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• Mock AIME 3 Pre 2005 Problems/Problem 1
  Three circles are mutually externally tangent. Two of the circles have radii <math>3</math> and <math>7</math>. If the area of the triangle f
  795 bytes (129 words) - 10:22, 4 April 2012
• Mock AIME 3 Pre 2005 Problems/Problem 14
  Circles <math>\omega_1</math> and <math>\omega_2</math> are centered on opposite si Let <math>r_1</math> and <math>r_2</math> be the radii of circles <math>\omega_1</math> and <math>\omega_2</math> respectively.
  3 KB (563 words) - 02:05, 25 November 2023
• 2007 AIME I Problems
  ...ent to the hypotenuse and to the extension of leg <math>CB</math>, and the circles are externally tangent to each other. The length of the radius of either c
  7 KB (1,218 words) - 15:28, 11 July 2022
• 2007 AIME I Problems/Problem 9
  ...to the hypotenuse and to the extension of [[leg]] <math>CB</math>, and the circles are externally tangent to each other. The length of the radius either circ ...tangents is <math>2.4x</math>. Note that if we connect the centers of the circles we have a rectangle with sidelengths 8x and 4x. So, <math>8x+2.4x+x=34</mat
  11 KB (1,851 words) - 12:31, 21 December 2021
• 2002 IMO Shortlist Problems/G4
  Circles <math> \displaystyle S_1 </math> and <math> \displaystyle S_2 </math> inter ...stay fixed, at half the measure of arc <math> \displaystyle QP </math> on circles <math> \displaystyle S_1 </math> and <math> \displaystyle S_2 </math>, resp
  3 KB (470 words) - 07:32, 28 March 2007
• 2007 AIME II Problems/Problem 15
  ...and <math>O_C</math>. Let the center of the circle tangent to those three circles be <math>O</math>. The [[homothety]] <math>\mathcal{H}\left(I, \frac{4-r}{4 ...h>, <math>B'</math>, <math>C'</math>, and <math>O</math> be the centers of circles <math>\omega_{A}</math>, <math>\omega_{B}</math>, <math>\omega_{C}</math>,
  11 KB (2,099 words) - 17:51, 4 January 2024
• 2007 AIME II Problems
  9 KB (1,435 words) - 01:45, 6 December 2021
• 2007 USAMO Problems/Problem 2
  ...e with radius greater than <math>\frac{1}{\sqrt{2}}</math> between those 3 circles. ...ath> must decrease in radius. Hence it is sufficient to consider 3 tangent circles. By lemma 1, there is always a circle of radius greater than <math>\frac{1}
  5 KB (754 words) - 03:41, 7 August 2014
• 2007 USAMO Problems/Problem 6
  ...and <math>Q_{A}</math> lie on <math>AO</math> since for a pair of tangent circles, the point of tangency and the two centers are [[collinear]].
  7 KB (1,274 words) - 15:11, 31 August 2017
• 2007 USA TST Problems
  Circles <math>\omega_{1}</math> and <math>\omega_{2}</math> intersect at <math>P</m
  3 KB (519 words) - 20:59, 24 May 2007
• 2004 IMO Shortlist Problems/C2
  ...ath> be positive integers. There are given <math> \displaystyle n </math> circles in the plane. Every two of them intersect at two distinct points, and all ...rcles a distinct color, which we use for both points common to the pair of circles.
  5 KB (861 words) - 10:05, 5 June 2007
• 2004 IMO Shortlist Problems/G2
  '''Lemma.''' Let <math> \displaystyle \omega_1 , \omega_2 </math> be two circles with centers <math> \displaystyle O_1, O_2 </math>, and common points <math
  10 KB (1,539 words) - 23:37, 6 June 2007
• 2007 Cyprus MO/Lyceum/Problem 21
  ...The curved parts of the band add up to a full circumference of one of the circles, so their sum is <math>20\pi\,\mathrm{ cm}</math>. The total length of the
  1 KB (175 words) - 22:21, 23 July 2020
• Pascal's Theorem
  '''Lemma.''' Let <math>\omega_1, \omega_2 </math> be two circles which intersect at <math>M, N </math>, let <math>AB </math> be a chord of <
  4 KB (712 words) - 21:57, 25 April 2020
• Cyclic
  Because two different circles intersect in at most two points, any polygon can be inscribed in at most on
  2 KB (291 words) - 05:30, 16 August 2011
• Common internal tangent line
  ...ent of both circles that intersects the segment joining the centers of two circles. ...ommon internal tangent line | common internal tangent]] [[intersect]]s the circles at <math>C</math> and <math>D</math>, respectively. [[Line]]s <math>AB</mat
  703 bytes (106 words) - 16:23, 10 March 2019
• 2006 AIME II Problems/Problem 9
  3 KB (553 words) - 10:45, 26 August 2015
• 2006 AIME II Problems
  Circles <math> \mathcal{C}_1, \mathcal{C}_2, </math> and <math> \mathcal{C}_3 </mat
  8 KB (1,350 words) - 12:00, 4 December 2022
• 2005 PMWC Problems
  The larger circle has radius 12 cm. Each of the six identical smaller circles smaller circles?
  9 KB (1,449 words) - 20:49, 2 October 2020
• 1998 PMWC Problems
  ...otated in the plane, they are different from one another. At most how many circles are painted with all three colors?
  11 KB (1,738 words) - 19:25, 10 March 2015

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