Difference between revisions of "1979 USAMO Problems/Problem 2"
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| − | Since <math>N</math> is the north pole, we define the Earth with a sphere of radius one in space with <math>N=(0,0,1)</math> | + | Since <math>N</math> is the north pole, we define the Earth with a sphere of radius one in space with <math>N=(0,0,1)</math> and sphere center <math>O=(0,0,0)</math> |
We then pick point <math>N</math> on the sphere and define the <math>xz-plane</math> as the plane that contains great circle points <math>A</math> , <math>B</math>, and <math>N</math> with the <math>x-axis</math> perpendicular to the <math>z-axis</math> and in the direction of <math>A</math>. | We then pick point <math>N</math> on the sphere and define the <math>xz-plane</math> as the plane that contains great circle points <math>A</math> , <math>B</math>, and <math>N</math> with the <math>x-axis</math> perpendicular to the <math>z-axis</math> and in the direction of <math>A</math>. | ||
| − | Using this coordinate system and <math>x</math>, <math>y</math>, and <math>z</math> axes <math>A=(cos(\phi),0,sin(\phi))</math> | + | Using this coordinate system and <math>x</math>, <math>y</math>, and <math>z</math> axes <math>A=(cos(\phi),0,sin(\phi))</math> where <math>\phi</math> is the angle from the <math>xy-plane</math> to <math>A</math> or latitude on this sphere with <math>-\dfrac{\pi}{2}\lt \phi\lt \dfrac{\pi}{2}</math> |
| + | Since <math>A</math> and <math>B</math> are points on a great circle through <math>N</math> equidistant from <math>N</math>, then <math>B=(-cos(\phi),0,sin(\phi))</math> | ||
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| + | Since <math>C</math> is a point on the equator, then <math>C=(cos(\theta),sin(\theta),0)</math> | ||
~Tomas Diaz | ~Tomas Diaz | ||
Revision as of 16:49, 15 September 2023
Contents
Problem
is the north pole. 
and 
are points on a great circle through equidistant from . 
is a point on the equator. Show that the great circle through and bisects the angle 
in the spherical triangle 
(a spherical triangle has great circle arcs as sides).
Hint
Draw a large diagram. A nice, large, and precise diagram. Note that drawing a sphere entails drawing a circle and then a dashed circle (preferably of a different color) perpendicular (in the plane) to the original circle.
Solution
Since is the north pole, we define the Earth with a sphere of radius one in space with 
and sphere center 
We then pick point on the sphere and define the 
as the plane that contains great circle points , , and with the 
perpendicular to the 
and in the direction of .
Using this coordinate system and 
, 
, and 
axes 
where 
is the angle from the 
to or latitude on this sphere with $-\dfrac{\pi}{2}\lt \phi\lt \dfrac{\pi}{2}$ (Error compiling LaTeX. Unknown error_msg)
Since and are points on a great circle through equidistant from , then 
Since is a point on the equator, then 
~Tomas Diaz
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
See Also
| 1979 USAMO (Problems • Resources) | ||
| Preceded by Problem 1 |
Followed by Problem 3 | |
| 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. 
