PC2025-P5 Best solutions and final results

Problem 5 – Satellite Orbit

In total, 137 solutions were submitted for problem 5, of which 104 were correct. Congratulations to everyone who successfully solved it! Of the 77 correct solutions that were submitted before the first hint was published, 30 were handwritten; only the 47 typeset submissions were considered for the best solution awards. The solutions were evaluated for “elegance”, i.e. how many and how mathematically complicated steps were needed to reach the result; for conceptual ingenuity, i.e. how out-of-standard were the used concepts; for the pedagogical value, i.e. how much will benefit less-prepared readers from reading it; for rigor and completeness..

A vast majority of the 77 solutions were based on the elegant method of the velocity hodograph, which transforms the physics problem into one of pure geometry. However, there were also solutions that tackled the problem directly from first principles, by applying conservation laws within a coordinate system and/or by employing advanced geometric properties of the ellipse.

Within the dominant hodograph-based approach, most solutions followed a standard path of establishing the circular hodograph, calculating its radius, and then finding the eccentricity. One of the key distinctions lies in the proof of the hodograph itself. While many expertly derived it from the Laplace-Runge-Lenz vector, Eric Wang provided a beautiful and unique proof from the most fundamental principles of dynamics.

The hodograph-based solutions can be divided into different groups depending on how the geometric problem was handled. Among the two shortest solution paths, the intended one (henceforth referred to as type-I) has two major steps: (a) the circumcircle’s radius R is obtained by expressing a triangle’s area as the half-product of its base and height, and as abc/4R; (b) the distance d of the hodograph’s origin O from the circumcircle’s center is found by expressing the power of point O for the chord AC, and for the diameter. The second path, Owen Zhou‘s solution, trades step (a) for expressing the power of point O (denoted E in his figure) for the chord BB’ by making use of the expression |EB’| = |EB| + 2d, which follows from the symmetry of the figure. Apart from these two most efficient methods, contestants employed a wide variety of trigonometric approaches.

The award winners who share the main prize with 1/3 pts each are Owen Zhou, Eric Wang, and Ilan Mihelja & Ivan Hegedić. Eric Wang’s type-I solution is clean and concise, featuring a unique dynamical proof of the hodograph. Ilan Mihelja & Ivan Hegedić’s type-I solution starts with a proof that the eccentricity vector (the normalized Laplace-Runge-Lenz vector) is constant, and follows the best textbook style by rigorously proving every tool used; as a bonus, they also provide an alternative solution.

The following seven solutions are also published for their unique merits and high quality. Robertas Juknevicius and Alexandru Bordei are recognized for their highly original geometric insights within the hodograph framework, reframing the problem around finding the circumcenter and applying Steward’s Theorem, respectively. Mikhail Lazarev‘s submission is published as the third and final type-I solution. Dženan Midžić and Marinus Lehmann represent the gold standard of the mainstream “Law of Cosines” hodograph approach, with their solutions being exemplars of exceptional clarity, rigor, and professional presentation. Turning to the non-hodograph solutions, Emerson F. A. Gavarrete merits publication for his brilliant and unique use of the director circle, presented with outstanding pedagogical clarity. Finally, Thrayambhakesh H stands as an outstanding example of the non-hodograph “first principles” method, demonstrating how to solve the problem with power and clarity using only coordinate geometry and conservation laws.

Final results for Problem 5

Pre-university students:

