PC2025-P2 Best solutions and final results

Problem 2 – Tetrahedral flock of birds

In total, 256 solutions were submitted for problem 2, of which 111 were correct. Congratulations to everyone who managed to solve it!

There were 61 correct solutions submitted before the first hint was published, and only those were considered as candidates for the best solution awards. Additionally, since there were so many well-written solutions to choose from, the 17 handwritten solutions were not considered this time, sorry about that!

All the solutions can be divided into three categories: (a) those that parameterized the problem through the side lengths of the cuboid defined by the diagonals AB and CD of its opposite faces (or almost equivalent variations of this); (b) those that parameterized the problem through the side lengths of the tetrahedron AB and AC; (c) brute force approach integrating the set of 12 differential equations, either simplifying it by using symmetry, or solving it as is. Note that the hints suggested approach (b).

In each category, the quality of the solution was assessed using the following criteria: clarity, i.e. how easy it is for a reader to follow the solution (useful figures increased the rating in this category); simplicity: solutions that involved simpler math had higher rating; rigor, i.e. any important aspects were left out from the discussions (typos decreased this rating, but very obvious and simple typos were overlooked; insight: if there were useful discussions of the analysis and results that went beyond what was asked explicitly.

In category (a), the best solutions were the ones submitted by Bruno Machado Feltran (0.4 pts) and by Alexandru Bordei (0.15 pts). In category (b), the best solution was submitted by Eric Wang (0.3 pts). In category (c), the best solution was submitted jointly by Adrian Serrano Capatina, Bruno Pontecorvo, and Daniel Gonzalez Filipov (0.15 pts).

The reason a brute force approach got awarded this time is that this particular brute-force solution has its own beauty and additional benefits: it utilizes symmetry to simplify calculations, and enables the analytical determination of bird trajectories. It could have been even better if the symmetry had been exploited fully and if there were no inaccuracies, as explained below. The solution starts with putting the starting point of the birds A, B, C, and D, respectively, to (1,1,1), (1,-1,-1), (-1,1,-1), and (-1,-1,1). They notice that if we rotate this system around the y-axis, mirror in the x-z plane (mirroring was dropped out from the solution), and re-label cyclically the birds, the initial situation is restored. Hence, x_A=z_D;\; x_B=z_A;\; x_c=z_B;\;x_D=z_C, for any subsequent moment of time. They also notice that if this symmetry transformation is applied twice, we can conclude that y_A=y_c;\; y_B=y_D. However, they fail to notice that the symmetry allows us to establish several additional equalities: everything can be described by three functions of time: y_A=y_C=-y_B=-y_D\equiv \eta(t); x_A=z_D=-x_C=-z_B\equiv \xi(t); x_B=z_A=-x_D=-z_C\equiv \xi(t). Instead, they derive the additional equalities in a more complicated way.


There are additional excellent solutions published here for various reasons. First, type-(a) solutions by Robertas Juknevičius (nice induction-based proof that the birds will remain forming a cuboid), Karlo Maksimović (python-generated plot of trajectories), Ilan Mihelja with Ivan Hegedić (efficient use of cylindrical coordinates and professional calculation of the integral), Teo Kai Wen (very good discussion of symmetries and simulations with a movie). Second, type-(b) solutions by Kanishk Jain (clean and efficient solution with the only downside that it is not shown how the differential equation is solved); Zander Li (clean solution with clear figure and full step-by-step mathematical calculations); Gleb Grachev (clean solution with full step-by-step mathematical calculations); Enej Jauk (clean solution with useful figures and graphs).

Final results for Problem 2

Pre-university students:

