1984 USAMO Problems/Problem 3

Problem

$P$ , $A$ , $B$ , $C$ , and $D$ are five distinct points in space such that $\angle APB = \angle BPC = \angle CPD = \angle DPA = \theta$ , where $\theta$ is a given acute angle. Determine the greatest and least values of $\angle APC + \angle BPD$ .

Solution

Greatest value is achieved when all the points are as close as possible to all being on a plane.

Since $\theta \le \frac{\pi}{2}$ , then $\angle APC + \angle BPD le \pi$

Smallest value is achieved when point P is above and the remaining points are as close as possible to colinear when $\theta \ge 0$ , then $\angle APC + \angle BPD \ge 0$

and the inequality for this problem is:

$0 \le \angle APC + \angle BPD le \pi$

~Tomas Diaz. orders@tomasdiaz.com

Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.

1984 USAMO (Problems • Resources)
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1984 USAMO Problems/Problem 3

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