First nameSurnameCountrySchool/UniversityP5 date time
DD/MM/YY 00:00:00		P5 Score
Kai WenTeoSingaporeRaffles Institution2/3/25 13:212.0
KanishkJainIndiaALLEN Kota2/3/25 13:321.85
MikhailLazarevRussiaLetovo School2/3/25 13:431.8
DariusGraurMoldovaLiceul Teoretic Orizont2/3/25 13:351.8
SaskiaPõldmaaEstoniaTallinn Secondary School of Sciences2/3/25 13:501.6
AlexSecrieruMoldovaLT “Orizont”2/3/25 13:541.5
TomasRazbadauskasLithuaniaKaunas University of Technology Gymnasium2/3/25 13:551.4
OwenZhouUnited StatesBasis Scottsdale2/3/25 17:101.3333
Tejas PenugondaTalha AshrafIndia/PakistanNixor College2/3/25 14:031.3
AndreiVilaRomaniaInternational Computer High School Bucharest2/3/25 14:341.2
Raiq BabazadeMurad Khalil, Ibrahim ValehliAzerbaijanPhysics, Mathematics and Informatics lyceum2/3/25 14:461.1
AlexandruBordeiMoldovaLT “Orizont” Durlesti2/3/25 15:361.1
ThrayambhakeshHIndiaReliable Institute, Kota3/3/25 19:311.1
RobertasJuknevičiusLithuaniaKaunas University of Technology Gymnasium2/3/25 14:581.1
MoritzRohnerLuxembourgLycée de Garçons Esch (LGE)2/3/25 18:481.0
KetanS HegdeIndiaVedantu3/3/25 15:341.0
ZanderLiCanadaLaurel Heights Secondary School2/3/25 15:371.0
EnejJaukSloveniaGimnazija Bežigrad4/3/25 11:591.0
OytunErTürkiyeKadıköy Anatolian Highschool3/3/25 20:211.0
DaniilsKoselevsLatviaJauno Lideru Vidusskola3/3/25 16:561.0
MahammadJafarliAzerbaijanTechnical-Humanitarian Lyceum3/3/25 15:501.0
AmritRajIndiaMaa Bharti Sr. Sec. School2/3/25 14:541.0
SubhanshuYadavIndiaFIITJEE2/3/25 17:201.0
DarshSikkaIndiaDPS RK Puram3/3/25 13:131.0
PrabhuduttaPrustiIndiaKIIT International School3/3/25 08:481.0
SiddhaarthDharaniSingaporeNUS High School5/3/25 07:201.0
FrancescoGariglioItalyIIS Majorana Moncalieri3/3/25 09:281.0
AljažErmanSloveniaGimnazija Kranj3/3/25 11:021.0
LucaGâleaRomaniaNational College Iasi5/3/25 18:081.0
GrigoriiGrechkinRussiaSchool CPM2/3/25 14:511.0
YisiLiuChinaThe First High School of Changsha3/3/25 04:441.0
OgtayLatifliAzerbaijanRepublican Physics-Mathematics and Informatics lyceum5/3/25 15:191.0
ShresthSuraiyaIndiaNarayana School2/3/25 20:381.0
ArtyomEryominRussiaLyceum №232/3/25 15:051.0
Kai XinChenChinaGuangDong ShiYan middle school – ShenZhen High school.3/3/25 12:531.0
AaravGuptaIndiaALLEN Marathahalli3/3/25 06:321.0
AtishayJainIndiaAllen Institute Marathahalli3/3/25 06:341.0
AnshuYadavIndiaAllen, kota2/3/25 15:521.0
Badekai Vishruth KrishnaRamachandra BhatIndiaMaa Bharati Senior Secondary School2/3/25 16:351.0
SushantAgarwalIndiaDelhi Public School RK Puram3/3/25 11:291.0
TanayTekadeIndiaSchool of Scholars14/3/25 05:351.0
JoshuaWangUnited StatesBrookfield Central High School2/3/25 20:301.0
CarloDi LorenzoItalyLiceo Newton Chivasso5/3/25 20:431.0
IlyaShpakovRussiaKapitza lyceum2/3/25 16:501.0
SatwikPatnaikIndiaAllen Career Institute3/3/25 04:561.0
一凡陆ChinaShanghai Caoyang NO. 2 High School3/3/25 10:131.0
Zeynep MinaOruçTürkiyeTübitak Fen Lisesi5/3/25 05:361.0
Vlad-DorianVulpeRomania“Tudor Vladimirescu” National College Târgu Jiu5/3/25 09:321.0
DavitHayrapetyanArmeniaPhysmath6/3/25 07:521.0
MASROOF AHMEDKHANINDIAMAA BHARTI SENIOR SECONDARY SCHOOL,KOTA8/3/25 16:001.0
RomanCheremnovKazakhstanMiras International School9/3/25 19:321.0
BrunoMachado FeltranBrazilColégio Etapa15/3/25 17:520.95
SauryaShresthaNepalSt. Xavier’s College, Nepal16/3/25 11:550.9
ZeynebMehemmedliAzerbaijanTTEL16/3/25 00:090.9
ItaloVidaurreEl SalvadorPrograma Jovenes Talento21/3/25 17:300.9
ChetanMalviyaINDIANarayana e-Techno School – Mumbai16/3/25 13:150.9
AlpTerliksizTürkiyeEnka Science and Technology Highschool16/3/25 14:050.8
Luis EdgarSánchez HenríquezEl SalvadorPrograma Jovenes Talentos30/3/25 11:030.8
AdityaDas JituBangladeshChattogram College30/3/25 11:370.8
Bahaa AldeenMelhemSyriaGap year30/3/25 17:400.72
TuncayAmiraslanliAzerbaijanSumgait school N2330/3/25 09:090.72
MəhəmmədUmarlıAzerbaijanFRITL5/3/25 23:330.7
MohammedAlarfajSaudi ArabiaDhahran Ahliyya Schools6/3/25 12:090.7
AryanPrakash VermaIndiaallen2/4/25 11:260.7
AarushMishraIndiaMaa Bharti Sr. Sec. School13/4/25 13:570.5
RahulNandiBangladeshChattogram College30/4/25 14:490.5
FedericoRoccaforteItalyIIS Concetto Marchesi24/4/25 08:490.5
AbdullahAlali BekSyriaAl-Maari School13/4/25 20:240.5
AdamElbelkasmiMoroccoTarik bnou ziad high school14/4/25 14:100.5
Serazus Salekin SaminSaminBangladeshCumilla Victoria Government College29/4/25 21:500.45
MaximOsatskiyRussiaLETOVO School22/4/25 12:500.45
NithilanRavikumarIndiaDelhi Public School Bangalore East20/4/25 05:520.45
AshutoshGuptaIndiaAllen Career Institute Pvt. Ltd.22/4/25 10:130.405