Age GroupFirst nameSurnameCountrySchool/UniversityP2 date time
DD/MM/YY 00:00:00		P2 Score
Pre-universityKanishkJainIndiaALLEN Kota17/11/24 13:512.1
Pre-universityDariusGraurMoldovaLiceul Teoretic Orizont17/11/24 17:251.9
Pre-universityPrabhuduttaPrustiIndiaKIIT International School17/11/24 19:211.8
Pre-universityAlexandruBordeiMoldovaLT “Orizont” Durlesti17/11/24 21:571.8
Pre-universityFedericoRoccaforteItalyIIS Concetto Marchesi17/11/24 22:011.6
Pre-universityOwenZhouUnited StatesBasis Scottsdale18/11/24 01:251.5
Pre-universityBrunoMachado FeltranBrazilColégio Etapa23/11/24 19:131.4
Pre-universityVijayLehtoUnited KingdomQueen Elizabeth’s School18/11/24 07:141.4
Pre-universityJingyang JamesHeSingaporeRaffles Institution18/11/24 11:031.3
Pre-universityRaiq BabazadeMurad Khalil, Ibrahim ValehliAzerbaijanPhysics, Mathematics and Informatics lyceum18/11/24 11:411.2
Pre-universityGlebGrachevSBGEI School № 154319/11/24 06:241.1
Pre-universityZanderLiCanadaLaurel Heights Secondary School1/12/24 04:081.1
Pre-universityEnejJaukSloveniaGimnazija Bežigrad20/11/24 13:461.1
Pre-universityHarikSodhiUnited KingdomQueen Elizabeth’s School, Barnet18/11/24 15:151.1
Pre-universityRobertasJuknevičiusLithuaniaKaunas University of Technology Gymnasium21/11/24 19:421.1
Pre-universityMert AndacKaranTürkiyeBahcesehir High School for Science and Technology21/11/24 16:591.0
Pre-universityMoritzRohnerLuxembourgLycée de Garçons Esch (LGE)19/11/24 19:591.0
Pre-universityTomasRazbadauskasLithuaniaKaunas University of Technology Gymnasium18/11/24 16:171.0
Pre-universityNikhilKodaliUnited StatesCupertino High School24/11/24 08:461.0
Pre-universityMirceaRebengiucRomania“Tudor Vianu” National High School18/11/24 21:091.0
Pre-universityMahammadJafarliAzerbaijanTechnical-Humanitarian Lyceum22/11/2024 15:13:001.0
Pre-universityKai WenTeoSingaporeRaffles Institution30/11/24 12:551.0
Pre-universityRichardYangUnited StatesIowa City West High School1/12/24 05:151.0
Pre-universityYi QuanTanSingaporeNUS High School of Mathematics and Science24/11/24 15:151.0
Pre-universityAgastyaGoelUnited StatesHenry M. Gunn High School19/11/24 04:201.0
Pre-universitySauryaShresthaNepalSt. Xavier’s College, Nepal23/11/24 18:481.0
Pre-universityLeanderGabriëlBelgiumVoskenslaan20/11/24 21:441.0
Pre-universityZeynebMehemmedliAzerbaijanTTEL27/11/24 19:461.0
Pre-universityLucaGâleaRomaniaNational College Iasi22/11/24 22:191.0
Pre-universityArtyomEryominLyceum №2321/11/2024 18:001.0
Pre-universityAlpTerliksizTürkiyeEnka Science and Technology Highschool20/11/2024 19:03:000.95
Pre-universityAlexSecrieruMoldovaLT “Orizont”24/11/24 11:240.95
Pre-universityAntonTrubitsynLyceum №23924/11/24 15:050.95
Pre-universitySaskiaPõldmaaEstoniaTallinn Secondary School of Sciences29/11/24 18:310.95
Pre-universityTejas PenugondaTalha AshrafIndia/PakistanNixor College18/11/24 21:490.95
Pre-universitySiddhaarthDharaniSingaporeNUS High School19/11/24 15:040.95
Pre-universityMihkelRannutEstoniaTallinna Reaalkool19/11/24 15:100.95
Pre-universityPiyushFalkeIndiaUnacademy28/11/24 15:050.9
Pre-universityOytunErTürkiyeKadıköy Anatolian Highschool6/12/24 18:050.9
Pre-universityDarshSikkaIndiaDPS RK Puram25/11/24 12:020.9
Pre-universityRahulNandiBangladeshChattogram College1/12/24 11:280.9
Pre-universityFrancescoGariglioItalyIIS Majorana Moncalieri8/12/24 11:010.9
Pre-universityAljažErmanSloveniaGimnazija Kranj24/11/24 21:010.9
Pre-universityYisiLiuChinaThe First High School of Changsha1/12/24 09:300.9
Pre-universityDarshilSrivastavaIndiaallen lajpat nagar4/12/24 14:390.9
Pre-universityWilliamZhaoUnited StatesDougherty Valley High School8/12/24 04:250.9
Pre-universityChinmaya ShankaraShastryIndiaAllen hebbal5/1/25 14:330.9
Pre-universityAmritRajIndiaMaa Bharti Sr. Sec. School6/12/24 17:500.81
Pre-universitySubhanshuYadavIndiaFIITJEE2/12/24 14:220.81
Pre-universityGrigoriiGrechkinSchool CPM7/12/24 22:020.81
Pre-universityOgtayLatifliAzerbaijanRepublican Physics-Mathematics and Informatics lyceum2/12/24 14:140.81
Pre-universityMəhəmmədUmarlıAzerbaijanFRITL14/12/24 08:310.8
Pre-universityAdityaDas JituBangladeshChattogram College9/12/24 05:010.8
Pre-universityAryanPrakash VermaIndiaallen8/12/24 13:420.8
Pre-universitySOURADEEPSHARMILA MANDALINDIAALLEN JAYANAGAR BENGALURU10/12/24 11:470.8
Pre-universityYun-ChiehSuTaiwanTaipei Fuhsing private school8/12/24 10:380.7695
Pre-universityAndreiCalarasuRomania“Tudor Vianu” National High School of Computer Science10/12/24 18:450.72
Pre-universityAndreiVilaRomaniaInternational Computer High School Bucharest9/12/24 14:360.72
Pre-universityDaniilsKoselevsLatviaJauno Lideru Vidusskola15/12/24 11:590.72
Pre-universityShresthSuraiyaIndiaNarayana School10/12/24 16:430.72
Pre-universityMatthewChenUnited StatesCupertino High School15/12/24 22:360.7
Pre-universityVadimArhipMoldovaAlexandru Ioan Cuza20/12/24 14:550.63
Pre-universityAaravGuptaIndiaALLEN Marathahalli18/12/24 07:220.63
Pre-universityJanosSimonHungaryCareer Academy South Bend25/12/24 22:180.6
Pre-universityBadekai Vishruth KrishnaRamachandra BhatIndiaMaa Bharati Senior Secondary School22/12/24 18:030.6
Pre-universityAryanRagupathyIndiaAllen Career Institute24/12/24 11:270.6
Pre-universityMohammedAlarfajSaudi ArabiaDhahran Ahliyya Schools5/1/25 09:400.5
Pre-universityAlexandraStefanova-Razumnaya23912/1/25 11:560.5
Pre-universityItaloVidaurreEl SalvadorPrograma Jovenes Talento16/1/25 02:220.5
Pre-universityTuncayAmiraslanliAzerbaijanSumgait school N2319/1/25 08:050.5
Pre-universityZiruChenChinaQingdao Academy11/1/25 15:410.5
Pre-universityTanayTekadeIndiaSchool of Scholars19/1/25 09:160.5
Pre-universityKai XinChenChinaGuangDong ShiYan middle school – ShenZhen High school.22/12/24 15:440.486
Pre-universityLuis EdgarSánchez HenríquezEl SalvadorPrograma Jovenes Talentos19/1/25 16:000.45
Pre-universityShaswatBanerjeeIndiaBodhicariya12/1/25 16:440.45
Pre-universityAshwindhRameshUnited StatesCampbell high school6/1/25 00:540.405

