 4
  defaultpen(linewidth(0.6)+fontsize(11)); pair O=(0,0),
  11 KB (1,741 words) - 22:40, 23 November 2023
• 1983 AIME Problems/Problem 12
  ...distance from their intersection point <math>H</math> to the center <math>O</math> is a positive rational number. Determine the length of <math>AB</mat draw(circle((0,0),4));
  2 KB (412 words) - 18:23, 1 January 2024
• 1983 AIME Problems/Problem 15
  pair O1=(0,0); pair A=(-0.91,-0.41);
  20 KB (3,497 words) - 15:37, 27 May 2024
• 1984 AIME Problems/Problem 13
  ...th>D_1=(5250,3150)</math>, <math>D_2=(5100,3400)</math>, and <math>H=(5100,0)</math>. We see that <math>\cot^{-1}(\angle A_2OA_1)=3</math>, <math>\cot^{
  3 KB (473 words) - 12:06, 18 December 2018
• 1985 AIME Problems/Problem 12
  Clearly, we have <math>P(0)=1.</math> For all <math>k\geq1,</math> note that after <math>k-1</math> cr ...bability that it crawls to vertex <math>A</math> on the next move is <math>0.</math></li><p>
  17 KB (2,837 words) - 13:34, 4 April 2024
• 1985 AIME Problems/Problem 9
  pointpen = black; pathpen = black + linewidth(0.8); real r = 8/15^0.5, a = 57.91, b = 93.135;
  5 KB (763 words) - 16:20, 28 September 2019

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