1973 USAMO Problems/Problem 1
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
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
1973 USAMO (Problems • Resources) | ||
Preceded by First Question |
Followed by Problem 2 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |