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• 1998 USAMO Problems/Problem 2
  Let <math>{\cal C}_1</math> and <math>{\cal C}_2</math> be concentric circles, with <math>{\cal C}_2</math> in the interior of <math>{\cal C}_1</math>.
  4 KB (656 words) - 17:26, 20 June 2019
• 2012 USAJMO Problems/Problem 1
  ...ass through <math>R</math> and <math>S</math>, so the radical axis of both circles is <math>RS</math>. Hence, <math>A</math> lies on <math>RS</math>, which i
  4 KB (613 words) - 20:50, 19 December 2023
• 2006 SMT/Geometry Problems
  ...us of a third circle that is tangent to one line and tangent to both other circles?
  5 KB (837 words) - 12:22, 27 May 2012
• 2006 SMT/Geometry Problems/Problem 4
  ...</math> must lie on circle <math> X </math>. We now want the area between circles <math> X </math> and <math> Y </math>. Let <math> M </math> be the midpoint First, let's try to find the radii of the circles. <math> \angle AYB=30^\circ\implies \angle AYM=15^\circ </math>. From the h
  4 KB (622 words) - 15:07, 28 May 2012
• 1952 AHSME Problems
  Two equal circles in the same plane cannot have the following number of common tangents.
  23 KB (3,556 words) - 15:35, 30 December 2023
• 1953 AHSME Problems
  ...ric circles. It is <math>10</math> feet wide. The circumference of the two circles differ by about: The diameters of two circles are <math>8</math> inches and <math>12</math> inches respectively. The rati
  21 KB (3,123 words) - 14:24, 20 February 2020
• 1954 AHSME Problems
  ...rline{AB}</math> is divided at <math>C</math> so that <math>AC=3CB</math>. Circles are described on <math>\overline{AC}</math>
  23 KB (3,535 words) - 16:29, 24 April 2020
• 1955 AHSME Problems
  The ratio of the areas of two concentric circles is <math>1: 3</math>. If the radius of the smaller is <math>r</math>, then
  22 KB (3,509 words) - 21:29, 31 December 2023
• 2003 AMC 8 Problems/Problem 22
  The following figures are composed of squares and circles. Which figure has a shaded region with largest area? ...f{B}</math> the area of the shaded region is the sum of the areas of the 4 circles subtracted from the area of the square. That is <math>2^2-4 \left( \left(
  3 KB (433 words) - 10:40, 14 June 2024
• 2012 AMC 10B Problems/Problem 16
  ...circles with radius 2 are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? ...find the area of this triangle to include the figure formed in between the circles. Since the equilateral triangle has two 30-60-90 triangles inside, we can f
  4 KB (675 words) - 03:58, 23 January 2023
• 1986 USAMO Problems
  Two distinct circles <math>K_1</math> and <math>K_2</math> are drawn in the plane. They intersec
  3 KB (500 words) - 19:00, 23 October 2019
• 1978 USAMO Problems/Problem 2
  ...center of the circle using two chords). The intersection point of the two circles other than X is the center of spiral similarity, and hence is O, which repr
  4 KB (712 words) - 21:57, 12 November 2023
• 2008 AMC 8 Problems
  Two circles that share the same center have radii <math>10</math> meters and <math>20</
  14 KB (2,035 words) - 15:23, 26 January 2024
• 2012 AMC 8 Problems/Problem 24
  ...e with radius <math> 2 </math>, meaning that the total area of the quarter circles is <math> 4\pi </math>. The area of the square is <math> 16 </math>. Thus,
  2 KB (310 words) - 16:18, 10 November 2023
• 2013 AMC 12A Problems/Problem 23
  The area also includes <math>4</math> circular segments. Two are quarter-circles centered at <math>P</math> of radii <math>\sqrt{2}</math> (the segment boun
  4 KB (603 words) - 16:51, 3 April 2020
• 1982 USAMO Problems/Problem 5
  Let <math>J</math> be the three circles radical center, meaning <math>JX</math> and <math>JZ</math> are tangent seg
  3 KB (524 words) - 16:08, 11 July 2018
• 1991 AHSME Problems
  Two circles are externally tangent. Lines <math>\overline{PAB}</math> and <math>\overli
  16 KB (2,451 words) - 04:27, 6 September 2021
• 2013 AIME II Problems/Problem 4
  ...ath>, while the other is centered at <math>B</math>. The equations of the circles are:
  8 KB (1,268 words) - 14:10, 31 January 2024
• 1967 IMO Problems/Problem 1
  If <math>\Delta ABD</math> is acute, prove that the four circles of radius <math>1</math> with centers <math>A</math>, <math>B</math>, <math ...th of <math>a</math> must not exceed 2 (the radius for each circle) or the circles will not meet and thus not cover the parallelogram.
  2 KB (360 words) - 00:10, 10 December 2022
• Rotation
  *Three concentric circles have radii <math>3,</math> <math>4,</math> and <math>5.</math> An equilater
  3 KB (432 words) - 23:22, 13 January 2021

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