Difference between revisions of "1973 USAMO Problems/Problem 1"
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{{USAMO box|year=1973|before=First Question|num-a=2}} | {{USAMO box|year=1973|before=First Question|num-a=2}} | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] | ||
Revision as of 07:26, 16 September 2012
Problem
Two points 
and 
lie in the interior of a regular tetrahedron 
. Prove that angle 
.
Solution
Let the side length of the regular tetrahedron be 
. Link and extend 
to meet the plane containing triangle 
at 
; link 
and extend it to meet the same plane at 
. We know that and are inside triangle and that 
Now let’s look at the plane containing triangle with points and inside the triangle. Link and extend 
on both sides to meet the sides of the triangle at 
and 
, on 
and on 
. We have 
But since and are interior of the tetrahedron, points and cannot be both at the vertices and 
, 
. Therefore, 
.
Solution with graphs posted at
http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.USA1973Problem1
See Also
| 1973 USAMO (Problems • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