University students:

Age GroupFirst nameSurnameCountrySchool/UniversityP2 date time
DD/MM/YY 00:00:00		P2 Score
UniversityEricWangUnited StatesMassachusetts Institute of Technology17/11/24 21:152.3
UniversityIlan MiheljaIvan HegedićCroatiaUniversity of Zagreb, EPFL17/11/24 22:251.95
UniversityAdrian, Bruno, DanielSerrano Capatina, Pontecorvo, Gonzalez FilipovSwitzerlandÉcole Polytechnique Fédérale de Lausanne18/11/24 00:161.9
UniversityIaroslavPozdniakMoscow Institute of Physics and Technology18/11/24 00:241.7
UniversityŠtěpánPlassCzech RepublicUniverzita Karlova18/11/24 12:421.6
UniversityDhruvShahIndiaMassachusetts Institute of Technology18/11/24 19:421.5
UniversityFedericoMastrofortiItalyUniversity of Bologna18/11/24 23:121.4
UniversityEthanLuUnited StatesOhio State University19/11/24 07:491.25
UniversityTornikeTsadzikidzeGeorgiaFree University of Tbilisi19/11/24 15:591.2
UniversityMarinusLehmannGermanyUniversity of Potsdam19/11/24 16:171.05
UniversitySifat AhmedTusharBangladeshUniversity of Dhaka19/11/24 19:151.0
UniversityHui XiangSimSingaporeNational University of Singapore20/11/24 06:541.0
UniversityYaroslavKondrachukUkraineNational University of Singapore22/11/24 16:401.0
UniversityKarloMaksimovićCroatiaUniversity of Zagreb27/11/24 06:201.0
UniversityFelixSchwarzfischerGermanyLudwig Maximilians Universität20/11/24 21:281.0
UniversityJakubKlimentCzech RepublicCharles University24/11/24 03:251.0
UniversityEmerson FranzuaAldana GavarreteHondurasUniversidad Nacional Autónoma de Honduras21/11/24 17:340.95
UniversityBayram AlpŞahinTürkiyeBoğaziçi University20/11/24 17:000.95
UniversityGuyGinzbergIsraelThe Open University of Israel29/11/24 13:150.9
UniversityFilippoSchelfiItalyUniversity of Trento27/11/24 15:550.9
UniversityMarianHaritonMoldovaTechnical University Of Moldova9/12/24 18:150.8
UniversityLarranceXingUnited StatesUniversity of Chicago8/12/24 07:250.729
UniversityKonstantinRodionenkoITMO University8/12/24 08:400.729
UniversityQiruiHEHong KongChinese University of Hong Kong15/12/24 12:510.72
UniversitySrirajChandraIndiaIndian Institute of Science09/12/2024 12:16:000.72
UniversityAbhiramCherukupalliUnited StatesCalifornia Institute of Technology16/12/24 02:370.7
UniversityAyaanMaanIndiaIIT Delhi18/12/24 13:590.7
UniversityMainakChandraIndiaIndian Institute of Science21/12/24 19:350.7
UniversityMichałKołodziejczykPolandWarsaw University of Technology27/12/24 00:540.6
UniversityKennethLuoCanadaUniversity of Toronto17/12/24 03:230.567
UniversityArmenGazazyanArmeniaYerevan State University18/12/24 19:390.567
UniversityJia XuanTanSingaporeUniversity of Cambridge14/1/25 05:130.5
UniversityGokulKrishnanIndiaIndian Institute of Technology Madras1/1/25 23:050.5
UniversityDženanMidžićBosnia and HerzegovinaUniversity of Rijeka10/3/25 09:350.45