Difference between revisions of "1984 USAMO Problems/Problem 3"
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Greatest value is achieved when all the points are as close as possible to all being on a plane. | Greatest value is achieved when all the points are as close as possible to all being on a plane. | ||
| − | Since <math>\theta \le \frac{\pi}{2}</math>, then <math>\angle APC + \angle BPD le \pi</math> | + | Since <math>\theta \le \frac{\pi}{2}</math>, then <math>\angle APC + \angle BPD \le \pi</math> |
Smallest value is achieved when point P is above and the remaining points are as close as possible to colinear when <math>\theta \ge 0</math>, then <math>\angle APC + \angle BPD \ge 0</math> | Smallest value is achieved when point P is above and the remaining points are as close as possible to colinear when <math>\theta \ge 0</math>, then <math>\angle APC + \angle BPD \ge 0</math> | ||
Revision as of 01:26, 21 November 2023
Problem
, 
, 
, 
, and 
are five distinct points in space such that 
, where 
is a given acute angle. Determine the greatest and least values of 
.
Solution
Greatest value is achieved when all the points are as close as possible to all being on a plane.
Since 
, then 
Smallest value is achieved when point P is above and the remaining points are as close as possible to colinear when 
, then 
and the inequality for this problem is:
~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.
See Also
| 1984 USAMO (Problems • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 | ||
| All USAMO Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
