A new problem will be published as soon as the latest problem gets at least 5 solvers. 2 more solvers are needed to release the new problem.
Arf League

Who is Cahit Arf? What is Arf League?
Cahit Arf (11 October 1910 â€“ 26 December 1997) was a Turkish mathematician. He is known for the Arf invariant of a quadratic form in characteristic 2 (applied in knot theory and surgery theory) in topology, the Hasseâ€“Arf theorem in ramification theory, Arf semigroups, and Arf rings. For more information please check Wikipedia.
Arf League is for enthusiasts who seeks challenging math problems. Please feel free to send your solutions or general comments to challengingproblems@ahmath.com.

Info and Announcements
Info
✔ There are 104 problems and 82 key facts, waiting for math enthusiasts! Dear Math Enthusiasts,
Thanks for your interest in Arf League.
✔ Among 1.12 billion websites, AhMath is now within the top 0.036%, and we hope its position will get better soon. We try to serve the best for all of you.
✔ Please feel free to send your comments and suggestions to challengingproblems@ahmath.com and help us to promote this lovely activity by distributing the
Arf League Poster and the QR Code.
✔ Arf League is looking for new problem and/or key fact writers. If you are interested in writing challenging math problems and/or key facts, please contact us. We'll be glad to meet with new contractors.
hide this To submit your answers you need to log in.
If you were not a member of this page and if your name is not listed in the score tables, you need to sign up first. Otherwise, please read the following instructions given below!
hide this This is the place where you can find the most valuable Mathematical Olympiads books!
 New Problem
 Previous Problems
 Score Tables
 PDR+Sources
 Contractors
 Testimonials
 Donate / Become a Patron
Format of the items of the selection menu: Problem Order, Problem Code (Difficulty Level). Red problems have a difficulty level of less than 0.75, thus, as an example, a 0.18 means only 18% of all answers submitted are correct.
Futured Math Enthusiasts
is solved firstly by
Mohamed Karamany
BA5D
by Hanelet Santos
Long Jumper
Marathon Runner
55709
Score Table  Math Enthusiasts (only the first 100)
Score depends on number of correct answers and number of trials.
 FS stands for First Solver. For each problem, there is a first solver. Each time you solve a problem as the first person your FS score increases by one.
 SF stands for Solution Files. Some of the Math Enthusiasts sends solutions to the problems. This is to promote their valuable contribution to the Arf League.
 SP stands for Sniper. You can get this badge if and only if you do not miss any problem in your first try and solve them correctly. Only snipers have the potential to get the full score.
 UT stands for User Type. "MT" is for Math Teacher and "S" is for Student.
hide this Please submit your country to be listed in this table!
Order  Score  # of Correct Answers  Names  Badges  Country  

