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Difference between revisions of "2007 USAMO Problems/Problem 6"

 
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== Problem ==
 
== Problem ==
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Let <math>ABC</math> be an acute triangle with <math>\omega</math>, <math>\Omega</math>, and <math>R</math> being its incircle, circumcircle, and circumradius, respectively.  Circle <math>\omega_A</math> is tangent internally to <math>\Omega</math> at <math>A</math> and tangent externally to <math>\omega</math>.  Circle <math>\Omega_A</math> is tangent internally to <math>\Omega</math> at <math>A</math> and tangent internally to <math>\omega</math>.  Let <math>P_A</math> and <math>Q_A</math> denote the centers of <math>\omega_A</math> and <math>\Omega_A</math>, respectively.  Define points <math>P_B</math>, <math>Q_B</math>, <math>P_C</math>, <math>Q_C</math> analogously.  Prove that
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<math>
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8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,
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</math>
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with equality if and only if triangle <math>ABC</math> is equilateral.
  
 
== Solution ==
 
== Solution ==
  
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== See also ==
 
 
{{USAMO newbox|year=2007|num-b=5|after=Last Question}}
 
{{USAMO newbox|year=2007|num-b=5|after=Last Question}}

Revision as of 17:09, 25 April 2007

Problem

Let $ABC$ be an acute triangle with $\omega$, $\Omega$, and $R$ being its incircle, circumcircle, and circumradius, respectively. Circle $\omega_A$ is tangent internally to $\Omega$ at $A$ and tangent externally to $\omega$. Circle $\Omega_A$ is tangent internally to $\Omega$ at $A$ and tangent internally to $\omega$. Let $P_A$ and $Q_A$ denote the centers of $\omega_A$ and $\Omega_A$, respectively. Define points $P_B$, $Q_B$, $P_C$, $Q_C$ analogously. Prove that

$8P_AQ_A \cdot P_BQ_B \cdot P_CQ_C \le R^3,$

with equality if and only if triangle $ABC$ is equilateral.

Solution

2007 USAMO (Problems • Resources)
Preceded by
Problem 5 		Followed by
Last Question
1 • 2 • 3 • 4 • 5 • 6
All USAMO Problems and Solutions
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