Difference between revisions of "2019 AIME I Problems/Problem 11"

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The 2019 AIME I takes place on March 13, 2019.
 
The 2019 AIME I takes place on March 13, 2019.
 
==Problem 11==
 
==Problem 11==
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In <math>\triangle ABC</math>, the sides have integers lengths and <math>AB=AC</math>. Circle <math>\omega</math> has its center at the incenter of <math>\triangle ABC</math>. An excircle of <math>\triangle ABC</math> is a circle in the exterior of <math>\triangle ABC</math> that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to <math>\overline{BC}</math> is internally tangent to <math>\omega</math>, and the other two excircles are both externally tangent to <math>\omega</math>. Find the minimum possible value of the perimeter of <math>\triangle ABC</math>.
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==Solution==
 
==Solution==
 
==See Also==
 
==See Also==
 
{{AIME box|year=2019|n=I|num-b=10|num-a=12}}
 
{{AIME box|year=2019|n=I|num-b=10|num-a=12}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 20:22, 14 March 2019

The 2019 AIME I takes place on March 13, 2019.

Problem 11

In $\triangle ABC$, the sides have integers lengths and $AB=AC$. Circle $\omega$ has its center at the incenter of $\triangle ABC$. An excircle of $\triangle ABC$ is a circle in the exterior of $\triangle ABC$ that is tangent to one side of the triangle and tangent to the extensions of the other two sides. Suppose that the excircle tangent to $\overline{BC}$ is internally tangent to $\omega$, and the other two excircles are both externally tangent to $\omega$. Find the minimum possible value of the perimeter of $\triangle ABC$.

Solution

See Also

2019 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 10
Followed by
Problem 12
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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