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:

First nameSurnameCountrySchoolP4 date time
DD/MM/YY 00:00:00
P4 Score
ValKaranCroatiaXV. Gimnazija Zagreb25/2/24 14:003.0135
Ralf RobertPaaboEstoniaHugo Treffneri Gümnaasium25/2/24 16:172.3579
Ilan MiheljaIvan HegedićCroatiaXV. gymnasium Zagreb, V. gymnasium Zagreb25/2/24 17:452.1436
GuyGinzbergIsraelOrt Irony Dalet Modiin25/2/24 18:001.9487
SaskiaPõldmaaEstoniaTallinna Reaalkool25/2/24 18:181.683
Mohamad EidAl-aktaaSyriaAl saadeh school25/2/24 19:401.6105
SainavaneetMukundIndiaALLEN Career Institute26/2/24 11:021.4848
MihkelRannutEstoniaTallinna Reaalkool25/2/24 21:471.4641
Rares FelixTudoseRomaniaLiceul Teoretic Internațional de Informatică București25/2/24 23:001.331
AyaanMaanIndiaOPG World School2/3/24 09:181.1618
AndreiVilaRomaniaInternational Computer High School Bucharest26/2/24 14:591.1618
AgastyaGoelUnited StatesHenry M. Gunn High School26/2/24 01:051.1495
OskariVirtanenFinlandLahden Lyseo6/3/24 20:081.0
PulkitGuptaIndiaOPG world school1/3/24 17:521.0
ArdaÜnverTurkeyKocaeli Gebze Bahçeşehir High School for Science and Technology26/2/24 22:571.0
ZeynebMehemmedliAzerbaijanTTEL9/3/24 20:571.0
FelixYewMalaysiaNational Junior College, Singapore27/2/24 10:231.0
TomasRazbadauskasLithuaniaKaunas University of Technology Gymnasium27/2/24 19:421.0
MatejGelevNorth MacedoniaYahja Kemal5/3/24 09:021.0
AnimeshPradhanIndiaModern Delhi International School29/2/24 12:311.0
OytunErTurkeyKadıköy Anatolian Highschool1/3/24 19:381.0
Bahaa AldeenMelhemSyriaAlbasel Mixed High School9/3/24 15:141.0
MendelMendelsohnRomaniaInternational Computer Highschool of Bucharest9/3/24 18:551.0
KaustubhKakaIndiaALLEN28/2/24 10:591.0
DariusGraurMoldovaLiceul Teoretic Orizont9/3/24 14:451.0
YangguangZhengChinaThe affilicated High School of South China normal university27/2/24 01:531.0
EricWangUnited StatesWest Windsor-Plainsboro High School North28/2/24 17:331.0
KutayKulbakTurkeyBahcesehir High School for Science and Technology16/3/24 22:230.9
Zhao-GuoWangTaiwanChing Cheng High School3/3/24 14:120.9
ShivanshJaiswalIndiaFIITJEE South Delhi10/3/24 18:230.9
YifuGuoChinaLicheng No. 2 Middle School6/3/24 22:440.9
TornikeTsadzikidzeGeorgiakomarovi Physics-Mathematics school15/3/24 18:160.81
LucaGâleaRomaniaNational College Iasi18/3/24 00:400.8
AntonTrubitsynRussiaFML №23923/3/24 17:100.8
DakshTayaliaIndiaSRGS OMR BLR24/3/24 10:550.8
TianyiDongChinaShanghai High School23/3/24 09:070.8
RuichengMaChinaHigh School Affiliated to Harbin Teachers College18/3/24 16:520.648
Vincent KarlWillisAustriaBRG Innsbruck APP2/4/24 20:040.6
Tirumala TejasPenugondaUnited Arab EmiratesGEMS United Indian School4/4/24 14:520.6
AffanKaknjoBosnia and HerzegovinaJU Prva gimnazija u Zenici4/5/24 18:390.5
Hung-YuShihTaiwanTainan First Senior High School23/4/24 13:380.5
JumadurdyNuryyevTurkmenistanSpecialized Comprehensive Boarding School named after Saparmyrat Turkmenbashy in the city of Ashgabat19/5/24 10:480.45

University students:

First nameSurnameCountryUniversityP4 date time
DD/MM/YY 00:00:00
P4 Score
LukasSchichtGermanyEberhard Karls Universität Tübingen25/2/24 14:192.7395
Bayram AlpSahinTurkeyBogazici University25/2/24 13:392.5937
LucaMilaneseItalyScuola Normale Superiore di Pisa25/2/24 17:232.1436
LukaPassek-KumeričkiCroatiaUniversity of Zagreb25/2/24 22:191.9487
KarloMaksimovićCroatiaUniversity of Zagreb26/2/24 07:421.7716
ViktorVukovićCroatiaUniversity of Zagreb26/2/24 11:431.6105
YUANFEIHong KongThe Hong Kong Polytechnic University26/2/24 11:571.4641
DhruvShahIndiaIndian Institute of Science26/2/24 18:251.331
Ahmet BahadırTrabzonTurkeyBoğaziçi University26/2/24 21:291.1495
TomasBabelisLithuaniaÉcole Polytechnique27/2/24 08:161.045
Adrian, Bruno, DanielSerrano Capatina, Pontecorvo, Gonzalez FilipovSwitzerlandEPFL9/3/24 10:371.0
MarinusLehmannGermanyUniversity of Potsdam27/2/24 23:591.0
DženanMidžićBosnia and HerzegovinaUniversity of Rijeka29/2/24 22:161.0
Kai-HsiangTuTaiwanNational Taiwan University10/3/24 11:441.0
VadimIonasMoldovaDenmark Technical University9/3/24 16:311.0
HexinGAOHong KongHong Kong Polytechnic University28/2/24 13:511.0
VictorVoronaMoldova29/2/24 07:381.0
SIfat AhmedTusharBangladeshUniversity of Dhaka16/3/24 14:130.9
Victor-IoanDumbravaRomaniaÉcole Polytechnique Paris10/3/24 19:400.9
KonstantinRodionenkoRussiaITMO University17/3/24 07:590.9
LarranceXingUnited StatesUniversity of Michigan Ann Arbor15/3/24 21:240.9
NickHartmannGermanyRWTH Aachen23/3/24 21:120.8
Md. BadhanMridhaBangladeshUniversity of Dhaka31/3/24 11:180.7
KerimCherkezovTurkmenistanMagtymguly Turkmen State University26/3/24 13:580.63
MusaArtykovTurkmenistanMagtymguly Turkmen State University28/3/24 06:570.63
MichałKołodziejczykPolandWarsaw University of Technology24/3/24 20:160.567
KonstantinTripunovskiNorth MacedoniaUniversity of Ljubljana – Faculty of Medicine8/4/24 13:340.5
MayankKumarIndiaIndian Institute of Science14/4/24 11:350.45
Elias KoschierMichail OgnyanovAustria14/4/24 12:570.405

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.