Difference between revisions of "1992 USAMO Problems/Problem 4"
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| + | == Resources == | ||
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| + | {{USAMO box|year=1992|num-b=3|num-a=5}} | ||
| + | * [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=356413#p356413 Discussion on AoPS/MathLinks] | ||
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| + | [[Category:Olympiad Algebra Problems]] | ||
Revision as of 10:52, 22 April 2010
Problem
Chords 
, 
, and 
of a sphere meet at an interior point 
but are not contained in the same plane. The sphere through 
, 
, 
, and is tangent to the sphere through 
, 
, 
, and . Prove that 
.
Solution
Consider the plane through 
. This plane, of course, also contains . We can easily find the 
is isosceles because the base angles are equal. Thus, 
. Similarly, 
. Thus, 
. By symmetry, 
and 
, and hence as desired.
Resources
| 1992 USAMO (Problems • Resources) | ||
| Preceded by Problem 3 |
Followed by Problem 5 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
