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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.

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✔ There are 98 problems and 78 key facts, waiting for math enthusiasts!
✔ Last solved problem: 95E3 (by Law Tik Shun).
✔ All problems are solved 1964 times.
✔ Number of page views: 50723.Dear Math Enthusiasts,
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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.
Score Table  Math Enthusiasts
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.
Order  Score  # of Correct Answers  Names  Badges  Country  

FS  SF  SP  UT  
1.  071.56  94  Caed Mark Medul Mendoza  6  109  MT  Philippines  
2.  070.35  77  Amirul Faiz Abdul Muthalib  11  0  MT  Malaysia  
3.  068.37  67  Lilanie Monique Torilla  0  55  ❂  S  Philippines 
4.  064.74  73  Jeffrey Robles  15  4  Philippines  
5.  059.88  65  Isaiah James de Dios Maling  4  27  Philippines  
6.  057.16  57  Sumet Ketsri  5  58  Thailand  
7.  055.25  62  Kurara Chibana  4  0  S  Japan  
8.  055.10  54  Willie Revillame Wowowin  0  0  ❂  Philippines  
9.  054.73  63  Russel J. Galanido  1  30  S  South Korea  
10.  046.88  61  Richard Phillip Dimaala Fernandez  4  64  MT  Philippines  
11.  046.55  63  Mertkan Simsek  8  0  MT  Turkey  
12.  044.75  50  Marvin Cato  1  30  MT  Philippines  
13.  038.54  48  Nixon Balandra  3  46  MT  Philippines  
14.  038.15  54  Randy Orton  6  0  United States of America  
15.  037.87  60  Rindell Mabunga  1  31  S  Philippines  
16.  034.69  34  Arjun Singh Rajawat  0  4  ❂  S  India 
17.  034.31  41  Jacob Sabido  0  5  S  Philippines  
18.  028.12  41  Joselito Torculas  4  26  Philippines  
19.  027.83  30  Radu Bogo  0  32  Romania  
20.  026.99  36  Lim Jing Ren  0  0  S  Malaysia  
21.  026.53  26  Alea Astrea  0  0  ❂  S  Philippines 
22.  026.12  32  Law Tik Shun  0  0  MT  Hong Kong  
23.  022.99  26  Kuba Helsztynski  0  0  MT  Poland  
24.  019.39  19  Lenard Guillermo  0  2  ❂  S  Philippines 
25.  018.55  20  Gluttony  0  0  Philippines  
26.  018.34  40  Roenz Joshlee Timbol  0  1  S  Philippines  
27.  016.82  26  Norwyn Nicholson Kah  0  38  S  Philippines  
28.  016.78  25  Jhepoy Dizon  0  0  S  Philippines  
29.  016.02  19  Mark Alvero  0  0  MT  Philippines  
30.  015.83  32  Daniel James Molina  0  1  S  Philippines  
31.  015.12  20  Ralph Macarasig  0  0  S  Philippines  
32.  014.37  26  Joem Canciller  1  0  Philippines  
33.  014.29  14  John Albert A. Reyes  0  6  ❂  S  Philippines 
34.  013.40  17  Nheil Ignacio  0  8  S  Philippines  
35.  012.85  23  Kumar Ayush  0  1  S  India  
36.  012.29  17  Ibrahim Demir  0  1  MT  Turkey  
37.  011.76  22  Christian Paul Patawaran  0  1  S  Philippines  
38.  011.51  19  Bikram Chohan  0  5  S  India  
39.  011.48  15  Srinivas Kanigiri  0  0  S  India  
40.  011.22  11  Melga Sonio  0  0  ❂  S  Philippines 
41.  010.78  13  Poetri Sonya Tarabunga  0  0  S  Indonesia  
42.  010.50  12  Andrew Chiu  0  0  Philippines  
43.  010.20  10  Monu Baba Rura Sirsa Up  0  0  ❂  S  India 
44.  010.00  14  Ewen Goisot  0  17  France  
45.  009.43  43  Kimi No Nawa  8  166  S  Japan  
46.  009.38  21  Christian Daang  1  25  S  Philippines  
47.  009.18  18  James Ericson  1  0  Thailand  
48.  009.09  14  Angelu G. Leynes  1  0  S  Philippines  
