Saudi Arabia Mathematical Competitions

Question

A Geostationary Earth Orbit is situated directly above the equator and has a period equal to the Earth's rotational period. It is at the precise distance of

22, 236

miles above the Earth that a satellite can maintain an orbit with a period of rotation around the Earth exactly equal to

24

hours. Because the satellites revolve at the same rotational speed of the Earth, they appear stationary from the Earth surface. That is why most station antennas (satellite dishes) do not need to move once they have been properly aimed at a target satellite in the sky. In an international project, a total of ten stations were equally spaced on this orbit (at the precise distance of

22, 236

miles above the equator). Given that the radius of the Earth is

3960

miles, find the exact straight distance between two neighboring stations. Write your answer in the form

a + b c

, where

a, b, c

are integers and

c > 0

is square-free.

Accepted Answer

Let A and B be two neighboring stations. We have A O B=5, hence A B=2 R 10, where R=22236+3960=26196. We will prove that 10=5-14. FIGURE_0 Since =0, then (2 5+3 5)=0. We have: gathered 2 5 3 5+ 2 5 3 5=0 2 5 5(4 ^3 5-3 5) +(2 ^2 5-1)(3 5-4 ^3 5)=0 gathered We may divide by 5 and we get gathered 2 5(4 ^3 5-3 5) +(2 ^2 5-1)(-1+4 ^2 5)=0 gathered Denote y= 5. Then, our equation becomes 16 y^4-12 y^2+1=0 with the solutions given by y^2=3 58. Since 6<5<4, then 5 (22, 32), so that y^2=3+58 and 5=3+58=5+14 Now, 10=1- 52=5-14. It follows A B=26196 5-12=-13098+13098 5 FIGURE_0