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A Different Method of Solving a Problem of IMO

Received: 30 July 2019     Accepted: 10 September 2019     Published: 1 April 2020
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Abstract

The IMO performs once a year, and has become an important activity in the field of mathematics. Because the problems in IMO are very difficult, and in general needs two days to finish the test of only six problems, therefore, it is significant to study how to solve and solve those IMO problems with various methods. With respect to question (a) of the problem of discussing, at first, using the so-called “exhaustive method” and the mathematical induction, the paper gets the conclusion of that if n is the integral multiple of 3, subtracting 1 from the nth power of 2 must be divisible by 7. Furthermore, it also proves by use of the disprove method that if n is not the integral multiple of 3, subtracting 1 from the nth power of 2 is impossible to be divisible by 7. The way of solving question (b) is similar to that of solving (a), in order to use the result of question (a) for the third step of the mathematical induction, the paper firstly consider the third power of that 1 added to (k+1)th power of 2 and applying the disprove method proves that it and hence that 1 added to the (k+1)th power of 2 are not divisible by 7, namely the question (b) is true.

Published in International Journal of Systems Science and Applied Mathematics (Volume 5, Issue 1)
DOI 10.11648/j.ijssam.20200501.11
Page(s) 1-3
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2020. Published by Science Publishing Group

Keywords

IMO, The Mathematical Induction, Algebraic Equation, Disprove Method

References
[1] A. C. McBride. The 2001 International Mathematical Olympiad [J]. Scottish Mathematical Council Journal, 2002, 31: 13-15.
[2] Cecil Rousseau and Gregg Patruno. The International Mathematical Olympiad Training Session [J]. College Mathematics Journal, 1985, 16 (5): 362-365. Online: 30n Jan. 2018.
[3] Titu Andreescu and Zuming Feng. News and Letters [J]. Mathematics Magazine, 2003, 76 (3): 242-247.
[4] M. Shifman. You failed your math test, Comrade Einstein: Adventures and misadventures of young mathematicians or test your skills in almost recreational mathematics [M]. USA: World Scientific, 2005, pp 232.
[5] Steve Olson. Beautiful Solutions at the International Mathematical Olympiad [J]. The Mathematics Teacher, 2006, 99 (7): 527-528.
[6] LLko Brnetic. Iequalities at the International Mathematical Olympiad [J]. Osjecki Matematicki List, 2008, 8: 1; 15-18.
[7] Kyong Mi Choi. Influences of Formal Schooling on International Mathematical Olympiad Winners from Korea [J]. Roeper Review, 2013, 35 (3): 187-196.
[8] Stan Dolan. 55th International Mathematical Olympiad, Cape Town, 3-13 [J]. The Mathematical Gazette, 2014, 98 (543): 546.
[9] Alison Higgs and Mary Twomey. Editorial [J]. Ethics and Social Wolfare, 2015, 9 (3): 223-224.
[10] C. P. Shao, H. B. Li and L. Huang. Chanllenging Theorem Proves with Mathematical Olympiad Problems in Solid Geometry [J]. Mathematics in Computer Science, 2016, 10 (1): 75-96.
[11] S. Agievich, A. Gorodiliva, V. Idrisova, N. Kolomeec, G. Shushuev and N. Tokreva. Mathematical Problems of the Second International Students’ Olympiad in Cryptography [J]. Cryptologia, 2017, 41 (6): 534-565.
[12] A. Gorodilova, S. Agievich, C. Carlet et al. Problems and Solutions from the Fourth International Students’ Olympiad in Cryptography [J]. Cryptologia, 2019, 43 (2): 138-174.
[13] Dusan Djukic, Vladimir Jankovic, Ivan Matic, and Nikola Petrovic. The IMO Compendium-A Collection of Problems Suggested for The International Olympiads: 1959-2009 (Second Edition) [M]. New York: Springer, 2011: pp 33.
[14] ROSEN, KENNETH H. Elementary Number Theory and Its Applications [M]. Massachusetts: ADDISON- WESLEY PUBLISHING COMPANY, 1984: PP 18-24.
[15] Dusan Djukic, Vladimir Jankovic, Ivan Matic, and Nikola Petrovic. The IMO Compendium-A Collection of Problems Suggested for The International Olympiads: 1959-2004 (First Edition) [M]. New York: Springer, 2006: pp 333-370.
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  • APA Style

