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.

There are 90 problems and 78 key facts, waiting for math enthusiasts!

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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,

1. Problem: 595A , proposed by Ahmet Arduc, Tip: Key Fact(s): 7695
1 is the multiplicative identity.
$$\small{A=\frac{1}{1\times2\times3}+\frac{2}{3\times4\times5}+\frac{3}{5\times6\times7}+\cdots+\frac{2016}{4031\times4032\times4033}}$$If $A$ is written in its simplest form as $\frac{a}{b}$, what is $a+b$ $?$ Ah Math

• 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

2. Problem: B28E , proposed by Ahmet Arduc
2 is the only even prime.
Find the smallest natural number $n$ such that the last five digits of the product of $\\$ n$$\times935\times972\times975 are all zero. Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Joseph Rodelas 3. Yavuz Selim Koseoglu 4. John Gamal Aziz Attia 5. Mahmut Cemrek 6. Βαρελάς Γεώρ𝛾ιος 7. Isaiah James de Dios Maling 8. Adrian Pilotos Burgos 9. Joselito Torculas 10. Nheil Ignacio 11. Jacob Sabido 12. Gerald M. Pascua 13. Russel J. Galanido 14. John Albert A. Reyes 15. Roenz Joshlee Timbol 16. Marvin Cato 17. Rindell Mabunga 18. Chris Norman Algo 19. Melga Sonio 20. Mark Elis Espiridion 21. Angelu G. Leynes 22. Caed Mark Medul Mendoza 23. Sumet Ketsri 24. Richard Phillip Dimaala Fernandez 25. Daniel James Molina 26. Nixon Balandra 27. Poetri Sonya Tarabunga 28. Lilanie Monique Torilla 29. Christian Daang 30. Emmanuel David 31. Christian Paul Patawaran 32. Jake Gacuan 33. Ibrahim Demir 34. Amirul Faiz Abdul Muthalib 35. Norwyn Nicholson Kah 36. Joem Canciller 37. Ralph Macarasig 38. Jhepoy Dizon 39. Lim Jing Ren 40. John Marco Latagan 41. Dan Lang 42. Lenard Guillermo 43. Hanelet Santos 44. Dreimuru Tempest 45. Dreimuru Tempest 46. Srinivas Kanigiri 47. Kurara Chibana 48. Alea Astrea 49. Kumar Ayush 50. Reymark Togno 51. Gluttony 52. Mark Alvero 53. Radu Bogo 54. Stefano Ongari 55. Randy Orton 56. Willie Revillame Wowowin 57. Mertkan Simsek 58. Arjun Singh Rajawat 59. Mark Lawrence P Velasco 3. Problem: A275 , proposed by Ahmet Arduc, Tip: Key Fact(s): 7695 3 is the only prime 1 less than a perfect square.$$A=\small{\frac{1}{2}+\frac{1}{2+4}+\frac{1}{2+4+6}+\cdots+\frac{1}{2+4+6+\cdots+4032}}$$If A is written in its simplest form as \frac{a}{b}, what is a+b ? Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Muhammed Aydo?du 3. Joseph Rodelas 4. Mahmut Cemrek 5. Yavuz Selim Koseoglu 6. John Gamal Aziz Attia 7. Isaiah James de Dios Maling 8. John Albert A. Reyes 9. Jacob Sabido 10. Gerald M. Pascua 11. Joselito Torculas 12. Nheil Ignacio 13. Russel J. Galanido 14. Angelu G. Leynes 15. Adrian Pilotos Burgos 16. Roenz Joshlee Timbol 17. Chris Norman Algo 18. Rindell Mabunga 19. Melga Sonio 20. Mark Elis Espiridion 21. Marvin Cato 22. Richard Phillip Dimaala Fernandez 23. Caed Mark Medul Mendoza 