1973 USAMO Problems/Problem 1

Problem

Two points and lie in the interior of a regular tetrahedron . Prove that angle .

Solution

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By Vo Duc Dien

Let the side length of the regular tetrahedron be . 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 < , ∠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 (Problems • Resources)
Preceded by First Question	Followed by Problem 2
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

By Vo Duc Dien

Let the side length of the regular tetrahedron be . 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 < , ∠IAJ < ∠BAD = 60°. Therefore, ∠PAQ < 60°.

Solution with graphs posted at

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