Difference between revisions of "1984 USAMO Problems/Problem 3"
(Created page with "== Problem == <math>P</math>, <math>A</math>, <math>B</math>, <math>C</math>, and <math>D</math> are five distinct points in space such that <math>\angle APB = \angle BPC = \a...") |
(→Solution) |
||
| (8 intermediate revisions by 2 users not shown) | |||
| Line 3: | Line 3: | ||
== Solution == | == Solution == | ||
| − | {{ | + | |
| + | Greatest value is achieved when all the points are as close as possible to all being on a plane. | ||
| + | |||
| + | 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 > 0</math>, then <math>\angle APC + \angle BPD > 0</math> | ||
| + | |||
| + | and the inequality for this problem is: | ||
| + | |||
| + | <math>0 < \angle APC + \angle BPD < \pi</math> | ||
| + | |||
| + | ~Tomas Diaz. orders@tomasdiaz.com | ||
| + | |||
| + | {{alternate solutions}} | ||
== See Also == | == See Also == | ||
| Line 10: | Line 23: | ||
[[Category:Olympiad Geometry Problems]] | [[Category:Olympiad Geometry Problems]] | ||
| + | [[Category:3D Geometry Problems]] | ||
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