24. Sumet Ketsri 25. Nixon Balandra 26. Daniel James Molina 27. Poetri Sonya Tarabunga 28. Lilanie Monique Torilla 29. Christian Daang 30. Christian Paul Patawaran 31. Jake Gacuan 32. Ibrahim Demir 33. Amirul Faiz Abdul Muthalib 34. Norwyn Nicholson Kah 35. Joem Canciller 36. Jhepoy Dizon 37. Dan Lang 38. Mark Alvero 39. Chayapol 40. Rosendo Parra Milian 41. Dreimuru Tempest 42. Srinivas Kanigiri 43. Kurara Chibana 44. Lim Jing Ren 45. James Ericson 46. Alea Astrea 47. Kumar Ayush 48. Gluttony 49. Radu Bogo 50. Stefano Ongari 51. Barry Villanueva 52. Reymark Togno 53. Randy Orton 54. Willie Revillame Wowowin 55. Mertkan Simsek 56. Mark Lawrence P Velasco 4. Problem: 2BE2 , proposed by Ahmet Arduc 4 is the smallest number of colors sufficient to color all planar maps. What is the units digit of the sum 1^2+2^3+3^4+\cdots+2016^{2017} ? Ah Math • Correct answers have been submitted by: 1. Angelu G. Leynes 2. Jeffrey Robles 3. Isaiah James de Dios Maling 4. Joselito Torculas 5. Russel J. Galanido 6. Marvin Cato 7. Rindell Mabunga 8. John Albert A. Reyes 9. Nheil Ignacio 10. Caed Mark Medul Mendoza 11. Sumet Ketsri 12. Richard Phillip Dimaala Fernandez 13. Daniel James Molina 14. Nixon Balandra 15. Lilanie Monique Torilla 16. Mark Elis Espiridion 17. Jacob Sabido 18. Emmanuel David 19. Jake Gacuan 20. Christian Paul Patawaran 21. Roenz Joshlee Timbol 22. Ibrahim Demir 23. Amirul Faiz Abdul Muthalib 24. Norwyn Nicholson Kah 25. John Lester Tan 26. Ralph Macarasig 27. Jhepoy Dizon 28. Chayapol 29. Dreimuru Tempest 30. Kurara Chibana 31. Lim Jing Ren 32. James Ericson 33. Alea Astrea 34. Lenard Guillermo 35. Radu Bogo 36. Randy Orton 37. Willie Revillame Wowowin 38. Keedgwh 39. Arjun Singh Rajawat 40. Monu Baba Rura Sirsa Up 41. Mertkan Simsek 5. Problem: 218D , proposed by Ahmet Arduc The sum of the first 73 odd primes is divisible by 73. Find the remainder when \underbrace{20172017\cdots2017}_{2017\text{ times}} is divided by 73. Ah Math • Correct answers have been submitted by: 1. Yavuz Selim Koseoglu 2. John Gamal Aziz Attia 3. Mahmut Cemrek 4. Joseph Rodelas 5. Jeffrey Robles 6. Isaiah James de Dios Maling 7. Nheil Ignacio 8. Roenz Joshlee Timbol 9. Marvin Cato 10. Russel J. Galanido 11. Joselito Torculas 12. Melga Sonio 13. Angelu G. Leynes 14. Caed Mark Medul Mendoza 15. Sumet Ketsri 16. John Albert A. Reyes 17. Nixon Balandra 18. Jacob Sabido 19. Daniel James Molina 20. Lilanie Monique Torilla 21. Mark Elis Espiridion 22. Poetri Sonya Tarabunga 23. Richard Phillip Dimaala Fernandez 24. Rindell Mabunga 25. Jake Gacuan 26. Grant Lewis Bulaong 27. Christian Paul Patawaran 28. Ibrahim Demir 29. Amirul Faiz Abdul Muthalib 30. Norwyn Nicholson Kah 31. John Lester Tan 32. Joem Canciller 33. Jhepoy Dizon 34. Lenard Guillermo 35. Dan Lang 36. Fred Gutierrez 37. Mark Alvero 38. Chayapol 39. Kurara Chibana 40. Alea Astrea 41. Lim Jing Ren 42. Gluttony 43. Radu Bogo 44. Stefano Ongari 45. Randy Orton 46. Willie Revillame Wowowin 47. Arjun Singh Rajawat 48. Monu Baba Rura Sirsa Up 49. Mertkan Simsek 6. Problem: A13E , proposed by Ahmet Arduc 2017 is a prime number. If a^2+b^2+c^2+d^2=2017, what is the minimum value of$$\small{(a+b+c)^2+(b+c+d)^2+(c+d+a)^2+(d+a+b)^2\text{ ?