Difference between revisions of "1979 USAMO Problems/Problem 2"
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Since <math>C</math> is a point on the equator, then <math>C=(cos(\theta),sin(\theta),0)</math> where <math>\theta</math> is the angle on the <math>xy-plane</math> from the origin to <math>C</math> or longitude on this sphere with <math>-\pi < \phi \le \pi</math> | Since <math>C</math> is a point on the equator, then <math>C=(cos(\theta),sin(\theta),0)</math> where <math>\theta</math> is the angle on the <math>xy-plane</math> from the origin to <math>C</math> or longitude on this sphere with <math>-\pi < \phi \le \pi</math> | ||
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+ | We note that vectors from the origin to points <math>N</math>, <math>A</math>, <math>B</math>, and <math>C</math> are all unit vectors because all those points are on the unit sphere. | ||
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+ | So, we're going to define points <math>N</math>, <math>A</math>, <math>B</math>, and <math>C</math> as unit vectors with their coordinates. | ||
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+ | We also define the following vectors as follows: | ||
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+ | Vector | ||
~Tomas Diaz | ~Tomas Diaz |
Revision as of 17:05, 15 September 2023
Contents
[hide]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
Since and are points on a great circle through equidistant from , then
Since is a point on the equator, then where is the angle on the from the origin to or longitude on this sphere with
We note that vectors from the origin to points , , , and are all unit vectors because all those points are on the unit sphere.
So, we're going to define points , , , and as unit vectors with their coordinates.
We also define the following vectors as follows:
Vector
~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.