FS  SF  SP  UT  
1.  072.06  87  Isaiah James de Dios Maling  4  27  Philippines  
2.  070.86  99  Caed Mark Medul Mendoza  8  109  MT  Philippines  
3.  069.42  95  Richard Phillip Dimaala Fernandez  6  90  MT  Philippines  
4.  066.48  78  Amirul Faiz Abdul Muthalib  11  0  MT  Malaysia  
5.  064.42  67  Lilanie Monique Torilla  0  55  ❂  S  Philippines 
6.  064.01  84  Jereco Dela Pena  0  0  MT  Philippines  
7.  063.84  79  Jeffrey Robles  15  4  Philippines  
8.  053.86  57  Sumet Ketsri  5  58  Thailand  
9.  052.06  62  Kurara Chibana  4  0  S  Japan  
10.  051.92  54  Willie Revillame Wowowin  0  0  ❂  Philippines  
11.  051.57  63  Russel J. Galanido  1  30  S  South Korea  
12.  043.87  63  Mertkan Simsek  8  0  MT  Turkey  
13.  042.17  50  Marvin Cato  1  30  MT  Philippines  
14.  041.98  49  Radu Bogo  0  40  Romania  
15.  038.81  52  Law Tik Shun  0  0  MT  Hong Kong  
16.  036.32  48  Nixon Balandra  3  46  MT  Philippines  
17.  035.95  54  Randy Orton  6  0  United States of America  
18.  035.69  60  Rindell Mabunga  1  31  S  Philippines  
19.  033.65  35  Arjun Singh Rajawat  0  4  ❂  S  India 
20.  032.33  41  Jacob Sabido  0  5  S  Philippines  
21.  027.97  40  Darwin Ojera  0  22  MT  Philippines  
22.  026.50  41  Joselito Torculas  4  26  Philippines  
23.  025.61  57  Roenz Joshlee Timbol  0  1  S  Philippines  
24.  025.43  36  Lim Jing Ren  0  0  S  Malaysia  
25.  025.00  26  Alea Astrea  0  0  ❂  S  Philippines 
26.  021.67  26  Kuba Helsztynski  0  0  MT  Poland  
27.  018.27  19  Lenard Guillermo  0  2  ❂  S  Philippines 
28.  017.48  20  Gluttony  0  0  Philippines  
29.  016.85  29  Joseph Rodelas  1  0  S  Philippines  
30.  015.85  26  Norwyn Nicholson Kah  0  38  S  Philippines  
31.  015.81  25  Jhepoy Dizon  0  0  S  Philippines  
32.  015.09  19  Mark Alvero  0  0  MT  Philippines  
33.  014.92  32  Daniel James Molina  0  1  S  Philippines  
34.  014.25  20  Ralph Macarasig  0  0  S  Philippines  
35.  013.54  26  Joem Canciller  1  0  Philippines  
36.  013.46  14  John Albert A. Reyes  0  6  ❂  S  Philippines 
37.  012.63  17  Nheil Ignacio  0  8  S  Philippines  
38.  012.11  23  Kumar Ayush  0  1  S  India  
39.  011.60  28  Mark Kenneth Dairo  1  26  S  Philippines  
40.  011.58  17  Ibrahim Demir  0  1  MT  Turkey  
41.  011.13  18  Francis Gerard Magtibay  0  0  S  Philippines  
42.  011.08  22  Christian Paul Patawaran  0  1  S  Philippines  
43.  010.85  19  Bikram Chohan  0  5  S  India  
44.  010.82  15  Srinivas Kanigiri  0  0  S  India  
45.  010.58  11  Melga Sonio  0  0  ❂  S  Philippines 
46.  010.16  13  Poetri Sonya Tarabunga  0  0  S  Indonesia  
47.  009.89  12  Andrew Chiu  0  0  Philippines  
48.  009.62  10  Monu Baba Rura Sirsa Up  0  0  ❂  S  India 
49.  009.42  14  Ewen Goisot  0  17  France  
50.  008.89  43  Kimi No Nawa  8  166  S  Japan  
