Difference between revisions of "1973 USAMO Problems/Problem 1"

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[[Category:Olympiad Geometry Problems]]
 
[[Category:Olympiad Geometry Problems]]
 
By Vo Duc Dien
 
 
Let the side length of the regular tetrahedron be <math>a</math>. Link and extend AP to meet the plane containing triangle BCD at E; link AQ and extend it to meet the same plane at F. We know that E and F are inside triangle BCD and that &#8736;PAQ = &#8736;EAF
 
 
Now let’s look at the plane containing triangle BCD with points E and F inside the triangle. Link and extend EF on both sides to meet the sides of the triangle BCD at I and J, I on BC and J on DC. We have  &#8736;EAF < &#8736;IAJ
 
 
But since I and J are on the sides and not on the vertices, IJ < <math>a</math>, &#8736;IAJ < &#8736;BAD = 60°. Therefore, &#8736;PAQ < 60°.
 
 
Solution with graphs posted at
 
 
http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.USA1973Problem1
 

Revision as of 02:20, 30 January 2010

Problem

Two points $P$ and $Q$ lie in the interior of a regular tetrahedron $ABCD$. Prove that angle $PAQ<60^o$.

Solution

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By Vo Duc Dien

Let the side length of the regular tetrahedron be $a$. Link and extend AP to meet the plane containing triangle BCD at E; link AQ and extend it to meet the same plane at F. We know that E and F are inside triangle BCD and that ∠PAQ = ∠EAF

Now let’s look at the plane containing triangle BCD with points E and F inside the triangle. Link and extend EF on both sides to meet the sides of the triangle BCD at I and J, I on BC and J on DC. We have ∠EAF < ∠IAJ

But since I and J are on the sides and not on the vertices, IJ < $a$, ∠IAJ < ∠BAD = 60°. Therefore, ∠PAQ < 60°.

Solution with graphs posted at

http://www.cut-the-knot.org/wiki-math/index.php?n=MathematicalOlympiads.USA1973Problem1

See also

1973 USAMO (ProblemsResources)
Preceded by
First Question
Followed by
Problem 2
1 2 3 4 5
All USAMO Problems and Solutions