Difference between revisions of "1973 USAMO Problems/Problem 1"
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==Solution== | ==Solution== | ||
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− | + | Let the side length of the regular tetrahedron be <math>a</math>. Link and extend <math>AP</math> to meet the plane containing triangle <math>BCD</math> at <math>E</math>; link <math>AQ</math> and extend it to meet the same plane at <math>F</math>. We know that <math>E</math> and <math>F</math> are inside triangle <math>BCD</math> and that <math>\angle PAQ = \angle EAF</math> | |
− | + | Now let’s look at the plane containing triangle <math>BCD</math> with points <math>E</math> and <math>F</math> inside the triangle. Link and extend <math>EF</math> on both sides to meet the sides of the triangle <math>BCD</math> at <math>I</math> and <math>J</math>, <math>I</math> on <math>BC</math> and <math>J</math> on <math>DC</math>. We have <math>\angle EAF < \angle IAJ</math> | |
− | + | But since <math>E</math> and <math>F</math> are interior of the tetrahedron, points <math>I</math> and <math>J</math> cannot be both at the vertices and <math>IJ < a</math>, <math>\angle IAJ < \angle BAD = 60</math>. Therefore, <math>\angle PAQ < 60</math>. | |
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− | But since E and F are interior of the tetrahedron, points I and J cannot be both at the vertices and | ||
Solution with graphs posted at | Solution with graphs posted at |
Revision as of 10:04, 8 July 2010
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 |