Difference between revisions of "2006 Romanian NMO Problems/Grade 9/Problem 3"

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==Problem==
 
==Problem==
We have a quadrilateral <math>ABCD</math> inscribed in a circle of radius <math>r</math>, for which there is a point <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</math>.
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We have a [[quadrilateral]] <math>ABCD</math> [[inscribe]]d in a [[circle]] of [[radius]] <math>r</math>, for which there is a [[point]] <math>P</math> on <math>CD</math> such that <math>CB=BP=PA=AB</math>.
  
 
(a) Prove that there are points <math>A,B,C,D,P</math> which fulfill the above conditions.
 
(a) Prove that there are points <math>A,B,C,D,P</math> which fulfill the above conditions.
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''Virgil Nicula''
 
''Virgil Nicula''
 
==Solution==
 
==Solution==
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{{solution}}
 
==See also==
 
==See also==
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*[[2006 Romanian NMO Problems/Problem 2 | Previous problem]]
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*[[2006 Romanian NMO Problems/Problem 4 | Next problem]]
 
*[[2006 Romanian NMO Problems]]
 
*[[2006 Romanian NMO Problems]]
 
[[Category: Olympiad Geometry Problems]]
 
[[Category: Olympiad Geometry Problems]]

Revision as of 00:10, 11 November 2006

Problem

We have a quadrilateral $ABCD$ inscribed in a circle of radius $r$, for which there is a point $P$ on $CD$ such that $CB=BP=PA=AB$.

(a) Prove that there are points $A,B,C,D,P$ which fulfill the above conditions.

(b) Prove that $PD=r$.

Virgil Nicula

Solution

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See also