AhMath Real than i, rational than π

# Arf League

• Who is Cahit Arf?
Cahit Arf (11 October 1910 – 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 Hasse–Arf theorem in ramification theory, Arf semigroups, and Arf rings. For more information please check Wikipedia.
• What is Arf League?
Arf League is for enthusiasts who seeks challenging math problems. Please feel free to send your solutions or general comments to
challenging-problems@ahmath.com.
• Announcements
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 challenging-problems@ahmath.com and help us to promote this lovely activity by distributing the Arf League Poster and the QR Code.

There are 95 problems and 78 key facts, waiting for math enthusiasts! All problems are solved 1810 times.

hide this Dear Math Enthusiasts,
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,

From now on, a new problem will be published as soon as the latest problem gets at least 10 solvers. 3 more solvers are needed to release the new problem.

95. Problem: 9EC7 , proposed by Ahmet Arduc
95 is the number of planar partitions of 10.

### Previous Problems

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.

Number of Responses: 2690
Number of Correct R.: 1803
Difficulty Level: 67.03
1. Problem: 595A , proposed by Ahmet Arduc, Tip: Key Fact(s): 7695
1 is the multiplicative identity.
Correct answers have been submitted by:
1. Mohamed Karamany
2. Jeffrey Robles
3. Joseph Rodelas
4. Mahmut Cemrek
5. Suleyman Akarsu
6. Yavuz Selim Koseoglu
7. John Gamal Aziz Attia
8. Serkan Callioglu
9. Βαρελάς Γεώρ𝛾ιος
10. Marvin Cato
11. Isaiah James de Dios Maling
12. Joselito Torculas
13. John Albert A. Reyes
14. Nheil Ignacio
15. Gerald M. Pascua
16. Angelu G. Leynes
17. Roenz Joshlee Timbol
18. Rindell Mabunga
19. Caed Mark Medul Mendoza
20. Russel J. Galanido
21. Melga Sonio
22. Mark Elis Espiridion
24. Sumet Ketsri
25. Daniel James Molina
26. Nixon Balandra
27. Jacob Sabido
28. Poetri Sonya Tarabunga
29. John Patrick
30. Lilanie Monique Torilla
31. Chris Norman Algo
32. Christian Daang
33. Grant Lewis Bulaong
34. Richard Phillip Dimaala Fernandez
35. Jake Gacuan
36. Ibrahim Demir
37. Christian Paul Patawaran
38. Amirul Faiz Abdul Muthalib
39. Norwyn Nicholson Kah
40. Jhepoy Dizon
41. Ralph Macarasig
42. Dan Lang
43. Chayapol
44. Dreimuru Tempest
45. Kurara Chibana
46. Lim Jing Ren
47. Alea Astrea
48. Kumar Ayush
49. Joem Canciller
50. Gluttony
51. Serdal Aslantas
53. Barry Villanueva
54. Randy Orton
55. Captain Magneto
56. Willie Revillame Wowowin
57. Mertkan Simsek
58. Mark Lawrence P Velasco

Please complete the following form to join to this activity. Be honest and do not register more than once! If you forget your user code please send an email to challenging-problems@ahmath.com.

I need a user code
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This is the second version, starting with the 13th problem. In this version, you'll see a full-functional 'Send Your Answer' section. Please feel free to send your solution(s) by using the textarea given in this section. We'll not publish any of the solutions here on this page.
In the 'Score Tables' section, we use different algorithms and/or parameters for different tables. In time, we may change the algorithm and/or parameters of any table. Therefore, your ranking and score may change even if you do not do any further activity.
By registering to Arf League, you give us the permission
• to use your solutions in any printed and/or published documents as a second/third/... way of the related problem.
By registering to Arf League, you agree that
• you are not allowed to copy, distribute, share or reproduce the problems and/or solutions without prior written approval from Ahmet Arduc.
Be honest and do not register more than once! If you forget your user code please send an email to challenging-problems@ahmath.com.
If you notice any problem (bug, typo in any question, incorrect answer, etc.) please feel free either to send us a message via Contact or to challenging-problems@ahmath.com.
Best Regards,
AhMath

