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• 1983 AIME Problems/Problem 14
  In the adjoining figure, two circles with radii <math>8</math> and <math>6</math> are drawn with their centers < ...math>QP = PR = x</math>. Extend the line containing the centers of the two circles to meet <math>R</math>, and to meet the other side of the large circle at a
  13 KB (2,151 words) - 17:48, 27 May 2024
• 1984 AIME Problems/Problem 6
  ...side of the line is equal to the total [[area]] of the parts of the three circles to the other side of it. What is the [[absolute value]] of the [[slope]] of ...e [[midpoint]] of <math>\overline{AC}</math> (the centers of the other two circles), and call it <math>M</math>. If we draw the feet of the [[perpendicular]]s
  6 KB (1,022 words) - 19:29, 22 January 2024
• 2006 AMC 10B Problems/Problem 4
  Circles of diameter <math>1</math> inch and <math>3</math> inches have the same cen
  1 KB (172 words) - 10:47, 19 December 2021
• 1991 AIME Problems/Problem 12
  ...en we found <math>AP</math>, the segment <math>OB</math> is tangent to the circles with diameters <math>AO,CO</math>.
  8 KB (1,270 words) - 23:36, 27 August 2023
• 1991 AIME Problems/Problem 11
  ...es must be tangent on the larger circle. Now consider two adjacent smaller circles. This means that the line connecting the radii is a segment of length <math ...3})^{2} = \pi (7 - 4 \sqrt {3})</math>, so the area of all <math>12</math> circles is <math>\pi (84 - 48 \sqrt {3})</math>, giving an answer of <math>84 + 48
  4 KB (740 words) - 17:46, 24 May 2024
• 1992 AIME Problems/Problem 10
  ...ique area of the two circles. We can do this by adding the area of the two circles and then subtracting out their overlap. There are two methods of finding th 2. Consider that the circles can be converted into polar coordinates, and their equations are <math>r =
  2 KB (323 words) - 12:05, 16 July 2019
• 2006 AMC 10B Problems/Problem 24
  ...at <math>AD</math> and <math>BC</math> are common external tangents to the circles. What is the area of the [[concave]] [[hexagon]] <math>AOBCPD</math>?
  4 KB (558 words) - 14:38, 6 April 2024
• 1993 AIME Problems/Problem 15
  ...</math>. Let <math>R\,</math> and <math>S\,</math> be the points where the circles inscribed in the triangles <math>ACH\,</math> and <math>BCH^{}_{}</math> ar
  3 KB (449 words) - 21:39, 21 September 2023
• 1994 AIME Problems/Problem 15
  ...he respective conditions for <math>P</math> is the region inside the (semi)circles with diameters <math>\overline{AB}, \overline{BC}, \overline{CA}</math>. ...</math> (shaded region below) is simply the sum of two [[segment]]s of the circles. If we construct the midpoints of <math>M_1, M_2 = \overline{AB}, \overline
  4 KB (717 words) - 22:20, 3 June 2021
• 1995 AIME Problems/Problem 4
  ...th>9</math> has a chord that is a common external tangent of the other two circles. Find the square of the length of this chord. We label the points as following: the centers of the circles of radii <math>3,6,9</math> are <math>O_3,O_6,O_9</math> respectively, and
  3 KB (605 words) - 11:30, 5 May 2024
• 1997 AIME Problems/Problem 14
  ...</math> with radius <math>\sqrt{2+\sqrt{3}}</math>. The equations of these circles are <math>(x-1)^2 = 1</math> and <math>x^2 + y^2 = 2 + \sqrt{3}</math>. Sol
  5 KB (874 words) - 22:30, 1 April 2022
• 1997 AIME Problems/Problem 4
  ...f the circle in question and the segment connecting the centers of the two circles of radii <math>5</math>. By the [[Pythagorean Theorem]], we now have two eq ...give the curvature for the circle internally tangent to each of the other circles. Using Descartes' theorem, we get <math>k_4=\frac15+\frac15+\frac18+2\sqrt{
  2 KB (354 words) - 22:33, 2 February 2021
• 1998 AIME Problems/Problem 10
  3 KB (496 words) - 13:02, 5 August 2019
• 1999 AIME Problems/Problem 6
  To find the area between the circles (actually, parts of the circles), we need to figure out the [[angle]] of the [[arc]]. This could be done by
  2 KB (354 words) - 16:42, 20 July 2021
• 2000 AIME I Problems/Problem 13
  ...way in a circle with this radius centered at <math>(x,0)</math>. All these circles are [[homothety|homothetic]] with respect to a center at <math>(5,0)</math>
  3 KB (571 words) - 00:38, 13 March 2014
• 2002 AIME I Problems/Problem 2
  ...gruent [[circle]]s arranged in three rows and enclosed in a rectangle. The circles are tangent to one another and to the sides of the rectangle as shown in th Let the [[radius]] of the circles be <math>r</math>. The longer dimension of the rectangle can be written as
  2 KB (287 words) - 19:54, 4 July 2013
• 2003 AIME I Problems/Problem 2
  ...dius <math>1</math> is colored red, and each region bounded by consecutive circles is colored either red or green, with no two adjacent regions the same color To get the green area, we can color all the circles of radius <math>100</math> or below green, then color all those with radius
  4 KB (523 words) - 15:49, 8 March 2021
• 2002 AIME II Problems/Problem 15
  ...he line <math>y = mx</math>, where <math>m > 0</math>, are tangent to both circles. It is given that <math>m</math> can be written in the form <math>a\sqrt {b ...respectively. We know that the point <math>(9,6)</math> is a point on both circles, so we have that
  7 KB (1,182 words) - 09:56, 7 February 2022
• Inradius
  ...that tangent point is equal to A to the other tangent point (explained in circles) and etc for B and C. After doing it for B and C, C (the hypotenuse) should
  2 KB (336 words) - 00:44, 23 April 2024
• Convex set
  1 KB (227 words) - 22:47, 1 March 2008

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