Difference between revisions of "2006 Cyprus MO/Lyceum/Problem 11"
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==Problem== | ==Problem== | ||
| − | + | [[Image:2006 CyMO-11.PNG|250px|right]] | |
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The lines <math>(\epsilon):x-2y=0</math> and <math>(\delta):x+y=4</math> intersect at the point <math>\Gamma</math>. If the line <math>(\delta)</math> intersects the axes <math>Ox</math> and <math>Oy</math> to the points <math>A</math> and <math>B</math> respectively, then the ratio of the area of the triangle <math>OA\Gamma</math> to the area of the triangle <math>OB\Gamma</math> equals | The lines <math>(\epsilon):x-2y=0</math> and <math>(\delta):x+y=4</math> intersect at the point <math>\Gamma</math>. If the line <math>(\delta)</math> intersects the axes <math>Ox</math> and <math>Oy</math> to the points <math>A</math> and <math>B</math> respectively, then the ratio of the area of the triangle <math>OA\Gamma</math> to the area of the triangle <math>OB\Gamma</math> equals | ||
| − | + | <math>\mathrm{(A)}\ \frac{1}{3}\qquad\mathrm{(B)}\ \frac{2}{3}\qquad\mathrm{(C)}\ \frac{3}{5}\qquad\mathrm{(D)}\ \frac{1}{2}\qquad\mathrm{(E)}\ \frac{4}{9}</math> | |
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| − | B | ||
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| − | C | ||
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| − | D | ||
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| − | E | ||
==Solution== | ==Solution== | ||
Latest revision as of 10:37, 27 April 2008
Problem
The lines 
and 
intersect at the point 
. If the line 
intersects the axes 
and 
to the points 
and 
respectively, then the ratio of the area of the triangle 
to the area of the triangle 
equals
Solution
We find some coordinates:
We find the area of triangles:
![$[OAB]=8$](http://latex.artofproblemsolving.com/a/d/c/adc725ea15c033d507efaff92d9aaa5d05fb2021.png)
![$[OA\Gamma]=\frac{\frac{4}{3}*4}{2}=\frac{8}{3}$](http://latex.artofproblemsolving.com/9/d/e/9de3ae98aa4e1561975ed2f9e1037ad34d2bf398.png)
![$[OB\Gamma]=[OAB]-[OA\Gamma]=\frac{16}{3}$](http://latex.artofproblemsolving.com/1/c/d/1cd74717074ad0368af9534e61a7da9f662f4c85.png)
![$\frac{[OA\Gamma]}{[OB\Gamma]}=\frac{1}{2} \Rightarrow \mathrm {(D)}$](http://latex.artofproblemsolving.com/3/0/e/30ecfd35a99fd407b740289a65cb5970efb127c3.png)
See also
| 2006 Cyprus MO, Lyceum (Problems) | ||
| Preceded by Problem 10 |
Followed by Problem 12 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
