Difference between revisions of "2011 USAMO Problems/Problem 5"
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==Solution== | ==Solution== | ||
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| + | First note that <math>\overline{Q_1 Q_2} \parallel \overline{AB}</math> if and only if the altitudes from <math>Q_1</math> and <math>Q_2</math> to <math>\overline{AB}</math> are the same, or <math>|Q_1B|\sin \angle ABQ_1 =|Q_2A|\sin \angle BAQ_2</math>. Similarly <math>\overline{Q_1 Q_2} \parallel \overline{CD}</math> iff <math>|Q_1C|\sin \angle DCQ_1 =|Q_2D|\sin \angle CDQ_2</math>. | ||
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| + | If we define <math>S =\dfrac{|Q_1B|\sin \angle ABQ_1}{|Q_2A|\sin \angle BAQ_2}\dfrac{|Q_2D|\sin \angle CDQ_2}{|Q_1C|\sin \angle DCQ_1}</math>, then we are done if we can show that S=1. | ||
==See also== | ==See also== | ||
Revision as of 16:13, 8 June 2011
Problem
Let 
be a given point inside quadrilateral 
. Points 
and 
are located within such that 
, 
, 
, 
. Prove that 
if and only if 
.
Solution
This problem needs a solution. If you have a solution for it, please help us out by adding it.
First note that if and only if the altitudes from and to 
are the same, or 
. Similarly iff 
.
If we define 
, then we are done if we can show that S=1.
See also
| 2011 USAMO (Problems • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
| 1 • 2 • 3 • 4 • 5 • 6 | ||
| All USAMO Problems and Solutions | ||
