# PC2024-P4 Best solutions and final results

Problem 4 – Satellite

In total, 91 solutions were submitted for problem 4, of which 71 were correct. Congratulations to everyone who managed to solve it!

The Best Solution Awards were selected from the 45 correct solutions sent before the first hint was published. Selecting a few solutions from such a large number, most of which were written very nicely and typeset in LaTeX, was an extremely difficult task.

A majority of the solutions made use of the fact that the hodograph of a body moving along an elliptical orbit in a coulombic field is a circle; however, there was a wide variety of how this fact was proved, and in the way how it was used. The easiest way to prove this fact is by using a geometrical interpretation of the Runge-Lenz vector. Namely, the Runge-Lenz vector is a sum of two vectors one of which is a unit vector pointing from the gravitation centre to the orbiting body (sometimes scaled with a constant factor) , and another one is proportional to the cross product of the velocity vector of the orbiting body with its angular momentum. Geometrically, the three vectors (the Runge-Lenz vector and the two vectors in the sum) form a triangle. Since (a) the Rungle-Lenz vector is a conserved vector, i.e. remains constant, and (b) the first vector in the sum has a constant length, when the starting point of the second vector is at rest and while the orbiting body performs a full cycle, the endpoint of the second vector draws a circle. The easiest way to use the fact that the hodograph is a circle is also geometrical: one needs to notice that if a point $A$ lies on a circle $a$, and a point $B$ lies on a circle $B$ then the largest possible distance between the points is achieved when the points lie in the diametrically opposite positions on the symmetry axis of the system.

The first best-solution award of 0.3 goes to Sainavaneet Mukund for a clean and short solution which was actually the only one to use a fully geometric approach as described above (a minor omission is the definition of $\hat n$ – a unit vector perpendicular to the plane).

Next, both Val Karan and Andrei Vila are awarded with a bonus of 0.2, for nicely structured and well-documented solutions with concise proofs of the Runge-Lenz vector. Their final results are obtained analytically using the triangle inequality. The two solutions differ significantly. (Val Karan does have a geometrical interpretation for the special case of $\alpha=90^\circ$, but that was added quite a bit later into his updated submission.)

The remaining bonus is shared equally, 0.15+0.15, between the two solutions handling correctly the case when the satellites orbit in opposite directions. Although not stated formally, it was implied that the satellites orbit in the same direction (almost all the satellites do orbit in the same direction as Earth rotates around its axis). However, with the current statement of the problem, a fully correct solution would need to consider separately the cases when the satellites rotate both in the same direction, and when one rotates clockwise while the other – counterclockwise. This can be done easily by copying the calculations performed for one case to the other case; as a result, the sign before $\cos\alpha$ in the answer is changed from minus to plus. And so, one solution correctly handling both cases was written by Lukas Schicht; this is also an example of a solution by which the circular shape of the hodograph was derived without using the Runge-Lenz vector. The other such solution was made by Ayaan Maan; this is also an example of a “brute-force” solution that does not use the fact that the hodographs are circles.

Final results for Problem 4

Pre-university students:

University students:

The list of countries:
Algeria, Argentina, Armenia, Australia, Austria, Azerbaijan, Bahrain, Bangladesh, Belarus, Belgium, Bosnia and Herzegovina, Brazil, Bulgaria, Cambodia, Cameroon, Canada, China, Croatia, Cuba, Czech Republic, Ecuador, Egypt, El Salvador, Estonia, Finland, France, Georgia, Germany, Greece, Guatemala, Honduras, Hong Kong, Hungary, India, Indonesia, Iran, Israel, Italy, Japan, Kazakhstan, Kenya, Kosovo, Kyrgyzstan, Latvia, Lithuania, Macau, Malaysia, Mexico, Moldova, Montenegro, Nepal, Netherlands, Nigeria, North Korea, North Macedonia, Pakistan, Peru, Philippines, Poland, Romania, Russia, Rwanda, Saudi Arabia, Serbia, Singapore, Slovakia, Slovenia, South Africa, South Korea, Spain, Sri Lanka, Suriname, Sweden, Switzerland, Syria, Taiwan, Tajikistan, Thailand, Turkey, Turkmenistan, Ukraine, United Arab Emirates, United Kingdom, United States, Uzbekistan, Vietnam.