AhMath Real than i, rational than π

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

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95. Problem: 9EC7 , proposed by Ahmet Arduc
95 is the number of planar partitions of 10.
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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
23. Adrian Pilotos Burgos
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
52. Radu Bogo
53. Barry Villanueva
54. Randy Orton
55. Captain Magneto
56. Willie Revillame Wowowin
57. Mertkan Simsek
58. Mark Lawrence P Velasco
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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.
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Order Score # of Correct Answers Names Badges Country
77Amirul Faiz Abdul Muthalib110 MalaysiaMalaysia
91Caed Mark Medul Mendoza5107 PhilippinesPhilippines
66Lilanie Monique Torilla055 PhilippinesPhilippines
72Jeffrey Robles144 PhilippinesPhilippines
64Isaiah James de Dios Maling426 PhilippinesPhilippines
57Sumet Ketsri558 ThailandThailand
62Kurara Chibana40 JapanJapan
54Willie Revillame Wowowin00 PhilippinesPhilippines
63Russel J. Galanido130 South KoreaSouth Korea
50Marvin Cato130 PhilippinesPhilippines
60Mertkan Simsek80 TurkeyTurkey
48Nixon Balandra346 PhilippinesPhilippines
54Randy Orton60 United States of AmericaUnited States of America
60Rindell Mabunga131 PhilippinesPhilippines
40Jacob Sabido05 PhilippinesPhilippines
44Richard Phillip Dimaala Fernandez445 PhilippinesPhilippines
41Joselito Torculas426 PhilippinesPhilippines
36Lim Jing Ren00 MalaysiaMalaysia
26Arjun Singh Rajawat04 IndiaIndia
26Alea Astrea00 PhilippinesPhilippines
25Radu Bogo025 RomaniaRomania
19Lenard Guillermo02 PhilippinesPhilippines
20Gluttony00 PhilippinesPhilippines
40Roenz Joshlee Timbol01 PhilippinesPhilippines
26Norwyn Nicholson Kah038 PhilippinesPhilippines
25Jhepoy Dizon00 PhilippinesPhilippines
19Mark Alvero00 PhilippinesPhilippines
32Daniel James Molina01 PhilippinesPhilippines
20Ralph Macarasig00 PhilippinesPhilippines
26Joem Canciller10 PhilippinesPhilippines
14John Albert A. Reyes06 PhilippinesPhilippines
17Nheil Ignacio08 PhilippinesPhilippines
23Kumar Ayush01 IndiaIndia
17Ibrahim Demir01 TurkeyTurkey
22Christian Paul Patawaran01 PhilippinesPhilippines
11Melga Sonio00 PhilippinesPhilippines
11Angelu G. Leynes10 PhilippinesPhilippines
13Poetri Sonya Tarabunga00 IndonesiaIndonesia
12Andrew Chiu00 PhilippinesPhilippines
14Ewen Goisot017 FranceFrance
13Srinivas Kanigiri00 IndiaIndia
43Kimi No Nawa7166 JapanJapan
18James Ericson10 ThailandThailand
11Mark Elis Espiridion00 PhilippinesPhilippines
20Christian Daang125 PhilippinesPhilippines
17Stefano Ongari00 ItalyItaly
8Monu Baba Rura Sirsa Up00 IndiaIndia
8Joseph Rodelas10 PhilippinesPhilippines
11Chris Norman Algo06 PhilippinesPhilippines
12Reymark Togno00 PhilippinesPhilippines
12Dreimuru Tempest20 PhilippinesPhilippines
7Adrian Pilotos Burgos00 PhilippinesPhilippines
6Yavuz Selim Koseoglu20 TurkeyTurkey
6Mahmut Cemrek00 TurkeyTurkey
6John Gamal Aziz Attia21 EgyptEgypt
10Ikemen30 JapanJapan
7Emmanuel David00 PhilippinesPhilippines
5Sigmund Dela Cruz05 PhilippinesPhilippines
4Chayapol02 ThailandThailand
4Βαρελάς Γεώρ𝛾ιος00 GreeceGreece
5John Lester Tan00 PhilippinesPhilippines
5Hanelet Santos00 PhilippinesPhilippines
5Grant Lewis Bulaong01 PhilippinesPhilippines
3Gerald M. Pascua00 PhilippinesPhilippines
3Barry Villanueva02 PhilippinesPhilippines
5Jake Gacuan09 PhilippinesPhilippines
19Keedgwh00 IndiaIndia
4Mark Lawrence P Velasco00 PhilippinesPhilippines
6Fred Gutierrez00 PhilippinesPhilippines
4Smahi Abdeslem04 AlgeriaAlgeria
5Dan Lang00 PhilippinesPhilippines
2Melek Cimen10 TurkeyTurkey
2Edge Ramos00 PhilippinesPhilippines
2Afshiram Muhammed00 TurkeyTurkey
5John Marco Latagan00 PhilippinesPhilippines
2Judy Ann Rubante00 PhilippinesPhilippines
2Evan Gruda00 United States of AmericaUnited States of America
1Suleyman Akarsu00 TurkeyTurkey
1Serkan Callioglu00 TurkeyTurkey
1Rosendo Parra Milian01 PeruPeru
1Rdvnaksu11 TurkeyTurkey
1Muhammed Aydogdu00 TurkeyTurkey
1Mohamed Karamany10 EgyptEgypt
1Captain Magneto00 GermanyGermany
1Avrila Frazier01 United States of AmericaUnited States of America
1Abhishek Singh00 IndiaIndia
2Serdal Aslantas01 RomaniaRomania
2John Rocel Perez00 PhilippinesPhilippines
1Ryan Quimo00 PhilippinesPhilippines
1Mark Allen Facun00 PhilippinesPhilippines
1John Patrick03 PhilippinesPhilippines
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    ✔   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)
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    ✔   The Art and Craft of Problem Solving, 2nd Ed. by Paul Zeitz (2007) instructor's manual.
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    ✔   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 [1969-1993] by Michael Doob (1993)
    ✔   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

    Key Fact ABAD
    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

    Key Fact EADA
    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

    Key Fact ADBA
    $\small{\textbf{Cauchy–Schwarz Inequality}}:\\[0pt]$ $\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

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

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

Please feel free to send your general comments to challenging-problems@ahmath.com.

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

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