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• 2015 AIME II Problems
  ...}</math> such that <math>BC</math> is a common external tangent of the two circles. A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{
  8 KB (1,326 words) - 19:15, 13 January 2024
• 2015 AIME II Problems/Problem 15
  ...math> so that line <math>BC</math> is a common external tangent of the two circles. A line <math>\ell</math> through <math>A</math> intersects <math>\mathcal{ ...we have <math>AM = BM</math> and <math>BM = CM</math> by equal tangents to circles, and since <math>BM = CM, M</math> is the midpoint of <math>\overline{BC},<
  31 KB (5,086 words) - 19:15, 20 December 2023
• 2016 AIME I Problems
  ...</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>. Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>X
  8 KB (1,360 words) - 12:19, 29 January 2022
• 1959 AHSME Problems/Problem 41
  ...tangent to the line and to the other two circles. The radius of the equal circles is: <math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qqu
  599 bytes (108 words) - 15:44, 28 December 2017
• 1959 AHSME Problems/Problem 40
  ...tangent to the line and to the other two circles. The radius of the equal circles is: <math>\textbf{(A)}\ 24 \qquad\textbf{(B)}\ 20\qquad\textbf{(C)}\ 18\qqu
  596 bytes (108 words) - 15:44, 28 December 2017
• 1983 AHSME Problems/Problem 2
  ...radius <math>3</math> cm. The maximum number of intersection points of two circles is <math>\boxed{\textbf{(B)} \ 2}</math>.
  620 bytes (98 words) - 00:39, 20 February 2019
• 1982 AHSME Problems/Problem 14
  ...diameters of circle <math>O, N</math>, and <math>P</math>, respectively. Circles <math>O, N</math>, and <math>P</math> all have radius <math>15</math> and t
  1 KB (248 words) - 22:35, 16 August 2022
• 2015 AMC 8 Problems/Problem 15
  <!--(to editors: this looks really weird)Venn Diagram (I couldn't make circles),
  2 KB (333 words) - 01:41, 15 January 2024
• 2016 AMC 12A Problems/Problem 15
  ...with center <math>Q</math> is externally tangent to each of the other two circles. What is the area of triangle <math>PQR</math>?
  8 KB (1,255 words) - 09:05, 5 September 2022
• 2019 AMC 12A Problems
  ...region, shaded in the figure, inside the larger circle but outside all the circles of radius <math>1</math>? Circles <math>\omega</math> and <math>\gamma</math>, both centered at <math>O</math
  14 KB (2,180 words) - 06:56, 9 June 2024
• 2018 UNCO Math Contest II Problems
  tangent to the leftmost circle. The three circles in the horizontal line. The leftmost and rightmost circles are
  6 KB (1,055 words) - 12:37, 30 July 2021
• 2018 UNCO Math Contest II Problems/Problem 5
  tangent to the leftmost circle. The three circles in the horizontal line. The leftmost and rightmost circles are
  973 bytes (148 words) - 02:13, 14 January 2019
• 2016 AMC 10B Problems/Problem 20
  ...equal to <math>\dfrac{3}{2}</math>, which is the ratio of the radii of the circles. Thus, we are looking for a point <math>(x,y)</math> such that <math>\dfrac Using the ratios of radii of the circles, <math>\frac{3}{2}</math>, we find that the scale factor is <math>1.5</math
  8 KB (1,011 words) - 11:57, 28 February 2024
• 2016 AMC 10B Problems/Problem 21
  ...e quadrants, you get a square of area <math>2</math>, along with four half-circles of diameter <math>\sqrt{2}</math>, for a total area of <math>2+2\cdot(\tfra
  5 KB (761 words) - 23:26, 7 September 2022
• 2016 AIME I Problems/Problem 14
  ...</math> intersects <math>m</math> of the squares and <math>n</math> of the circles. Find <math>m + n</math>. ...one square and one circle, hence this counts <math>288</math> squares and circles. Thus <math>m + n = 286 + 288 = \boxed{574}</math>.
  6 KB (967 words) - 21:08, 22 November 2022
• 2016 AIME I Problems/Problem 15
  Circles <math>\omega_1</math> and <math>\omega_2</math> intersect at points <math>X ...math>\omega_{1},\omega_{2}</math>, it has equal power with respect to both circles, thus <cmath>AZ^{2}=\text{Pow}_{\omega_{1}}(Z)=ZX\cdot ZY=\text{Pow}_{\omeg
  14 KB (2,427 words) - 17:12, 8 January 2024
• 1953 AHSME Problems/Problem 11
  ...ric circles. It is <math>10</math> feet wide. The circumference of the two circles differ by about: ...f a circle is directly proportional to its diameter, the difference in the circles' diameters is simply <math>20\pi </math> feet. Using <math>\pi \approx 3</m
  814 bytes (124 words) - 12:23, 22 April 2020
• 1953 AHSME Problems/Problem 12
  The diameters of two circles are <math>8</math> inches and <math>12</math> inches respectively. The rati ...}</math> where <math>r</math> is the radius. We know that the radii of the circles are <math>4</math> and <math>6</math> inches (half the diameter) so the rat
  848 bytes (134 words) - 14:04, 25 June 2024
• 1986 USAMO Problems/Problem 4
  Two distinct circles <math>K_1</math> and <math>K_2</math> are drawn in the plane. They intersec
  845 bytes (142 words) - 09:59, 19 July 2016
• 1977 AHSME Problems/Problem 15
  ...is then constructed, such that each side of the triangle is tangent to two circles, as shown below. Find the perimeter of the triangle. ...the circles to the sides of the triangle, and lines from the radii of the circles to the vertices of the triangle. Because the triangle is equilateral, the l
  1 KB (200 words) - 21:30, 1 May 2024

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