## 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.

FSSFSP
1.072.57
77Amirul Faiz Abdul Muthalib110 Malaysia
2.072.04
91Caed Mark Medul Mendoza5107 Philippines
3.069.47
66Lilanie Monique Torilla055 Philippines
4.065.75
72Jeffrey Robles144 Philippines
5.060.73
64Isaiah James de Dios Maling426 Philippines
6.058.97
57Sumet Ketsri558 Thailand
7.056.99
62Kurara Chibana40 Japan
8.056.84
54Willie Revillame Wowowin00 Philippines
9.056.46
63Russel J. Galanido130 South Korea
10.046.17
50Marvin Cato130 Philippines
11.045.66
60Mertkan Simsek80 Turkey
12.039.76
48Nixon Balandra346 Philippines
13.039.35
54Randy Orton60 United States of America
14.039.07
60Rindell Mabunga131 Philippines
15.034.37
40Jacob Sabido05 Philippines
16.033.41
44Richard Phillip Dimaala Fernandez445 Philippines
17.029.01
41Joselito Torculas426 Philippines
18.027.84
36Lim Jing Ren00 Malaysia
19.027.37
26Arjun Singh Rajawat04 India
20.027.37
26Alea Astrea00 Philippines
21.025.30
22.020.00
19Lenard Guillermo02 Philippines
23.019.14
20Gluttony00 Philippines
24.018.92
40Roenz Joshlee Timbol01 Philippines
25.017.36
26Norwyn Nicholson Kah038 Philippines
26.017.31
25Jhepoy Dizon00 Philippines
27.016.52
19Mark Alvero00 Philippines
28.016.33
32Daniel James Molina01 Philippines
29.015.59
20Ralph Macarasig00 Philippines
30.014.82
26Joem Canciller10 Philippines
31.014.74
14John Albert A. Reyes06 Philippines
32.013.83
17Nheil Ignacio08 Philippines
33.013.26
23Kumar Ayush01 India
34.012.68
17Ibrahim Demir01 Turkey
35.012.13
22Christian Paul Patawaran01 Philippines
36.011.58
11Melga Sonio00 Philippines
37.011.58
11Angelu G. Leynes10 Philippines
38.011.12
13Poetri Sonya Tarabunga00 Indonesia
39.010.83
12Andrew Chiu00 Philippines
40.010.32
14Ewen Goisot017 France
41.009.88
13Srinivas Kanigiri00 India
42.009.83
43Kimi No Nawa7166 Japan
43.009.47
18James Ericson10 Thailand
44.009.10
11Mark Elis Espiridion00 Philippines
45.008.96
20Christian Daang125 Philippines
46.008.69
17Stefano Ongari00 Italy
47.008.42
8Monu Baba Rura Sirsa Up00 India
48.008.42
8Joseph Rodelas10 Philippines
49.007.96
11Chris Norman Algo06 Philippines
50.007.58
12Reymark Togno00 Philippines
51.007.58
12Dreimuru Tempest20 Philippines
52.007.37
53.006.32
6Yavuz Selim Koseoglu20 Turkey
54.006.32
6Mahmut Cemrek00 Turkey
55.006.32
6John Gamal Aziz Attia21 Egypt
56.005.85
10Ikemen30 Japan
57.005.16
7Emmanuel David00 Philippines
58.004.39
5Sigmund Dela Cruz05 Philippines
59.004.21
4Chayapol02 Thailand
60.004.21
4Βαρελάς Γεώρ𝛾ιος00 Greece
61.003.76
5John Lester Tan00 Philippines
62.003.29
5Hanelet Santos00 Philippines
63.003.29
5Grant Lewis Bulaong01 Philippines
64.003.16
3Gerald M. Pascua00 Philippines
65.003.16
3Barry Villanueva02 Philippines
66.002.92
5Jake Gacuan09 Philippines
67.002.88
19Keedgwh00 India
68.002.81
4Mark Lawrence P Velasco00 Philippines
69.002.71
6Fred Gutierrez00 Philippines
70.002.41
4Smahi Abdeslem04 Algeria
71.002.19
5Dan Lang00 Philippines
72.002.11
2Melek Cimen10 Turkey
73.002.11
2Edge Ramos00 Philippines
74.002.11
2Afshiram Muhammed00 Turkey
75.001.55
5John Marco Latagan00 Philippines
76.001.40
2Judy Ann Rubante00 Philippines
77.001.05
2Evan Gruda00 United States of America
78.001.05
1Suleyman Akarsu00 Turkey
79.001.05
1Serkan Callioglu00 Turkey
80.001.05
1Rosendo Parra Milian01 Peru
81.001.05
1Rdvnaksu11 Turkey
82.001.05
1Muhammed Aydogdu00 Turkey
83.001.05
1Mohamed Karamany10 Egypt
84.001.05
1Captain Magneto00 Germany
85.001.05
1Avrila Frazier01 United States of America
86.001.05
1Abhishek Singh00 India
87.000.84
2Serdal Aslantas01 Romania
88.000.70
2John Rocel Perez00 Philippines
89.000.35
1Ryan Quimo00 Philippines
90.000.35
1Mark Allen Facun00 Philippines
91.000.35
1John Patrick03 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.