51.  008.83  21  Christian Daang  1  25  S  Philippines  
52.  008.65  18  James Ericson  1  0  Thailand  
53.  008.57  14  Angelu G. Leynes  1  0  S  Philippines  
54.  008.31  11  Mark Elis Espiridion  0  0  Philippines  
55.  007.94  17  Stefano Ongari  0  0  S  Italy  
56.  007.85  14  Gideon Tomas  0  0  S  Philippines  
57.  007.39  13  Chris Norman Algo  0  10  S  Philippines  
58.  006.92  12  Reymark Togno  0  0  Philippines  
59.  006.92  12  Dreimuru Tempest  2  0  S  Philippines  
60.  006.73  7  Adrian Pilotos Burgos  0  0  ❂  Philippines  
61.  006.01  10  Mythreya Lakshman  0  3  India  
62.  005.77  6  Yavuz Selim Koseoglu  2  0  ❂  S  Turkey 
63.  005.77  6  Mahmut Cemrek  0  0  ❂  Turkey  
64.  005.77  6  John Gamal Aziz Attia  2  1  ❂  S  Egypt 
65.  005.56  9  Blake Herrera  0  0  S  United States of America  
66.  005.34  10  Ikemen  3  0  Japan  
67.  004.71  7  Emmanuel David  0  0  S  Philippines  
68.  004.12  27  Keedgwh  0  0  India  
69.  004.01  5  Sigmund Dela Cruz  0  5  S  Philippines  
70.  003.85  4  Chayapol  0  2  ❂  S  Thailand 
71.  003.85  4  Βαρελάς Γεώρ𝛾ιος  0  0  ❂  Greece  
72.  003.43  5  John Lester Tan  0  0  S  Philippines  
73.  003.25  9  Anjelo V. Racho  0  0  S  Philippines  
74.  003.00  5  Grant Lewis Bulaong  0  1  S  Philippines  
75.  002.93  8  Hanelet Santos  0  0  S  Philippines  
76.  002.88  3  Gerald M. Pascua  0  0  ❂  Philippines  
77.  002.88  3  Barry Villanueva  0  2  ❂  Philippines  
78.  002.88  3  Andrei Alido  0  0  ❂  S  Philippines 
79.  002.67  5  Jake Gacuan  0  9  Philippines  
80.  002.56  4  Mark Lawrence P Velasco  0  0  S  Philippines  
81.  002.47  6  Fred Gutierrez  0  0  S  Philippines  
82.  002.20  4  Smahi Abdeslem  0  4  S  Algeria  
83.  002.00  5  Dan Lang  0  0  Philippines  
84.  001.92  2  Nathanael Joshua Balete  0  0  ❂  Philippines  
85.  001.92  2  Jaydee Lucero  0  2  ❂  S  Philippines 
86.  001.92  2  Edge Ramos  0  0  ❂  S  Philippines 
87.  001.92  2  Avishek Yadav  1  0  ❂  S  India 
88.  001.92  2  Afshiram Muhammed  0  0  ❂  S  Turkey 
89.  001.41  5  John Marco Latagan  0  0  S  Philippines  
90.  001.28  2  Judy Ann Rubante  0  0  S  Philippines  
91.  001.03  4  Gerlie Emata Mantua  0  1  S  Philippines  
92.  000.96  2  Evan Gruda  0  0  S  United States of America  
93.  000.96  1  Suleyman Akarsu  0  0  ❂  Turkey  
94.  000.96  1  Serkan Callioglu  0  0  ❂  Turkey  
95.  000.96  1  Rosendo Parra Milian  0  1  ❂  S  Peru 
96.  000.96  1  Muhammed Aydogdu  0  0  ❂  Turkey  
97.  000.96  1  Mohamed Karamany  1  0  ❂  Egypt  
98.  000.96  1  Kwok Ho Yin, Jaffy  0  1  ❂  S  Hong Kong 
99.  000.96  1  Captain Magneto  0  0  ❂  S  Germany 
100.  000.96  1  Avrila Frazier  0  1  ❂  MT  United States of America 
hide this All problems are solved 2368 times.
Score Table  Countries
Country  # of Correct Answers  # of Trials  Performance 

