https://artofproblemsolving.com/wiki/index.php?title=2013_Mock_AIME_I_Problems/Problem_1&feed=atom&action=history2013 Mock AIME I Problems/Problem 1 - Revision history2024-03-28T10:03:06ZRevision history for this page on the wikiMediaWiki 1.31.1https://artofproblemsolving.com/wiki/index.php?title=2013_Mock_AIME_I_Problems/Problem_1&diff=84407&oldid=prevRocketscience: Created page with "== Problem 1 == Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <math>A_1</math> and <math>A_2</math> such that <math>A_1A_2=6</math>. Le..."2017-03-05T04:05:38Z<p>Created page with "== Problem 1 == Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <math>A_1</math> and <math>A_2</math> such that <math>A_1A_2=6</math>. Le..."</p>
<p><b>New page</b></p><div>== Problem 1 ==<br />
Two circles <math>C_1</math> and <math>C_2</math>, each of unit radius, have centers <math>A_1</math> and <math>A_2</math> such that <math>A_1A_2=6</math>. Let <math>P</math> be the midpoint of <math>A_1A_2</math> and let <math>C_3</math> be a circle externally tangent to both <math>C_1</math> and <math>C_2</math>. <math>C_1</math> and <math>C_3</math> have a common tangent that passes through <math>P</math>. If this tangent is also a common tangent to <math>C_2</math> and <math>C_1</math>, find the radius of circle <math>C_3</math>.<br />
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== Solution ==<br />
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Let <math>A_3</math> be the center of circle <math>C_3</math> and <math>Q</math> be the point of tangency between <math>C_3</math> and <math>C_2</math>. Note that triangles <math>PQA_2</math> and <math>A_3PA_2</math> are similar, so <math>\frac{A_3A_2}{PA_2}=\frac{PA_2}{A_2Q}=3</math> and <math>A_3A_2=r+1=9</math>. Thus the radius of <math>C_3</math> is <math>\boxed{8}</math>.</div>Rocketscience