• Books, Websites, etc.
✔   Solving Equations in Integers by A. O. Gelfond (1981)
✔   Solving Mathematical Problems - A Personal Perspective by Terence Tao (2006)
✔   Street-Fighting 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 Béla 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 Cauchy-Schwarz 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 [1959-2009] 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 Carroll’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 [1938-1964] (1980)
✔   The William Lowell Putnam Mathematical Competition - Problems and Solutions [1965–1984] (1985)
✔   The William Lowell Putnam Mathematical Competition - Problems, Solutions, and Commentary [1985-2000] (2002)
✔   The William Lowell Putnam Mathematical Competition - Problems, Solutions, and Commentary [2001-2008] (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 [2006-2007] by Zuming Feng, Yufei Zhao
✔   USA Mathematical Olympiads [1972-1986] 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 7446
The product of a rational number and an irrational number is irrational. Ah Math

Key Fact 7695
For a fraction to be in lowest terms (or, to be written in its simplest form), its numerator and denominator must be relatively prime. Ah Math

The square of any even integer is of form $4k$. Ah Math

Key Fact BAB3
The square of any odd integer is of form $4k+1$. Ah Math

Key Fact 988C
The product of four consecutive natural numbers is never a perfect square. Ah Math

Key Fact 9A34
$$1001=7\cdot11\cdot13$$ Ah Math

Key Fact B664
$$n\cdot n!=(n+1)!-n!$$ Ah Math

Key Fact 969B
$$\frac{1}{n\cdot(n+1)}=\frac{1}{n}-\frac{1}{n+1}$$ Ah Math

Key Fact CE81
$$\frac{1}{n \cdot (n+m)}=\frac{1}{m} \left( \frac{1}{n}-\frac{1}{n+m} \right)$$ Ah Math

Key Fact B945
$$\frac{1}{n(n+1)(n+2)}=\frac{1}{2}\cdot \left[ \frac{1}{n(n+1)} - \frac{1}{(n+1)(n+2)} \right]$$ Ah Math

Key Fact BCC8
$$\frac{n}{(n+1)!}=\frac{1}{n!}-\frac{1}{(n+1)!}$$ Ah Math

Key Fact 965A
$$n^4+n^2+1=(n^2+1-n)\cdot(n^2+1+n)$$ Ah Math

Key Fact E856
$$(n+1)^2-(n+1)+1=n^2+n+1$$ Ah Math

Key Fact 7C45
$\small{\textbf{Sophie-Germain Identity}}$: \begin{align}a^4+4b^4&\cssId{Step1}{=\left[(a+b)^2+b^2\right]\left[(a-b)^2+b^2\right]}\\&\cssId{Step1}{=(a^2+2ab+2b^2)(a^2-2ab+2b^2)} \end{align} Ah Math

