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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 104 problems and 82 key facts, waiting for math enthusiasts!
✔ Last solved problem: 1DAA (by Richard Phillip Dimaala Fernandez).
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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.
Futured Math Enthusiasts
is solved firstly by
Mohamed Karamany
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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.  069.96  98  Caed Mark Medul Mendoza  8  109  MT  Philippines  
3.  066.48  78  Amirul Faiz Abdul Muthalib  11  0  MT  Malaysia  
4.  066.00  90  Richard Phillip Dimaala Fernandez  6  87  MT  Philippines  
5.  064.42  67  Lilanie Monique Torilla  0  55  ❂  S  Philippines 
6.  063.84  79  Jeffrey Robles  15  4  Philippines  
7.  057.46  70  Jereco Dela Pena  0  0  MT  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.  039.74  45  Radu Bogo  0  39  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  Hanelet Santos  0  0  S  Philippines  
75.  003.00  5  Grant Lewis Bulaong  0  1  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  Afshiram Muhammed  0  0  ❂  S  Turkey 
88.  001.41  5  John Marco Latagan  0  0  S  Philippines  
89.  001.28  2  Judy Ann Rubante  0  0  S  Philippines  
90.  001.03  4  Gerlie Emata Mantua  0  1  S  Philippines  
91.  000.96  2  Evan Gruda  0  0  S  United States of America  
92.  000.96  1  Suleyman Akarsu  0  0  ❂  Turkey  
93.  000.96  1  Serkan Callioglu  0  0  ❂  Turkey  
94.  000.96  1  Rosendo Parra Milian  0  1  ❂  S  Peru 
95.  000.96  1  Muhammed Aydogdu  0  0  ❂  Turkey  
96.  000.96  1  Mohamed Karamany  1  0  ❂  Egypt  
97.  000.96  1  Kwok Ho Yin, Jaffy  0  1  ❂  S  Hong Kong 
98.  000.96  1  Captain Magneto  0  0  ❂  S  Germany 
99.  000.96  1  Avrila Frazier  0  1  ❂  MT  United States of America 
100.  000.96  1  Avishek Yadav  1  0  ❂  S  India 
hide this All problems are solved 2339 times.
Score Table  Countries
Country  # of Correct Answers  # of Trials  Performance 

Turkey  100  140  71.43% 
Philippines  1439  2095  68.69% 
Hong Kong  53  68  77.94% 
India  142  330  43.03% 
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  47  54  87.04% 
Germany  1  1  100% 
France  14  20  70% 
Poland  26  30  86.67% 
Iran  2  8  25% 
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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

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.
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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!!!
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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