Difference between revisions of "1979 USAMO Problems/Problem 2"
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+ | <math>\overrightarrow{V_{CA}}=\left\langle V_{CA_{x}},V_{CA_{y}},sin(\phi) \right\rangle</math> | ||
Revision as of 18:34, 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
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 is the unit vector in the direction of arc
and tangent to the great circle of
at
Vector is the unit vector in the direction of arc
and tangent to the great circle of
at
Vector is the unit vector in the direction of arc
and tangent to the great circle of
at
To calculate each of these vectors we shall use the cross product as follows:
Vector :
Since we're only interested in the component of the 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.