- Problem 2 with the first hint and the results after one week
- Problem 2 with two hints and the results after two weeks
- Problem 2 with three hints and the results after three weeks

This problem was aimed, first and foremost, to demonstrate that in some cases, geometrical approach to relativistic kinematics using x-ict diagram is much more efficient than algebraic approach based on Lorentz transform. More or less as expected, none of the participants were able to figure out the geometric approach without hints. Even with the hints, only one contesant was able to work out fully geometrical solution, and five more found semi-geometrical solutions in which case they relied on algebraic results.

The best solution award goes to the only fully-geometric solution:**Prathyush Poduval**. As hinted, the trajectory of the spaceship in *x*-i*ct*-diagram is a circle; Prathyush derives this fact from the finding that in proper time *τ*, the tangent of the *x*-i*ct*-trajectory rotates at a constant angular speed. Alternatively, one could have expressed the curvature radius of the trajectory as the ratio of an infinitesimal arc length, i.e. an increment of the interval d*s*=d(i*cτ*), and the corresponding rotation angle of the tangent d*α*=d*gτ*/i*c*; the result *R=c ^{2}/g* is independent of time, so the curve is a circle. One should also notice that due to equality d

*s*=d(i

*cτ*), the length of a

*x*-i

*ct*-trajectory is given by the corresponding proper time,

*s*=i

*cτ*. What is left to do is to apply the theorem of inscribed angles; more detailed explanations of this last step can be found in other solutions, see below. For a reference, here is his brute-force solution, which is also among the most economically written brute-force solutions.

Half-geometric solutions making use of the inscribed angles theorem together with equality *s*=i*cτ* (earning bonus factor 1.1).**Balázs Németh** – a detailed and nicely written solution;**Navneel Singhal**;**Peter Elek**;**Thomas Bergamaschi** – another well-documented solution;**Alkın Kaz**.

While the brute-force approach is mathematically longer, it is still useful to have a look on some of such solutions. I have selected the best-documented solutions from the first week.**Tóbiás Marozsák**;**Balázs Németh**;**Dylan Toh**;**Satoshi Yoshida**;**Elvinas Ribinskas**;

Finally, here is a selection of correct answers (assuming c=1 and g=1) – these might look different, but are equivalent, nevertheless.

And here are the results. Number of fully correct solutions: 53. Names in italic correspond to unofficial participants (they get their deserved speed bonus, but do not advance the count for the next speed bonuses

name | school | country | Pr 2: solved; | score |

Tóbiás Marozsák | Óbudai Árpád Gimnázium | Hungary | 28 Jan 17:06 | 2.853 |

Eftime Andrei | International Computer Highschool Bucharest | Romania | 28 Jan 17:15 | 2.358 |

Thomas Bergamaschi | Colegio Etapa Valinhos-Brazil | Brazil | 28 Jan 17:51 | 2.358 |

Navneel Singhal | ALLEN Kota | India | 28 Jan 18:12 | 2.144 |

Gabriel Golfetti | Colégio Etapa | Brazil | 28 Jan 18:14 | 1.772 |

Victor Hugo O Bastos | Ari de Sá Cavalcante | Brazil | 28 Jan 21:11 | 1.611 |

Luciano Rodriges | Christus | Brazil | 28 Jan 22:14 | 1.464 |

Rafael Timbó | Colégio Antares S/S LTDA | Brazil | 28 Jan 23:04 | 1.331 |

Carlos Henricco Queiroz | Farias Brito Colegio | Brazil | 29 Jan 02:53 | 1.21 |

Soma Nagahama | Osaka Seiko High School | Japan | 29 Jan 10:26 | 1.1 |

Artur Soares Rodrigues | Colégio Farias Brito | Brazil | 29 Jan 10:53 | 1 |

Takamasa Ando | Okayama Asahi High School | Japan | 29 Jan 11:38 | 1 |

Dylan Toh | NUS High School | Singapore | 29 Jan 14:51 | 1.1 |

Balázs Németh | Budapesti Fazekas Gimnázium | Hungary | 29 Jan 18:44 | 1.21 |

Paulo Kitayama | Farias Brito Colegio | Brazil | 29 Jan 20:30 | 1 |

Peter Elek | DRK Dóczy Gimnázium | Hungary | 29 Jan 21:26 | 0.99 |

Bulcsu Fajszi | Fazekas Secondary School, Budapest | Hungary | 30 Jan 14:05 | 1 |