University students:

First nameSurnameCountrySchool/UniversityP5 date time
DD/MM/YY 00:00:00		P5 Score
Ilan MiheljaIvan HegedićCroatiaUniversity of Zagreb, EPFL2/3/25 13:582.3333
DženanMidžićBosnia and HerzegovinaUniversity of Rijeka2/3/25 16:401.9
ArmenGazazyanArmeniaYerevan State University2/3/25 14:451.9
EricWangUnited StatesMassachusetts Institute of Technology2/3/25 17:581.7333
Adrian, Bruno, DanielSerrano Capatina, Pontecorvo, Gonzalez FilipovSwitzerlandÉcole Polytechnique Fédérale de Lausanne2/3/25 16:521.7
IaroslavPozdniakRussiaMoscow Institute of Physics and Technology2/3/25 17:281.55
LarranceXingUnited StatesUniversity of Chicago2/3/25 17:401.5
Sifat AhmedTusharBangladeshUniversity of Dhaka3/3/25 02:421.3
SOUMYADEEPDASIndiaINDIAN INSTITUTE OF SCIENCE3/3/25 13:111.2
Emerson FranzuaAldana GavarreteHondurasUniversidad Nacional Autónoma de Honduras4/3/25 21:081.1
DhruvShahIndiaMassachusetts Institute of Technology3/3/25 15:441.1
MarinusLehmannGermanyUniversity of Potsdam5/3/25 21:481.1
MichałKołodziejczykPolandWarsaw University of Technology8/3/25 00:101.0
KonstantinRodionenkoRussiaITMO University9/3/25 10:051.0
SrirajChandraIndiaIndian Institute of Science16/3/25 05:271.0
Hui XiangSimSingaporeNational University of Singapore5/3/25 11:331.0
AHMETUĞUZTürkiyeYahya Akel Science High School, teacher6/3/25 16:341.0
FrancescoGherardiItalyUniversità di Padova14/3/25 19:231.0
Jun-XiWuChinaSouthern University of Science and Technology4/3/25 11:491.0
MainakChandraIndiaIndian Institute of Science3/3/25 17:321.0
AkarshSahayIndiaUniversity of Oxford10/3/25 18:101.0
GuyGinzbergIsraelThe Open University of Israel3/3/25 22:290.95
ViníciusFerreira RodriguesBrazilUniversidade de São Paulo (IFSC USP)18/3/25 03:550.9
Vincent KarlWillisAustriaETH Zürich22/3/25 09:560.9
FedericoMastrofortiItalyUniversity of Bologna16/3/25 17:020.9
ŠtěpánPlassCzech RepublicUniverzita Karlova17/3/25 16:410.9
FilippoSchelfiItalyUniversity of Trento10/3/25 10:500.9
GokulKrishnanIndiaIndian Institute of Technology Madras29/3/25 13:490.8
QiruiHEHong KongChinese University of Hong Kong6/4/25 10:020.7
VaibhavVIndiaUIT10/4/25 04:460.6
Aditya TakkarSumit Kumar Pandey, Vikash KalaliyaIndiaIndian Institute of Technology Jammu28/4/25 13:490.5