# 2021-P2 Best Solutions and Final Results

Problem 2 – Comet’s orbit

By 7th March 2021, there are 992 registered participants from 71 countries (the list of countries is given below). During the five weeks, 151 solutions were submitted for problem 2, out of which 95 were correct.

This problem was clearly a big success as there was a big number of correct solutions with completely different approaches which showed that the problem was a great source of inspiration for the contestants. As usual, the main contenders for the best solutions were the solutions submitted before the publication of the first hint – there were 65 such solutions. Out of those 65 solutions, I picked 11 well-documented not-too-long solutions which demonstrated different approaches. The ranking list gives preference to the shortest solutions and to creative ideas.

1. Lotus Chen, 45% of the bonus – by far the shortest solution, not surprisingly based on the eccentricity vector.
2. Eddie Chen, 25% of the bonus – also a very short solution, bypasses the eccentricity vector, mostly geometrical.
3. Sangyul Lee, 15% of the bonus  – an analytical solution based on the formula of the ellipse in Cartesian coordinates and generalizing the result.
4. Nguyen Manh Quan, 15% of the bonus – a solution bypassing the eccentricity vector and using a series of clever geometrical constructions, and application of Apollonian circles.

Apart from these, there are solutions that are also very interesting and deserve publication, hence earning the publication bonus factor of 1.1. Jianzheng Deng and Oliwier Urbanski – solutions that start with the eccentricity vector and then make use of clever geometry. Yu Lu – a mostly analytical solution based on the formula of the ellipse in polar coordinates and making use of the Lagrange multiplier. Absur Khan Siam and Fang Tzu, Hsu – solutions making use of the fact that the locus of the intersection point of two mutually perpendicular tangents to an ellipse is a circle. Arhaan Ahmad –  a solution making use of the fact that the locus of the intersection point of a tangent to an ellipse with its perpendicular drawn through a focus is a circle. Ricards Kristers Knipsisa nicely written solution based on the fact that the hodograph is a circle (which, as a matter of fact, has been used in many solutions and follows immediately from the eccentricity vector, here proved in a different way).

Finally, we are publishing here the solutions by the author of the problem, Thomas Foster, and by the academic head of Physics Cup, Jaan Kalda.

Final results for Problem 2

Pre-university students

University students

Participating countries: Algeria, Azerbaijan, Australia, Bangladesh, Belarus, Belgium, Bosnia and Herzegovina, Bolivia, Brazil, Bulgaria, Cambodia, Canada, China, Croatia, Czechia, Egypt, El Salvador, Estonia, Finland, Georgia, Germany, Greece, Hungary, India, Indonesia, Iran, Israel, Italy, Japan, Kazakhstan, Kenya, Korea, Kyrgyzstan, Latvia, Lebanon, Lithuania, Macedonia, Malaysia, Mexico, Moldova, Mongolia, Nepal, Nigeria, North Macedonia, Pakistan, Peru, Philippines, Poland, Romania, Russia, Saudi Arabia, Serbia, Singapore, Slovakia, Slovenia, South Africa, South Korea, Spain, Sri Lanka, Sweden, Switzerland, Syria, Taiwan, Thailand, Turkey, Turkmenistan, United Arab Emirates, United Kingdom, United States, Uzbekistan, Vietnam.