Key Fact EA1C
$$x^4+y^4+z^4=(x^2+y^2+z^2)^2-2(x^2y^2+y^2z^2+x^2z^2)$$ Ah Math

Key Fact 3E17
For odd $n \in N$, $$a^n+b^n=(a+b)(a^{n-1}-a^{n-2} \cdot b \,+\,...\,-\,a \cdot b^{n-2}+b^{n-1})$$ Ah Math

Key Fact 2545
For all $n \in N$, $$a^n-b^n=(a-b)(a^{n-1}+a^{n-2} \cdot b \,+\,...\,+\,a \cdot b^{n-2}+b^{n-1})$$ Ah Math

Key Fact A376
For $a=k\cdot (2k+1)$ where $k \in N$,$$a^2+(a+1)^2+⋯+(a+k)^2\\=(a+k+1)^2+(a+k+2)^2+⋯+(a+2k)^2$$ Ah Math

Key Fact 3278
A composite number is a positive integer greater than 1 that has more than two positive divisors. Ah Math

Key Fact 2DCC
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. Ah Math

Key Fact 87D4
All positive integers greater than 1 are either prime or composite. Ah Math

Key Fact 9E11
1 is the only positive integer that is neither prime nor composite. Ah Math

Key Fact A478
The only even prime number is 2. Ah Math

Key Fact 2AB2
No prime number greater than 5 ends in a 5. Ah Math

Key Fact DAEA
List of prime numbers up to 100: $$2,\,3,\,5,\,7,\,11,\,13,\,17,\,19,\,23,\,29,\,31,\,37,\,41,$$$$\,43,\,47,\,53,\,59,\,61,\,67,\,71,\,73,\,79,\,83,\,89,\,97$$ Ah Math

Key Fact 9ECE
Every integer greater than 1 has a unique prime factorization up to the order of the factors. Ah Math

Key Fact 525C
If $n$ is a composite number, then it must be divisible by a prime $p$ such that $p \le \sqrt{n}$. Ah Math

Key Fact 163E
$\small{\textbf{ Wilson's Theorem}}:\\[5pt]\text{ A positive integer }p>1 \text{ is prime if and only if }$ $$(p-1)! \equiv -1 \hspace{0.3cm} (\text{mod } p)$$ Ah Math

Key Fact 5221
$\small{\textbf{ Bertrand's Postulate}}:\\[5pt]\text{ For any integer } n>3 \text{, there always exists at least one prime number} \\ \, p \text{ with }$ $$n < p < 2n-2$$ Ah Math

Key Fact 24D6
$\small{\textbf{ Bonse's Inequality}}:\\[5pt]\text{ If } p_1, \, ...,\, p_n, \,p_{n+1} \text{ are the smallest } n+1 \text{ prime numbers and } n \ge4 \text{,} \\ \, \text{then}$ $$p_1 \, \cdot \, ... \, \cdot \, p_n < \, p_{n+1}^2$$ Ah Math

Key Fact C181
If $p$ is a prime number, then the prime power $p^a$ has $a+1$ divisors. Ah Math

Key Fact 8A33
The square of any odd integer leaves remainder 1 upon division by 8. Ah Math

Key Fact 7B55
When $n$ is a positive even integer,$$n⋅(n+4)⋅(n+8)⋅(n+12)$$is divisible by 8. Ah Math

Key Fact 8C41
$n^3-n$ is always divisible by 6, where $n \in Z$. Ah Math

Key Fact 8632
The product of any $n$ consecutive integer is always divisible by $n!$ Ah Math

If $n$ is a positive integer,$$(n+1)\cdot(n+2)⋅…⋅(2n)$$is divisible by $2^n$. Ah Math

Key Fact A4A7
The binomial expansion of $(x+y)^n$ has $n+1$ terms. Ah Math

Key Fact 3845
The binomial expansion of $$(a_1+a_2+⋯+a_{r-1}+a_r )^n$$has $\large {n+r-1 \choose r-1}$ terms. Ah Math