    Zhang Yue. (2020). A Different Method of Solving a Problem of IMO. International Journal of Systems Science and Applied Mathematics, 5(1), 1-3. https://doi.org/10.11648/j.ijssam.20200501.11

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    ACS Style

    Zhang Yue. A Different Method of Solving a Problem of IMO. Int. J. Syst. Sci. Appl. Math. 2020, 5(1), 1-3. doi: 10.11648/j.ijssam.20200501.11

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    AMA Style

    Zhang Yue. A Different Method of Solving a Problem of IMO. Int J Syst Sci Appl Math. 2020;5(1):1-3. doi: 10.11648/j.ijssam.20200501.11

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  • @article{10.11648/j.ijssam.20200501.11,
      author = {Zhang Yue},
      title = {A Different Method of Solving a Problem of IMO},
      journal = {International Journal of Systems Science and Applied Mathematics},
      volume = {5},
      number = {1},
      pages = {1-3},
      doi = {10.11648/j.ijssam.20200501.11},
      url = {https://doi.org/10.11648/j.ijssam.20200501.11},
      eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijssam.20200501.11},
      abstract = {The IMO performs once a year, and has become an important activity in the field of mathematics. Because the problems in IMO are very difficult, and in general needs two days to finish the test of only six problems, therefore, it is significant to study how to solve and solve those IMO problems with various methods. With respect to question (a) of the problem of discussing, at first, using the so-called “exhaustive method” and the mathematical induction, the paper gets the conclusion of that if n is the integral multiple of 3, subtracting 1 from the nth power of 2 must be divisible by 7. Furthermore, it also proves by use of the disprove method that if n is not the integral multiple of 3, subtracting 1 from the nth power of 2 is impossible to be divisible by 7. The way of solving question (b) is similar to that of solving (a), in order to use the result of question (a) for the third step of the mathematical induction, the paper firstly consider the third power of that 1 added to (k+1)th power of 2 and applying the disprove method proves that it and hence that 1 added to the (k+1)th power of 2 are not divisible by 7, namely the question (b) is true.},
     year = {2020}
    }
    

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    N1  - https://doi.org/10.11648/j.ijssam.20200501.11
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    T2  - International Journal of Systems Science and Applied Mathematics
    JF  - International Journal of Systems Science and Applied Mathematics
    JO  - International Journal of Systems Science and Applied Mathematics
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    AB  - The IMO performs once a year, and has become an important activity in the field of mathematics. Because the problems in IMO are very difficult, and in general needs two days to finish the test of only six problems, therefore, it is significant to study how to solve and solve those IMO problems with various methods. With respect to question (a) of the problem of discussing, at first, using the so-called “exhaustive method” and the mathematical induction, the paper gets the conclusion of that if n is the integral multiple of 3, subtracting 1 from the nth power of 2 must be divisible by 7. Furthermore, it also proves by use of the disprove method that if n is not the integral multiple of 3, subtracting 1 from the nth power of 2 is impossible to be divisible by 7. The way of solving question (b) is similar to that of solving (a), in order to use the result of question (a) for the third step of the mathematical induction, the paper firstly consider the third power of that 1 added to (k+1)th power of 2 and applying the disprove method proves that it and hence that 1 added to the (k+1)th power of 2 are not divisible by 7, namely the question (b) is true.
    VL  - 5
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Author Information
  • Department of Physics, Hunan Normal University, Changsha, China

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