49.  008.82  11  Mark Elis Espiridion  0  0  Philippines  
50.  008.43  17  Stefano Ongari  0  0  S  Italy  
51.  008.16  8  Joseph Rodelas  1  0  ❂  S  Philippines 
52.  007.84  13  Chris Norman Algo  0  10  S  Philippines  
53.  007.35  12  Reymark Togno  0  0  Philippines  
54.  007.35  12  Dreimuru Tempest  2  0  S  Philippines  
55.  007.14  7  Adrian Pilotos Burgos  0  0  ❂  Philippines  
56.  006.38  10  Mythreya Lakshman  0  3  India  
57.  006.12  6  Yavuz Selim Koseoglu  2  0  ❂  S  Turkey 
58.  006.12  6  Mahmut Cemrek  0  0  ❂  Turkey  
59.  006.12  6  John Gamal Aziz Attia  2  1  ❂  S  Egypt 
60.  005.67  10  Ikemen  3  0  Japan  
61.  005.00  7  Emmanuel David  0  0  S  Philippines  
62.  004.38  27  Keedgwh  0  0  India  
63.  004.25  5  Sigmund Dela Cruz  0  5  S  Philippines  
64.  004.08  4  Chayapol  0  2  ❂  S  Thailand 
65.  003.64  5  John Lester Tan  0  0  S  Philippines  
66.  003.19  5  Hanelet Santos  0  0  S  Philippines  
67.  003.19  5  Grant Lewis Bulaong  0  1  S  Philippines  
68.  003.06  3  Gerald M. Pascua  0  0  ❂  Philippines  
69.  003.06  3  Barry Villanueva  0  2  ❂  Philippines  
70.  002.83  5  Jake Gacuan  0  9  Philippines  
71.  002.72  4  Mark Lawrence P Velasco  0  0  S  Philippines  
72.  002.62  6  Fred Gutierrez  0  0  S  Philippines  
73.  002.33  4  Smahi Abdeslem  0  4  S  Algeria  
74.  002.13  5  Dan Lang  0  0  Philippines  
75.  002.04  2  Nathanael Joshua Balete  0  0  ❂  Philippines  
76.  002.04  2  Edge Ramos  0  0  ❂  S  Philippines 
77.  002.04  2  Afshiram Muhammed  0  0  ❂  S  Turkey 
78.  001.50  5  John Marco Latagan  0  0  S  Philippines  
79.  001.36  2  Judy Ann Rubante  0  0  S  Philippines  
80.  001.09  4  Gerlie Emata Mantua  0  1  S  Philippines  
81.  001.02  2  Evan Gruda  0  0  S  United States of America  
82.  001.02  1  Suleyman Akarsu  0  0  ❂  Turkey  
83.  001.02  1  Serkan Callioglu  0  0  ❂  Turkey  
84.  001.02  1  Rosendo Parra Milian  0  1  ❂  S  Peru 
85.  001.02  1  Rdvnaksu  1  1  ❂  S  Turkey 
86.  001.02  1  Muhammed Aydogdu  0  0  ❂  Turkey  
87.  001.02  1  Mohamed Karamany  1  0  ❂  Egypt  
88.  001.02  1  Kwok Ho Yin, Jaffy  0  1  ❂  S  Hong Kong 
89.  001.02  1  Captain Magneto  0  0  ❂  S  Germany 
90.  001.02  1  Avrila Frazier  0  1  ❂  MT  United States of America 
91.  001.02  1  Abhishek Singh  0  0  ❂  S  India 
92.  000.82  2  Serdal Aslantas  0  1  MT  Romania  
93.  000.68  2  John Rocel Perez  0  0  S  Philippines  
94.  000.51  1  Prince Ivan Paraiso  0  0  S  Philippines  
95.  000.51  1  Dale Angelo Azas Cortez  0  1  S  Philippines  
96.  000.41  2  Melek Cimen  1  1  S  Turkey  
97.  000.34  1  Ryan Quimo  0  0  S  Philippines  
98.  000.34  1  Mark Allen Facun  0  0  S  Philippines  
99.  000.34  1  John Patrick  0  3  S  Philippines 
 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.
✔ Singapore Mathematical Olympiads, from 2005 to 2013
✔ Solving Equations in Integers by A. O. Gelfond (1981)
✔ Solving Mathematical Problems  A Personal Perspective by Terence Tao (2006)
✔ StreetFighting Mathematics  The Art of Educated Guessing and Opportunistic Problem Solving by Sanjoy Mahajan (2010)
✔ Techniques of Problem Solving by Luis Fernandez, Steven G. Krantz (1997) (Solutions Manual)
✔ Techniques of Problem Solving by Steven G. Krantz (1997)
✔ The Art and Craft of Problem Solving, 2nd Ed. by Paul Zeitz (2007) instructor's manual.