Key Fact D8CD
$$1+2+3\,+\,...\,+\,n=\frac{n(n+1)}{2}$$ Ah Math

Key Fact DEA5
$$1^2+2^2+3^2\,+\,...\,+\,n^2=\frac{n(n+1)(2n+1)}{6}$$ Ah Math

Key Fact 7B38
$$1^3+2^3+3^3+...+\,n^3=\left[\frac{n(n+1)}{2}\right]^2$$ Ah Math

Key Fact 2EBB
$$1^4+2^4+3^4+...+\,n^4=\frac{pst}{30}$$ where $p=n(n+1)$, $s=2n+1$, and $t=3p-1$. Ah Math

Key Fact 89DD
$$\sum_{k=1}^n k \cdot (k+1) \cdot (k+2) = \frac{n\cdot(n+1)\cdot(n+2)\cdot(n+3)}{4}$$ Ah Math

Key Fact ECD9
There are $$\frac{n\cdot(n+1)\cdot(2n+1)}{6}$$squares on an $n\times n$ chessboard. Ah Math

Key Fact A86B
The $\small{\textbf{Arithmetic Mean}}$ ($\small{\textbf{AM}}$) of positive real numbers $x_1, x_2, ...,x_n$ is defined as$$\frac{x_1 + x_2 + ... + x_n}{n}$$ Ah Math

Key Fact B65C
The $\small{\textbf{Geometric Mean}}$ ($\small{\textbf{GM}}$) of positive real numbers $x_1, x_2, ...,x_n$ is defined as$$\root n \of {x_1 \cdot x_2 \cdot ...\cdot x_n}$$ Ah Math

Key Fact D477
For any set of positive real numbers $x_1,\ldots,x_n$, the arithmetic mean is greater than or equal to the geometric mean. That is,$$\frac{x_1 + x_2 + ... + x_n}{n}\ge\root n \of {x_1 \cdot x_2 \cdot ...\cdot x_n}$$ Ah Math

Key Fact DDC5
The $\small{\textbf{Harmonic Mean}}$ ($\small{\textbf{HM}}$) of positive real numbers $x_1, x_2, ...,x_n$ is defined to be$$\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$$ Ah Math

Key Fact 29D9
For any set of positive real numbers $x_1,\ldots,x_n$, the arithmetic mean is greater than or equal to the harmonic mean. That is,$$\frac{x_1 + x_2 + ... + x_n}{n}\ge\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$$ Ah Math

Key Fact 5375
For non-zero real numbers $a$ and $b$,$$\frac{a}{b}+\frac{b}{a}\ge2$$ Ah Math

Key Fact 1E7C
For real numbers $a$ and $b$, $$a^2+b^2+1 \ge ab+a+b$$ Ah Math

Key Fact 3981
For real numbers $a$ and $b$, $$a^4+b^4+8 \ge 8ab$$ Ah Math

Key Fact 1EB9
For positive real numbers $a$, $b$, and $c$, $$\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\le\frac{a^2+b^2+c^2}{a\cdot b\cdot c}$$ Ah Math

Key Fact D7B8
For positive real numbers $a$, $b$, and $c$, $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge3$$. Ah Math

Key Fact 2483
For real numbers $a,b,c \ge 0$, $$(a+b)\cdot(a+c)\cdot(b+c) \ge 8abc$$ Ah Math

Key Fact 6C13
For real numbers $a,b,c \ge 0$, $$(a^2+1)\cdot(b^2+1)\cdot(c^2+1) \ge 8abc$$ Ah Math

Key Fact 7185
For real numbers $a,b,c\ge0$, $$ab+ac+bc\ge a \cdot \sqrt{bc}+b \cdot \sqrt{ac}+c \cdot \sqrt{ab}$$ Ah Math

Key Fact 8C62
For positive real numbers $a$, $b$, $c$, and $d$, $$(a+b+c+d)\cdot \left( \frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{d} \right)\ge16$$ Ah Math

Key Fact 913D
$$(1+x)^n\ge(1+n\cdot x)$$ Ah Math

$\small{\textbf{Cauchy–Schwarz Inequality}}:\\[0pt]$ $\text{For all sequences of real numbers$a_i$and$b_i$, we have}$ $$\left(\sum_{i=1}^n a_i^2\right)\cdot\left(\sum_{i=1}^n b_i^2\right)\ge\left(\sum_{i=1}^n a_i\cdot b_i\right)^2$$ Ah Math

