Difference between revisions of "1972 USAMO Problems/Problem 5"
(New page: ==Problem== A given convex pentagon <math>ABCDE</math> has the property that the area of each of the five triangles <math>ABC</math>, <math>BCD</math>, <math>CDE</math>, <math>DEA</math>, ...) |
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| − | {{USAMO box|year=|num-b= | + | {{USAMO box|year=|num-b=4|after=Last Question}} |
| − | [[Category:Olympiad Problems]] | + | [[Category:Olympiad Geometry Problems]] |
Revision as of 15:52, 30 December 2008
Problem
A given convex pentagon 
has the property that the area of each of the five triangles 
, 
, 
, 
, and 
is unity. Show that all pentagons with the above property have the same area, and calculate that area. Show, furthermore, that there are infinitely many non-congruent pentagons having the above area property.
![[asy] size(80); defaultpen(fontsize(7)); pair A=(0,7), B=(5,4), C=(3,0), D=(-3,0), E=(-5,4), P; P=extension(B,D,C,E); draw(D--E--A--B--C--D--B--E--C); label("A",A,(0,1));label("B",B,(1,0));label("C",C,(1,-1));label("D",D,(-1,-1));label("E",E,(-1,0));label("P",P,(0,1)); [/asy]](http://latex.artofproblemsolving.com/3/2/3/323bdb5c3f41250e3db1af79ed1bd258fad55799.png)
Solution
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See also
| USAMO (Problems • Resources) | ||
| Preceded by Problem 4 |
Followed by Last Question | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
