Physics Cup - 2019, Problem 2

To begin with, note that the assumption n>>1 was actually not needed, the result T_0+sqrt(n)t remains valid even if n is not small. This assumption was included mostly because the result becomes interesting only if n>>1: in theory, it is possible to achieve a significant cooling of a small portion of something by making use of minute temperature differences.

In total, 46 correct solutions of problem 2 were submitted.

The solutions of this problem can be divided into two categories: (a) based on the second law of thermodynamics (the total entropy of the system will either remain constant (if the processes are reversible) or increase; (b) showing how the Carnot cycles should be used, and using the formula for the efficiency of the Carnot cycle. The former is much clearer and concise, so the best solution awards go to solutions using the first approach, assuming it was submitted before the respective hint, i.e. during the first week. Overall, there were three such solutions (there were other solutions which made use of the entropy calculations, but calculated unnecessarily the work of the heat engines and heat pumps).

The best solution awards go to: Thomas Foster and Ivan Ridkokasha both of whom earn 40% of the bonus, and to Siddharth Tiwary who gets the remaining 20%.

The essential part of the Thomas' solution concludes basically on page 3 where he shows that some water will remain in the liquid phase. Further he proceeds to show how heat engines and heat pumps need to be connected to achieve the desired result. To be entirely rigorous, it is indeed necessary to show that there is a way to cool the small part of ice down to the given temperature. However, it would have been shorter (and equally sufficient) to just state that as long as there are some temperature differences between the main portion of ice and water, the temperature differences could be used to produce work by a heat engine, and thereby drive a heat pump to further freeze the cold ice.

Ivan's solution is very similar to that of Thomas; he has moved the non-obligatory parts into Appendix.

Siddharth's solution is very short and laconic, with very few comments (this makes it less suitable for those who want to learn); His solution is also less rigorous in not analysing if the desired result can be really achieved (see above). Additionally, he makes a statement which is not correct: he believes that the temperature T_0+sqrt(n)t could be also achieved (it cannot be achieved because of the phase transition, and because c_v for water is not proportional to T). These shortcomings explain why he gets a smaller part of the bonus.

The additionally published solutions are well-written and brief solutions based on the approach (b) (if you want to improve the chances of your solution getting published, write in LaTeX!).
Vladislav Polyakov - the second-fastest solution; well written and good, too!
Johanes Suhardjo - amended his original solution during the second week, adding an alternative, entropy-based solution (using the first hint). Both solutions are well written.
Felix Bekir Christensen - another nice solution using the approach (b).

Results for Problem 2, university students.
name country school solved score
Siddharth Tiwary India Indian Inst. of Techn. Bombay 13 Jan. 2019 14:32 3.168
Thomas Foster UK University of Oxford 13 Jan. 2019 15:10 3.518
Ivan Ridkokasha Ukraine Shevchenko University Kyiv13 Jan. 2019 16:55 3.198
Johanes Suhardjo Indonesia Hong Kong Univ Sci & Tech 13 Jan. 2019 18:57 2.144
Samarth Hawaldar India Indian Institute of Science 15 Jan. 2019 4:16 1.772
Felix Bekir Christensen Germany University of Oxford 19 Jan. 2019 22:38 1.772
Petra Brčić Croatia University of Zagreb 20 Jan. 2019 18:59 1.464
Damiano Tietto Italy Università degli Studi di Padova  03 Feb. 2019 17:32 1.331
Nur Mohammad Imran Bangladesh University Of Chittagong 19 Feb. 2019 10:44 0.968
Results for Problem 2, high school students.
name country school solved score
Vladislav PolyakovRussia School No 23 St Petersburg 13 Jan. 2019 14:50 2.853
Oliver Lindström Sweden Minervagymnasium 13 Jan. 2019 15:54 2.358
Stefan Dolteanu Romania Intern Computer HS Bucharest 13 Jan. 2019 16:14 2.144
Domagoj Perković Croatia III. Gigmazija Split 13 Jan. 2019 18:40 1.949
Gabriel Trigo Brazil Colegio ETAPA, Valinhos 13 Jan. 2019 20:39 1.772
Ivander J.M. Waskito Indonesia Mawar Sharon Christian School 14 Jan. 2019 3:17 1.611
Eduard Burlacu Romania Intern Computer HS Bucharest 14 Jan. 2019 8:49 1.391
Bartłomiej Sikorski Poland I Liceum O. Starachowicach 15 Jan. 2019 21:20 1.331
Oliwier Urbański Poland Szkoła Nr 28 w Poznaniu 18 Jan. 2019 11:56 1.21
Zăhărăchescu Mihai Romania Intern Computer HS Bucharest 19 Jan. 2019 9:44 1.1
Tùng Trần Xuân Vietnam HN-Amsterdam 21 Jan. 2019 2:54 1
Muhamed Sokolović Bosnia & Herz. Richmond Park ISS Sarajevo 21 Jan. 2019 19:31 1
Mateusz Kapusta Poland Secondary Sch No 3, Wroclaw 22 Jan. 2019 20:26 1
Stephen Catsamas Australia Marcellin College 27 Jan. 2019 13:21 1
Gusti P.S.G. Atmaja Indonesia Kesatuan Bangsa SHS 27 Jan. 2019 15:58 1
Morteza Mudrick Indonesia SMA Negeri 1 Surakarta 30 Jan. 2019 14:08 0.9
Yunus Emre Parmaksız Turkey Bahcesehir High 03 Feb. 2019 21:42 0.8
Roberto Marín Delgado Costa Rica Colegio Científico San Carlos 04 Feb. 2019 0:20 0.9
Masuk Ridwan Bangladesh St. Joseph Higher Secondary S. 04 Feb. 2019 13:44 1
Jianzhi Wang Singapore Raffles Institution 04 Feb. 2019 14:56 1
Siam Absur Khan Bangladesh Ideal School And College 05 Feb. 2019 18:02 1
Péter Elek Hungary DRK Dóczy Gimnáziuma 05 Feb. 2019 20:36 1
Marvin Janini Brazil Colégio Etapa 05 Feb. 2019 23:10 1
Imtiaz Tanweer Bangladesh Notre Dame College 06 Feb. 2019 13:50 1
Ionel-Emilian Chiosa Romania Intern Computer HS Bucharest 07 Feb. 2019 19:21 1
Kaarel Kivisalu Estonia Tallinn Secondary S. of Science 09 Feb. 2019 22:49 1
Vinicius de A. Névoa  Brazil Colégio Arena 10 Feb. 2019 4:48 1
Kaviraj Prithvi India Base PU College 10 Feb. 2019 11:30 1
Arhaan Ahmad India St. Mary's Academy 12 Feb. 2019 14:40 1
Raihan Amin Rana Bangladesh Notre Dame College 12 Feb. 2019 22:59 0.9
Hisham Almalki Saudi Arabia Manarat Al-Riyadh School 13 Feb. 2019 8:02 0.855
Rashedul Islam Bangladesh Ideal School and College 14 Feb. 2019 11:32 1
Md Atikul Islam Bangladesh St. Joseph Higher Secondary S. 15 Feb. 2019 17:56 1
Ričards Kristers Knipšis Latvia Riga State Gymnasium No.1 16 Feb. 2019 18:56 1
Jonathan Gräfe Germany Gymnasium Dresden-Bühlau16 Feb. 2019 21:01 1
Janis Huns Latvia Riga State Gymnasium No.1 16 Feb. 2019 21:40 1
Lucas Harada Brazil Colegio Etapa - Ana Rosa 16 Feb. 2019 22:21 1

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