Key Fact 2169
$\small{\textbf{ Chebyshev Inequality}}:\\[5pt]\text{ Let }x_1, x_2,..., x_n \text{ and } y_1, y_2,..., y_n \text{ be two sequences of real }\\ \text{ numbers, such that } x_1 \le x_2 \le \cdots \le x_n \text{ and } y_1 \le y_2 \le \cdots \le y_n. \\ \text{ Then, }$$\frac{(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)}{n} \le x_1y_1+x_2y_2+...+x_ny_n$$ Ah Math Key Fact E1B9 Let$y=k$be any line which intersects$y=ax^2+bx+c$at two points, say$P$and$Q$, where$a, b, c, k \in R$and$a \neq 0$. Then the abscissa of the midpoint of the line segment$PQ$is the abscissa of the vertex of the parabola. Ah Math Key Fact 72A4$\small{\textbf{De Moivre's Formula}}:$For any angle$\alpha$and for any integer$n$, $$(\text{cos } \alpha + i\cdot \text{sin } \alpha)^n=\text{cos } n\alpha + i\cdot \text{sin } n\alpha$$ Ah Math Key Fact 5DB3$\small{\textbf{Fibonacci Numbers}}:\\[0pt]\text{Sequence defined recursively by } F_1=F_2=1 \text{ and } F_{n+2}=F_{n+1}+F_n \\ \text{for all } n \in N.$Ah Math Key Fact 47EB A finite series is given by all the terms of a finite sequence, added together. Ah Math Key Fact 13D3 An infinite series is given by all the terms of an infinite sequence, added together. Ah Math Key Fact EAA4 The$n$th partial sum of a series is the sum of the first$n$terms. Ah Math Key Fact 3332 The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series. Ah Math Key Fact 2B69 The sequence of partial sums of a series sometimes tends to a real limit. If this does not happen, we say that the series has no sum. Ah Math Key Fact C62B A series can have a sum only if the individual terms tend to zero. But there are some series (with individual terms tending to zero) that do not have sums. Ah Math Key Fact 1529 The sum of an infinite series of the form $$a_1+a_1\cdot r+a_1\cdot r^2+a_1\cdot r^3+...$$ is$\large{\frac{a_1}{1-r}}$where$\left| r \right| < 1$. Ah Math Key Fact E6E7 Relative error = error / measurement Ah Math Key Fact 8B56 The bisector of an angle of the triangle divides the opposite side into segments proportional to the other two sides. Ah Math Key Fact B4C3 The sum of the exterior angles of any (convex) polygon is a constant, namely two straight angles (360$^{\circ}$). Ah Math Key Fact C4CC Of all triangles inscribable in a semi-circle, with the diameter as base, the one with the greatest area is the one with the largest altitude (the radius); that is, isosceles triangle. Ah Math Key Fact 8146 Two equal circles in a plane cannot have only one common tangent. Ah Math Key Fact B13A The number of lattice points on the segment from the origin to$(a,b)$is one more than the greatest common factor of$a$and$b\$. Ah Math

Key Fact 4BC8
A triangle is a right triangle if and only if one of its medians is half as long as the side to which it is drawn. Ah Math

If you want to be a contractor (problem writer, key fact writer, etc.), please contact AhMath by sending an email to challenging-problems@ahmath.com.

### Amirul Faiz Abdul Muthalib

Problem Writer

Number of problems: 10

With over 5 years of hands-on, successful teaching experience, Amirul is an enthusiastic Pre-University Mathematics Lecturer in INTEC Education College, Malaysia. Being a versatile individual who love Maths, he has earned a Master’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.

### İhsan Yücel

Problem Writer

Number of problems: 3

I 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 TÜBİTAK National Secondary School Research Projects Contest. One of my students, Read More 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". Read Less

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 non-mathematical 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 self-paced 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 non-students to further enhance their math skills. God bless and More power AHMATH FTW!!!

- Isaiah James de Dios Maling / Philippines