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2016 AMC 8 Problems
...to intersect the circles at points <math>C</math> and <math>D</math>. The circles intersect at two points, one of which is <math>E</math>. What is the degree

12 KB (1,665 words) - 06:00, 8 July 2024
2016 AMC 8 Problems/Problem 23
...to intersect the circles at points <math>C</math> and <math>D</math>. The circles intersect at two points, one of which is <math>E</math>. What is the degree

1 KB (238 words) - 10:29, 24 July 2024
Mock AIME 4 Pre 2005/Problems
<math>10</math> lines and <math>10</math> circles divide the plane into at most <math>n</math> disjoint regions. Compute <mat

7 KB (1,094 words) - 15:39, 24 March 2019
2017 AMC 12A Problems/Problem 16
Connect the centers of the tangent circles! (call the center of the large circle <math>C</math>) Notice that we don't even need the circles anymore; thus, draw triangle <math>\Delta ABP</math> with cevian <math>PC</

13 KB (1,982 words) - 17:12, 20 December 2022
2017 AMC 10B Problems
...idpoint of <math>\overline{BC}</math>. What is the sum of the radii of the circles inscribed in <math>\triangle ADB</math> and <math>\triangle ADC</math>?

12 KB (1,930 words) - 20:23, 9 September 2022
2017 AMC 10B Problems/Problem 21
...idpoint of <math>\overline{BC}</math>. What is the sum of the radii of the circles inscribed in <math>\triangle ADB</math> and <math>\triangle ADC</math>? ...using Heron’s formula. We can use that area to find the inradius of the circles by the inradius formula <math>A=sr.</math> Therefore, we get <math>\boxed{\

4 KB (601 words) - 00:34, 8 August 2023
2017 AMC 10B Problems/Problem 18
...cles <math>2</math> and <math>3</math> must be the same, and the colors of circles <math>4</math> and <math>6</math> must be the same. This gives us <math>4</ ...th>4</math>, and <math>6</math> will be the same. Similarly, the colors of circles <math>2</math>, <math>3</math>, and <math>5</math> will be the same. This i

7 KB (1,057 words) - 23:27, 27 August 2022
2017 AMC 12B Problems
.../math>. The line passing through the two points of intersection of the two circles has equation <math>x + y = c</math>. What is <math>c</math>?

15 KB (2,343 words) - 18:26, 25 December 2020
2017 AMC 12B Problems/Problem 9
.../math>. The line passing through the two points of intersection of the two circles has equation <math>x+y=c</math>. What is <math>c</math>? The equations of the two circles are <math>(x+10)^2+(y+4)^2=169</math> and <math>(x-3)^2+(y-9)^2=65</math>.

2 KB (366 words) - 13:54, 15 February 2021
2018 AMC 10A Problems
Two circles of radius <math>5</math> are externally tangent to each other and are inter

14 KB (2,171 words) - 21:10, 4 November 2023
2013 Mock AIME I Problems/Problem 1
Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <m

1 KB (193 words) - 09:01, 30 July 2024
2017 AIME II Problems
...in the figure below. There is one point <math>B</math> inside all of these circles. When <math>r = \frac{11}{60}</math>, the distance from the center <math>C_

7 KB (1,200 words) - 15:02, 8 September 2020
2017 AIME II Problems/Problem 12
...in the figure below. There is one point <math>B</math> inside all of these circles. When <math>r = \frac{11}{60}</math>, the distance from the center <math>C_

6 KB (908 words) - 02:35, 23 January 2024
Circular Inversion
...rsion can be a very useful tool in solving problems involving many tangent circles and/or lines. ...h>. Extend the three semicircles to full circles. Label the resulting four circles as shown in the diagram:

16 KB (2,516 words) - 23:48, 15 January 2024
2017 USAJMO Problems/Problem 5
...e measure, but they point to the same line <math>BC</math>! Hence, the two circles must be congruent. (This is also a well-known result) ...bisector, it must also hit the circle at the point <math>P</math>. The two circles are congruent, which implies <math>MN=MP\implies ND=DP\implies</math> NDP i

6 KB (968 words) - 12:05, 7 June 2024
2017 UNCO Math Contest II Problems
...e has radius 24, a second circle has radius 15, and the centers of the two circles are 52 units apart. A line tangent to both circles crosses the line connecting the two centers at a point

7 KB (1,192 words) - 15:14, 20 August 2020
2017 UNCO Math Contest II Problems/Problem 1
...e has radius 24, a second circle has radius 15, and the centers of the two circles are 52 units apart. A line tangent to both circles crosses the line connecting the two centers at a point

1 KB (211 words) - 18:16, 16 January 2023
2000 IMO Problems/Problem 1
...math> and <math>N</math>. Let <math>AB</math> be the line tangent to these circles at <math>A</math> and <math>B</math>, respectively, so that <math>M</math> ...ct <math>AB</math> at <math>X</math>. By our lemma, <math>\textit{(the two circles are tangent to AB)}</math>, <math>X</math> bisects <math>AB</math>. Since <

3 KB (465 words) - 15:12, 21 June 2024
1954 AHSME Problems/Problem 26
...ne{AB}</math> is divided at <math> C</math> so that <math> AC=3CB</math>. Circles are described on <math> \overline{AC}</math> and <math> \overline{CB}</math ...e radius of the small circle. Draw the line from the center of each of the circles to the point of contact of the tangent of the circle. By similar triangles,

1 KB (184 words) - 17:31, 19 September 2022
2018 AMC 10A Problems/Problem 16
==Solution 2 - Circles== ...th>20,</math> that each contribute <b>two</b> integer lengths (since these circles intersect the hypotenuse twice) from <math>B</math> to <math>\overline{AC}<

3 KB (541 words) - 03:32, 23 January 2023

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