✔ The Art and Craft of Problem Solving, 2nd Ed. by Paul Zeitz (2007)
✔ The Art of Mathematics  Coffee Time in Memphis by BÃ©la Bollob?s (2006)
✔ The Art of Problem Posing, 3rd Ed. by Stephen I. Brown, Marion I. Walter (2005)
✔ The Art of Problem Solving  A Resource for the Mathematics Teacher by Alfred S. Posamentier, Wolfgang Schulz (1996)
✔ The Art of Problem Solving, Vol. 1  The Basics by Sandor Lehoczky, Richard Rusczyk (2006)
✔ The Best Problems From Around the World by Cao Minh Quang (2006)
✔ The Canadian Mathematical Olympiad [19691993] by Michael Doob (1993)
✔ The CauchySchwarz Master Class  An Introduction to the Art of Mathematical Inequalities (2004)
✔ The Colorado Mathematical Olympiad and Further Explorations From the Mountains of Colorado to the Peaks of Mathematics (2011)
✔ The Colossal Book of Mathematics  Classic Puzzles, Paradoxes, and Problems by Martin Gardner (2001)
✔ The Geometry of Numbers by C. D. Olds, Anneli Lax (2000)
✔ The Green Book of Mathematical Problems by Kenneth Hardy, Kenneth S. Williams (1985)
✔ The Higher Arithmetic  An Introduction to the Theory of Numbers, 8th Ed. by H. Davenport (2008)
✔ The IMO Compendium  A Collection of Problems Suggested for the International Mathematical Olympiads [19592009] 2nd Ed. (2011)
✔ The Last Recreations  Hydras, Eggs, and Other Mathematical Mystifications by Martin Gardner (1997)
✔ The Magic Numbers of Dr. Matrix by Martin Gardner (1985)
✔ The Math Problems Notebook by Valentin Boju, Louis Funar (2007)
✔ The Mathemagician and Pied Puzzler  A Collection in Tribute to Martin Gardner by Elwyn R. Berlekamp, Tom Rodgers (1999)
✔ The Mathematical Recreations of Lewis Carroll  Pillow Problems and a Tangled Tale (1958)
✔ The Mathematics of Ciphers  Number Theory and RSA Cryptography by S.C. Coutinho (1999)
✔ The Mathscope  All the Best From Vietnamese Problem Solving Journals (2007)
✔ The Method of Mathematical Induction by I. S. Sominskii (1961)
✔ The Moscow Puzzles  359 Mathematical Recreations by Boris A. Kordemsky (1972)
✔ The New Mathlete Problem Book with Sample Solutions and Appendices (1977)
✔ The New York City Contest Problem Book  Problems and Solutions from the New York City Interscholastic Mathematics..(1986)
✔ The Penguin Book of Curious and Interesting Geometry by David Wells (1991)
✔ The Penguin Book of Curious and Interesting Numbers by David Wells (1986)
✔ The Penguin Book of Curious and Interesting Puzzles by David Wells (1992)
✔ The Pleasures of Pi,e and Other Interesting Numbers by Y. E. O. Adrian (2006)
✔ The Quest for Functions  Functional Equations for the Beginners by Paul Vaderlind (2005)
✔ The Red Book of Mathematical Problems by Kenneth S. Williams, Kenneth Hardy (1988)
✔ The Square Root of 2  A Dialogue Concerning a Number and a Sequence by David Flannery (2006)
✔ The Stanford Mathematics Problem Book  With Hints and Solutions by George Polya, Jeremy Kilpatrick (1974)
✔ The Theory of Numbers  A Text and Source Book of Problems by Andrew Adler, John E. Cloury (1995)
✔ The Unexpected Hanging and Other Mathematical Diversions by Martin Gardner (1991)
✔ The Universe in a Handkerchief  Lewis Carrollâ€™s Mathematical Recreations, Games, Puzzles, and Word Plays (1996)
✔ The USSR Olympiad Problem Book  Selected Problems and Theorems of Elementary Mathematics (1993)
✔ The William Lowell Putnam Mathematical Competition  Problems and Solutions [19381964] (1980)
✔ The William Lowell Putnam Mathematical Competition  Problems and Solutions [1965â€“1984] (1985)
✔ The William Lowell Putnam Mathematical Competition  Problems, Solutions, and Commentary [19852000] (2002)
✔ The William Lowell Putnam Mathematical Competition  Problems, Solutions, and Commentary [20012008] (2008)
✔ The Wohascum County Problem Book by George T. Gilbert (1993)
✔ The Wonders of Magic Squares by Jim Moran (1982)
✔ Time Travel and Other Mathematical Bewilderments by Gardner Martin (1988)
✔ Topics in Algebra and Analysis  Preparing for the Mathematical Olympiad by Radmila Bulajich Manfrino (2015)