Turkey  100  140  71.43% 
Philippines  1462  2140  68.32% 
Hong Kong  53  68  77.94% 
India  143  331  43.2% 
Canada  1  2  50% 
Egypt  7  7  100% 
South Korea  63  74  85.14% 
Greece  4  4  100% 
Thailand  79  98  80.61% 
United States of America  66  97  68.04% 
Indonesia  13  16  81.25% 
Japan  115  289  39.79% 
Malaysia  104  127  81.89% 
Peru  1  1  100% 
Algeria  4  7  57.14% 
Italy  17  35  48.57% 
Romania  51  60  85% 
Germany  1  1  100% 
France  14  20  70% 
Poland  26  30  86.67% 
Iran  2  8  25% 
Nigeria  1  1  100% 
 Arf League Problems Discussion Room (It takes a couple of seconds to load! Please wait...)

Worked Solutions to the Problems
Solutions to all Challenging Math Problems are ready!
If you want to get the pdf of all the questions with solutions please do the payment by using the paypal button given on the left. For more documents and info about olympiads, please click Olympiads.
 Books, Websites, etc.
Key Facts

1. Key Fact Tags: product,rational,number,irrational

2. Key Fact Tags: fraction,lowest term,numerator,denominator,relatively prime

3. Key Fact Tags: square,even,integer,divisibility

4. Key Fact Tags: square,odd,integer,divisibility

5. Key Fact Tags: product,perfect square,consecutive numbers

6. Key Fact Tags: divisibility

7. Key Fact Tags: identity,factorial

8. Key Fact Tags: identity

9. Key Fact Tags: identity

10. Key Fact Tags: identity

11. Key Fact Tags: factorial,inequality

12. Key Fact Tags: identity

13. Key Fact Tags: identity

14. Key Fact Tags: SophieGermain,identity

15. Key Fact Tags: identity

16. Key Fact Tags: identity,factorization

17. Key Fact Tags: identity,factorization

18. Key Fact Tags: identity

19. Key Fact Tags: composite number

20. Key Fact Tags: prime number,definition

21. Key Fact Tags: prime number,composite number,1

22. Key Fact Tags: prime number,composite number,1

23. Key Fact Tags: prime number,even

24. Key Fact Tags: prime number

25. Key Fact Tags: prime numbers

26. Key Fact Tags: prime factorization

27. Key Fact Tags: prime number,composite number

28. Key Fact Tags: Wilson&#39;s Theorem

29. Key Fact Tags: Bertrand&#39;s Postulate,prime number

30. Key Fact Tags: Bonse&#39;s Inequality,prime numbers

31. Key Fact Tags: prime,divisor

32. Key Fact Tags: divisibility,square,odd,integer

33. Key Fact Tags: divisibility

34. Key Fact Tags: divisibility

35. Key Fact Tags: product,consecutive,integer,divisibility

36. Key Fact Tags: divisibility

37. Key Fact Tags: binomial expansion

38. Key Fact Tags: binomial expansion,number of terms

39. Key Fact Tags: identity

40. Key Fact Tags: identity

41. Key Fact Tags: identity

42. Key Fact Tags: identity

43. Key Fact Tags: identity

44. Key Fact Tags: chessboard,square

45. Key Fact Tags: arithmetic mean,real numbers

46. Key Fact Tags: geometric mean,real numbers

47. Key Fact Tags: arithmetic mean,geometric mean,inequality

48. Key Fact Tags: harmonic mean,real numbers

49. Key Fact Tags: arithmetic mean,harmonic mean,inequality

50. Key Fact Tags: inequality

51. Key Fact Tags: inequality

52. Key Fact Tags: inequality

53. Key Fact Tags: inequality,identity

54. Key Fact Tags: inequality

55. Key Fact Tags: inequality

56. Key Fact Tags: inequality

57. Key Fact Tags: inequality

58. Key Fact Tags: inequality

59. Key Fact Tags: inequality,identity

60. Key Fact Tags: Cauchy–Schwarz,inequality

61. Key Fact Tags: Chebyshev,inequality

62. Key Fact Tags: line,intersection,parabola,abscissa,vertex,midpoint,line segment

63. Key Fact Tags: De Moivre,formula

64. Key Fact Tags: fibonacci,numbers

65. Key Fact Tags: finite serie,finite sequence

66. Key Fact Tags: infinite serie,infinite sequence

67. Key Fact Tags: partial sum,series

68. Key Fact Tags: sequence,partial sum,series,limit

69. Key Fact Tags: sequence,partial sum,series,limit

70. Key Fact Tags: series,sum

71. Key Fact Tags: sum,infinite serie

72. Key Fact Tags: relative error,measurement

73. Key Fact Tags: angle,bisector,triangle

74. Key Fact Tags: sum,exterior angles,convex,polygon

75. Key Fact Tags: triangle,circle,greatest area,isosceles triangle

76. Key Fact Tags: circle,plane,common tangent

77. Key Fact Tags: lattice points,gcd,greatest common divisor

78. Key Fact Tags: right triangle,median

79. Key Fact Tags: minimal polynomial

80. Key Fact Tags: british flag theorem, rectangle, geometry

81. Key Fact Tags: pythagorean triple, triangle

82. Key Fact Tags: primitive pythagorean triples, triangle
If you want to be a contractor (problem writer, key fact writer, etc.), please contact AhMath by sending an email to challengingproblems@ahmath.com.
