Difference between revisions of "1979 USAMO Problems/Problem 2"
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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> where <math>\phi</math> is the angle from the <math>xy-plane</math> to <math>A</math> or latitude on this sphere with <cmath> | + | 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 <cmath>\frac{-\pi}{2}\lt \phi \lt \frac{\pi}{2}</cmath> |
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> | 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> |
Revision as of 16:53, 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
\[\frac{-\pi}{2}\lt \phi \lt \frac{\pi}{2}\] (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.