Satoshi Yoshida | The University of Tokyo | Japan | 30 Jan 16:57 | 1.1 |

Levy Batista | Farias Brito | Brazil | 31 Jan 02:05 | 1 |

Muhammad Farhan Husain | Kharisma Bangsa High School | Indonesia | 01 Feb 12:57 | 1 |

Juan Sheikh Mohammad | Al Bassel school | Syria | 01 Feb 21:07 | 1 |

Vinicius Alcântara Névoa | Colégio Visão | Brazil | 03 Feb 17:00 | 0.9 |

Sabina Dragoi | International Computer Highschool Bucharest | Romania | 04 Feb 09:49 | 0.9 |

Elvinas Ribinskas | University of Cambridge | Lithuania | 04 Feb 11:16 | 0.99 |

Radosław Grabarczyk | Marynarki Wojennej RP w Gdyni | Poland | 04 Feb 14:56 | 0.8 |

Flavio Salvati | I.I.S. Leonardo da Vinci | Italy | 05 Feb 22:23 | 0.8 |

Leonardo Martins Pires | Colégio Objetivo Integrado | Brazil | 07 Feb 00:36 | 0.8 |

Otávio Bittencourt | Colégio Objetivo Integrado | Brazil | 07 Feb 20:08 | 1 |

Prathyush Poduval | Canara PU College | India | 08 Feb 07:08 | 2.175 |

Gabriel Capelo | Colégio Ari de Sá Cavalcante | Brazil | 09 Feb 01:27 | 0.8 |

Nozomi Sakura | Hiroo Gakuen High School | Japan | 09 Feb 21:17 | 1 |

Dolteanu Stefan | International Computer Highschool Bucharest | Romania | 11 Feb 09:11 | 0.8 |

Davit Mdinaradze | Komarovi Tbilisi N199 | Georgia | 11 Feb 15:47 | 0.64 |

Kosuke Yoshimi | Nada High School | Japan | 12 Feb 07:35 | 0.64 |

Abrar Al Shadid | Chittagong College | Bangladesh | 14 Feb 05:45 | 0.8 |

Gabriel Trigo | Colegio Etapa | Brazil | 15 Feb 01:49 | 1 |

Gabriel Domingues | Colégio Etapa | Brazil | 15 Feb 20:34 | 0.72 |

Faisal AlSallom | Ibn Khaldun School | Saudi Arabia | 18 Feb 17:18 | 0.8 |

Md. Ijtihad Abtahi | Chittagong College | Bangladesh | 19 Feb 09:41 | 0.8 |

Francisco Dahab | Colégio Etapa | Brazil | 22 Feb 09:55 | 1 |

Caique Corrêa | Colégio objetivo integrado | Brazil | 22 Feb 10:15 | 1 |

Marcio Imanishi de Moraes | Colégio Objetivo Integrado | Brazil | 22 Feb 15:40 | 0.9 |

Leonardo Menegon | Colégio Objetivo Integrado | Brazil | 22 Feb 15:52 | 1 |

Marco Ambrosini | Liceo Scientifico Statale ‘Plinio Seniore’ | Italy | 22 Feb 22:40 | 0.8 |

Konstantine Gagnidze | Komarovi Tbilisi N199 | Georgia | 23 Feb 21:34 | 0.64 |

Yunus Emre Parmaksiz | Bahçeşehir High School for Sci. & Techn. | Turkey | 26 Feb 11:10 | 1 |

Mert Unsal | Bahçeşehir High School for Sci. & Techn. | Turkey | 26 Feb 12:02 | 1 |

Mustafa Tugtekin | Bahçeşehir High School for Sci. & Techn. | Turkey | 27 Feb 09:47 | 1 |

Berkin Binbaş | Bahçeşehir High School for Sci. & Techn. | Turkey | 28 Feb 07:57 | 1 |

Nícolas Lopes | Colégio Ari de Sá Cavalcante | Brazil | 28 Feb 19:28 | 0.8 |

Alkın Kaz | Bahçeşehir High School for Sci. & Techn. | Turkey | 28 Feb 20:35 | 1.1 |

Chiosa Ionel-Emilian | International Computer Highschool Bucharest | Romania | 28 Feb 21:05 | 1 |

Ícaro Bacelar | Colégio Farias Brito | Brazil | 28 Feb 21:42 | 0.8 |