✔ Trigonometric Delights by Eli Maor (1998)
✔ USA and International Mathematical Olympiads [2004] by Titu Andreescu, Zuming Feng (2005)
✔ USA and International Mathematical Olympiads [20062007] by Zuming Feng, Yufei Zhao
✔ USA Mathematical Olympiads [19721986] Compiled and with Solutions by Murray S. Klamkin (1988)
✔ Variance on Topics of Plane Geometry by Florentin Smarandache (2013)
✔ What Is the Name of This Book  The Riddle of Dracula and Other Logical Puzzles by Raymond M. Smullyan (1978)
✔ What to Solve  Problems and Suggestions for Young Mathematicians by Judita Cofman (1990)
✔ Wheels, Life and Other Mathematical Amusements by Martin Gardner (1983)
✔ When Less is More  Visualizing Basic Inequalities by Claudi Alsina, Roger Nelsen (2009)
✔ Which Way Did the Bicycle Go by Konhauser, Velleman, Wagon (1996)
✔ Winning Solutions by Edward Lozansky, Cecil Rousseau (1996)

Key Facts
Key Fact Tags: product,rational,number,irrationalKey Fact Tags: fraction,lowest term,numerator,denominator,relatively primeKey Fact Tags: square,even,integer,divisibilityKey Fact Tags: square,odd,integer,divisibilityKey Fact Tags: product,perfect square,consecutive numbersKey Fact Tags: divisibilityKey Fact Tags: identity,factorialKey Fact Tags: identityKey Fact Tags: identityKey Fact Tags: identityKey Fact Tags: factorial,inequalityKey Fact Tags: identityKey Fact Tags: identityKey Fact Tags: SophieGermain,identityKey Fact Tags: identityKey Fact Tags: identity,factorizationKey Fact Tags: identity,factorizationKey Fact Tags: identityKey Fact Tags: composite numberKey Fact Tags: prime number,definitionKey Fact Tags: prime number,composite number,1Key Fact Tags: prime number,composite number,1Key Fact Tags: prime number,evenKey Fact Tags: prime numberKey Fact Tags: prime numbersKey Fact Tags: prime factorizationKey Fact Tags: prime number,composite numberKey Fact Tags: Wilson&#39;s TheoremKey Fact Tags: Bertrand&#39;s Postulate,prime numberKey Fact Tags: Bonse&#39;s Inequality,prime numbersKey Fact Tags: prime,divisorKey Fact Tags: divisibility,square,odd,integerKey Fact Tags: divisibilityKey Fact Tags: divisibilityKey Fact Tags: product,consecutive,integer,divisibilityKey Fact Tags: divisibilityKey Fact Tags: binomial expansionKey Fact Tags: binomial expansion,number of termsKey Fact Tags: identityKey Fact Tags: identityKey Fact Tags: identityKey Fact Tags: identityKey Fact Tags: identityKey Fact Tags: chessboard,squareKey Fact Tags: arithmetic mean,real numbersKey Fact Tags: geometric mean,real numbersKey Fact Tags: arithmetic mean,geometric mean,inequalityKey Fact Tags: harmonic mean,real numbersKey Fact Tags: arithmetic mean,harmonic mean,inequalityKey Fact Tags: inequalityKey Fact Tags: inequalityKey Fact Tags: inequalityKey Fact Tags: inequality,identityKey Fact Tags: inequalityKey Fact Tags: inequalityKey Fact Tags: inequalityKey Fact Tags: inequalityKey Fact Tags: inequalityKey Fact Tags: inequality,identityKey Fact Tags: Cauchy–Schwarz,inequalityKey Fact Tags: Chebyshev,inequalityKey Fact Tags: line,intersection,parabola,abscissa,vertex,midpoint,line segmentKey Fact Tags: De Moivre,formulaKey Fact Tags: fibonacci,numbersKey Fact Tags: finite serie,finite sequenceKey Fact Tags: infinite serie,infinite sequenceKey Fact Tags: partial sum,seriesKey Fact Tags: sequence,partial sum,series,limitKey Fact Tags: sequence,partial sum,series,limitKey Fact Tags: series,sumKey Fact Tags: sum,infinite serieKey Fact Tags: relative error,measurementKey Fact Tags: angle,bisector,triangleKey Fact Tags: sum,exterior angles,convex,polygonKey Fact Tags: triangle,circle,greatest area,isosceles triangleKey Fact Tags: circle,plane,common tangentKey Fact Tags: lattice points,gcd,greatest common divisorKey Fact Tags: right triangle,median
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.
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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