A Different Method of Solving a Problem of IMO

Zhang Yue

doi:doi:10.11648/j.ijssam.20200501.11

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A Different Method of Solving a Problem of IMO

Zhang Yue

Published in International Journal of Systems Science and Applied Mathematics (Volume 5, Issue 1)

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), 2024. Published by Science Publishing Group

Keywords

IMO, The Mathematical Induction, Algebraic Equation, Disprove Method

References

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[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.
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[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.
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[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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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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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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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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TY  - JOUR
T1  - A Different Method of Solving a Problem of IMO
AU  - Zhang Yue
Y1  - 2020/04/01
PY  - 2020
N1  - https://doi.org/10.11648/j.ijssam.20200501.11
DO  - 10.11648/j.ijssam.20200501.11
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
SP  - 1
EP  - 3
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20200501.11
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
IS  - 1
ER  -

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Author Information

Zhang Yue

Department of Physics, Hunan Normal University, Changsha, China

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Plain Text BibTeX RIS

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

Copy | Download

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

Copy | Download

@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}
}

Copy | Download

TY  - JOUR
T1  - A Different Method of Solving a Problem of IMO
AU  - Zhang Yue
Y1  - 2020/04/01
PY  - 2020
N1  - https://doi.org/10.11648/j.ijssam.20200501.11
DO  - 10.11648/j.ijssam.20200501.11
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
SP  - 1
EP  - 3
PB  - Science Publishing Group
SN  - 2575-5803
UR  - https://doi.org/10.11648/j.ijssam.20200501.11
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
IS  - 1
ER  -

Copy | Download