Amirul Faiz Abdul Muthalib
Problem Writer
Number of problems: 10With over 5 years of handson, successful teaching experience, Amirul is an enthusiastic PreUniversity Mathematics Lecturer in INTEC Education College, Malaysia. Being a versatile individual who love Maths, he has earned a Masterâ€™s Degree in Mechanical Engineering in Japan. He had opportunity to teach Maths in Japanese Syllabus for scholarship sponsored students who will further their study in KOSEN (National College of Technology in Japan). He also train some of his students for the National Olympiad every year.
Ä°hsan YÃ¼cel
Problem Writer
Number of problems: 3I am currently in the thesis phase of my postgraduate education which is about mathematics teaching education. I got some notable national and international achievements in various secondary school mathematics national and international project competitions. Likewise, my students also got many elementary and secondary level mathematical olympics achievements. In the recent past, my students got the first and the second place in the TÃœBÄ°TAK National Secondary School Research Projects Contest. One of my students, a team member of the national math team, received a bronze medal in IMO. I'm an author of many scientific articles about mathematics in the quarterly magazine "Mathematical World". I'm the founder of Matematik MeraklÄ±sÄ±, which is a website dedicated to Mathematics.
You?
Subject?
Leave a Comment!
Please use the form below to send your general comments about Arf League. All fields are required!
It's a good way to practice maths and also the site encourages you to submit your answer with an accompanying solution. Participants can not just simply guess the answer or predict what will be the answer based on the given, they need to know and understand the topic. Also this refrains participants to just simply ask the answer from other people. So the method was excellent and people will definitely learn many things about math.
 Rindell Mabunga / Philippines
I want to say my deepest gratitude in creating this website. The questions are not the typical type and not normally taught to nonmathematical major degrees. I really hope that you'll continue your mission and vision in creating this site, as this serves as a platform to enhance our mathematical skills. Hoping for the best good luck!!!
 Russel J. Galanido / South Korea
I see AhMath as the one encouraging selfpaced learning or practice in solving math problems. It does not put me into any kind of pressure like requiring myself to be the first one to solve the problems correctly and giving answers with a limited time. Also, it gives me excitement for the next problems to solve as well as for the appearance of my name on the list of the people who submitted the correct solutions and answers for those math problems. Overall, it gives me confidence to be better in math.
Thank you for everything, AhMath. Marvin Cato / Philippines
I knew AhMath from a facebook group, "Math Enthusiasts Quiz Group", that I just joined a few weeks ago. Then I found that there are the "challenging math problems" here that refresh my Math skill.
I like many ideas on this website, the word "Math Enthusiasts" exactly describes what I'm and it makes me recall the feeling about Math when I was young, it was fun and exciting!
I like the motto "Real than i, rational than n", it's playing in my head and I'm still doubt about its meaning :D
There is no time constraint for solving the problems so we can think about the solutions in the different ways. Sometimes the new solutions reveal the beauty of Math and that only happens when we have time to think about other solutions.
Thank you the creator and moderator of this website, especially Ahmet Arduc, who brings me back to the feeling of fun and exciting with his creative problems. Sumet Ketsri / Thailand
Ahmath is basically a great site. The problems are really challenging and mind boggling. I really like the way that the Math enthusiasts should show the solution as well for each problem. You need to learn on your own. This site greatly help me improve my Math skills and writing solutions as well. Thank you much Ahmath for an amazing job. More power and God bless. "Train HARD, win EASY. Train Easy, win HARD". Hoping you all the best out there. Just continue on your aim to help the students as well as the nonstudents to further enhance their math skills. God bless and More power AHMATH FTW!!!
 Isaiah James de Dios Maling / Philippines