1984 USAMO Problems/Problem 3: Difference between revisions
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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 | Since <math>\theta < \frac{\pi}{2}</math>, then <math>\angle APC + \angle BPD < \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 | Smallest value is achieved when point P is above and the remaining points are as close as possible to colinear when <math>\theta > 0</math>, then <math>\angle APC + \angle BPD > 0</math> | ||
and the inequality for this problem is: | and the inequality for this problem is: | ||
<math>0 | <math>0 < \angle APC + \angle BPD < \pi</math> | ||
~Tomas Diaz. orders@tomasdiaz.com | ~Tomas Diaz. orders@tomasdiaz.com | ||
Latest 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 | ||
These problems are copyrighted © by the Mathematical Association of America. 
