PC2025-P4 Best solutions and final results

Problem 4 – Axis of a lens

In total, 93 solutions were submitted for problem 4, of which 79 were correct. Congratulations to everyone who successfully solved it!

There were 31 correct solutions submitted before the first hint was published, and only those were considered as candidates for the best solution awards. Additionally, while handwritten solutions are generally not considered (due to the fact that recently, a dominating majority of the submissions had been excellently typeset in LaTeX), an exception was made this time for two solutions that demonstrated exceptionally short construction paths. The key criterion for this problem was construction efficiency combined with mathematical elegance—solutions that recognized the most direct geometric insights to minimize the number of construction steps required.

A dominating majority of the 31 correct solutions were based on homothety. More specifically, while it is very difficult to find the exact position of the original circle, one can easily draw its homothety relative to O; if we further limit ourselves only to those construction elements that undergo the same homothety, the exact location of the circle will not matter. There were only five solutions that used a different approach: Dhruv Shah, Alexandra Stefanova-Razumnaya and Qirui He smartly used angle bisectors to directly determine the lens’s plane, and two submissions constructed an auxiliary hyperbola to find the main focus.

Most homothety-based solutions follow a standard pattern: they construct the focal plane closer to the ellipse by identifying the convergence points of initially parallel rays, either by finding first the image of the circle’s center (reducing the problem thereby to Physics Cup 2017 Problem 4), or by finding alternative ways of identifying pairs of initially parallel rays. However, Joshua Wang and Emerson Franzua employ distinctly different strategies: Wang finds the focal plane closer to the circle rather than the ellipse, while Emerson constructs the lens plane directly, bypassing focal plane construction. The third solution that deviates from the mainstream within this class of solutions is that of Sim Hui Xiang who makes use of inversion theory.

The award winners are as follows. Alexandra Stefanova-Razumnaya (.ggb), Dhruv Shah (.ggb), and Qirui He (.ggb) share the main prize, with Alexandra and Dhruv receiving slightly smaller bonuses (0.21) than Qirui He (0.3) due to their handwritten format. All three solutions (15-16 construction steps) demonstrate the key insight of using the intended-solution’s bisector method, avoiding the need for auxiliary circles and homothety arguments. A small comment to Qirui He’s solution that might help in understanding: his construction of the angle α is based on the fact that α is the inscribed angle of the arc AD in his Fig. 2, and as such, can be expressed in terms of the angle between the tangents of the ellipse drawn from the point O.

Joshua Wang (.ggb) and Emerson Franzua Aldana Gavarrete (.ggb) receive honorable mentions (0.14 pts each). They share the shortest construction sequence among the homothety-based solutions, with 20-steps, and both employ ideas not found in the other homothety-based solutions.

Other published solutions are: Siddhaarth Dharani (.ggb), whose systematic tabular methodology and progressive GeoGebra documentation exemplify clear pedagogical presentation in standard homothety-based constructions, and Moritz Rohner (.ggb), who provides theoretical rigor by proving the independence of circle choice in homothety methods. Both of them use 22 steps to reach their goal; with that, they 6th and 7th place in the length ranking. Teo Kai Wen‘s solution (.ggb; 25 steps) merits publication for a clear presentation with pedagogically useful structuring. Next, Sim Hui Xiang‘s solution (.ggb) merits publication not because of its length (32 steps), but because of educational merit of showing how to use the inversion theory. Eric Wang (.ggb; 34 steps) demonstrates how to use the concept of cross-ratio and the theory of projective geometry, as well as how to keep rigor and clarity through mathematical lemmas. Finally, the solution of Francesco Gariglio (.ggb; 41 steps) represents the third type of solutions that reaches the goal via constructing first the locus of the main focus in the form of an hyperbola.

Finally, here is my own (JK) solution (11 steps).

Final results for Problem 4

Pre-university students:

First nameSurnameCountrySchool/UniversityP4 date time
DD/MM/YY 00:00:00		P4 Score
RobertasJuknevičiusLithuaniaKaunas University of Technology Gymnasium26/1/25 17:552.0
AndreiVilaRomaniaInternational Computer High School Bucharest26/1/25 18:041.85
GrigoriiGrechkinRussiaSchool CPM26/1/25 21:491.8
Kai WenTeoSingaporeRaffles Institution27/1/25 03:251.7
DariusGraurMoldovaLiceul Teoretic Orizont26/1/25 22:171.7
JoshuaWangUnited StatesBrookfield Central High School27/1/25 07:291.54
Badekai Vishruth KrishnaRamachandra BhatIndiaMaa Bharati Senior Secondary School27/1/25 05:341.45
SiddhaarthDharaniSingaporeNUS High School27/1/25 14:021.4
AlexandraStefanova-RazumnayaRussia2392/2/25 12:051.21
AlexandruBordeiMoldovaLT “Orizont” Durlesti27/1/25 14:251.2
MoritzRohnerLuxembourgLycée de Garçons Esch (LGE)28/1/25 07:351.1
FrancescoGariglioItalyIIS Majorana Moncalieri8/2/25 15:191.1
KanishkJainIndiaALLEN Kota27/1/25 19:121.1
TomasRazbadauskasLithuaniaKaunas University of Technology Gymnasium2/2/25 19:471.0
AlexSecrieruMoldovaLT “Orizont”30/1/25 08:471.0
DamjanStojanovNorth MacedoniaYahya Kemal Skopje31/1/25 21:241.0
ShresthSuraiyaIndiaNarayana School29/1/25 07:301.0
PavelBulanovRussiaSchool #5331/2/25 19:151.0
OwenZhouUnited StatesBasis Scottsdale28/1/25 06:130.95
ZanderLiCanadaLaurel Heights Secondary School9/2/25 18:580.9
EnejJaukSloveniaGimnazija Bežigrad16/2/25 11:330.9
DaniilsKoselevsLatviaJauno Lideru Vidusskola15/2/25 21:380.9
AmritRajIndiaMaa Bharti Sr. Sec. School16/2/25 06:280.9
SauryaShresthaNepalSt. Xavier’s College, Nepal15/2/25 21:280.9
AljažErmanSloveniaGimnazija Kranj1/2/25 21:340.9
ZeynebMehemmedliAzerbaijanTTEL15/2/25 21:580.9
LucaGâleaRomaniaNational College Iasi10/2/25 21:460.9
ArtyomEryominRussiaLyceum №238/2/25 16:420.9
Kai XinChenChinaGuangDong ShiYan middle school – ShenZhen High school.11/2/25 14:190.9
MikhailLazarevRussiaLetovo School15/2/25 09:150.9
IlyaPavlovRussiaПФМЛ №23912/2/25 22:140.855
ChetanMalviyaINDIANarayana e-Techno School – Mumbai10/2/25 07:520.81
MahammadJafarliAzerbaijanTechnical-Humanitarian Lyceum22/2/25 10:400.8
OytunErTürkiyeKadıköy Anatolian Highschool28/2/25 18:380.7
SubhanshuYadavIndiaFIITJEE23/2/25 20:570.7
TuncayAmiraslanliAzerbaijanSumgait school N232/3/25 11:060.7
TanayTekadeIndiaSchool of Scholars1/3/25 08:220.7
IlyaShpakovRussiaKapitza lyceum1/3/25 23:430.7
RahulNandiBangladeshChattogram College2/3/25 12:570.665
YisiLiuChinaThe First High School of Changsha22/2/25 16:230.648
AlpTerliksizTürkiyeEnka Science and Technology Highschool24/2/25 11:510.63
Raiq BabazadeMurad Khalil, Ibrahim ValehliAzerbaijanPhysics, Mathematics and Informatics lyceum28/2/25 18:580.63
Tejas PenugondaTalha AshrafIndia/PakistanNixor College1/3/25 19:150.63
CarloDi LorenzoItalyLiceo Newton Chivasso8/3/25 17:300.6
MaximOsatskiyRussiaLETOVO School3/3/25 20:000.6
SaskiaPõldmaaEstoniaTallinn Secondary School of Sciences9/3/25 10:460.57
DarshSikkaIndiaDPS RK Puram30/3/25 11:440.5
AdityaDas JituBangladeshChattogram College10/3/25 17:390.5
ItaloVidaurreEl SalvadorPrograma Jovenes Talento14/3/25 18:200.5
AryanPrakash VermaIndiaallen28/3/25 20:570.5
Luis EdgarSánchez HenríquezEl SalvadorPrograma Jovenes Talentos30/3/25 22:200.45

University students:

Age GroupFirst nameSurnameCountrySchool/UniversityP4 date time
DD/MM/YY 00:00:00		P4 Score
UniversityDženanMidžićBosnia and HerzegovinaUniversity of Rijeka26/1/25 14:342.0
UniversityGuyGinzbergIsraelThe Open University of Israel26/1/25 15:581.85
UniversityQiruiHEHong KongChinese University of Hong Kong28/1/25 03:031.8
UniversityIlan MiheljaIvan HegedićCroatiaUniversity of Zagreb, EPFL26/1/25 16:531.8
UniversityIaroslavPozdniakRussiaMoscow Institute of Physics and Technology27/1/25 01:321.7
UniversityAdrian, Bruno, DanielSerrano Capatina, Pontecorvo, Gonzalez FilipovSwitzerlandÉcole Polytechnique Fédérale de Lausanne27/1/25 16:551.6
UniversityDhruvShahIndiaMassachusetts Institute of Technology2/2/25 16:271.51
UniversityEricWangUnited StatesMassachusetts Institute of Technology30/1/25 00:451.5
UniversityEmerson FranzuaAldana GavarreteHondurasUniversidad Nacional Autónoma de Honduras7/2/25 01:351.34
UniversityHui XiangSimSingaporeNational University of Singapore7/2/25 04:531.1
UniversityKonstantinRodionenkoRussiaITMO University23/2/25 12:500.8
UniversityŠtěpánPlassCzech RepublicUniverzita Karlova25/2/25 18:250.7
UniversityMarcoAltamuraItalyUniversità di Padova23/2/25 12:560.7
UniversitySifat AhmedTusharBangladeshUniversity of Dhaka5/3/25 17:220.6
UniversityMichałKołodziejczykPolandWarsaw University of Technology22/3/25 20:260.5
UniversityLarranceXingUnited StatesUniversity of Chicago15/3/25 04:430.5
UniversityMarinusLehmannGermanyUniversity of Potsdam15/3/25 16:290.5