}}$$Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Βαρελάς Γεώρ𝛾ιος 3. Isaiah James de Dios Maling 4. Joselito Torculas 5. Marvin Cato 6. Russel J. Galanido 7. Melga Sonio 8. Nheil Ignacio 9. Sumet Ketsri 10. Lilanie Monique Torilla 11. Nixon Balandra 12. Caed Mark Medul Mendoza 13. Poetri Sonya Tarabunga 14. Rindell Mabunga 15. Richard Phillip Dimaala Fernandez 16. Christian Paul Patawaran 17. Amirul Faiz Abdul Muthalib 18. Daniel James Molina 19. Norwyn Nicholson Kah 20. Jhepoy Dizon 21. Lenard Guillermo 22. Kurara Chibana 23. Afshiram Muhammed 24. Jacob Sabido 25. Roenz Joshlee Timbol 7. Problem: 52D2 , proposed by Ahmet Arduc 7 is the smallest number of sides of a regular polygon that is not constructible by straightedge and compass. Find the value of$$\small{\frac{1}{2}+\frac{1}{3}+\frac{2}{3}+\frac{1}{4}+\frac{2}{4}+\frac{3}{4}+\frac{1}{5}+\cdots+\frac{2015}{2017}+\frac{2016}{2017}}$$Ah Math • Correct answers have been submitted by: 1. Melek Cimen 2. Joseph Rodelas 3. Yavuz Selim Koseoglu 4. Mahmut Cemrek 5. Jeffrey Robles 6. Βαρελάς Γεώρ𝛾ιος 7. Isaiah James de Dios Maling 8. John Albert A. Reyes 9. Nheil Ignacio 10. Adrian Pilotos Burgos 11. Joselito Torculas 12. Rindell Mabunga 13. Roenz Joshlee Timbol 14. Marvin Cato 15. Russel J. Galanido 16. Angelu G. Leynes 17. Melga Sonio 18. Caed Mark Medul Mendoza 19. Sumet Ketsri 20. Daniel James Molina 21. Nixon Balandra 22. Lilanie Monique Torilla 23. Mark Elis Espiridion 24. Christian Daang 25. Emmanuel David 26. Jacob Sabido 27. Poetri Sonya Tarabunga 28. Richard Phillip Dimaala Fernandez 29. Grant Lewis Bulaong 30. Ibrahim Demir 31. Amirul Faiz Abdul Muthalib 32. Norwyn Nicholson Kah 33. Christian Paul Patawaran 34. Joem Canciller 35. Ralph Macarasig 36. Jhepoy Dizon 37. Lim Jing Ren 38. Lenard Guillermo 39. Mark Alvero 40. Dreimuru Tempest 41. Kurara Chibana 42. James Ericson 43. Alea Astrea 44. Reymark Togno 45. Gluttony 46. Radu Bogo 47. Randy Orton 48. Willie Revillame Wowowin 49. Evan Gruda 50. Arjun Singh Rajawat 51. Mark Allen Facun 52. Mertkan Simsek 53. Mark Lawrence P Velasco 8. Problem: 79B2 , proposed by Ahmet Arduc 8 is the largest cube in the Fibonacci sequence. If n is a positive even multiple of 5 and$$\small{8^2+10^2+12^2+18^2+20^2+22^2+\cdots+(n-2)^2+n^2+(n+2)^2}$$is divisible by 9, what is the minumum value of n ? Ah Math • Correct answers have been submitted by: 1. Yavuz Selim Koseoglu 2. Mahmut Cemrek 3. Jeffrey Robles 4. Nheil Ignacio 5. Isaiah James de Dios Maling 6. Joselito Torculas 7. Rindell Mabunga 8. Richard Phillip Dimaala Fernandez 9. Melga Sonio 10. Angelu G. Leynes 11. Adrian Pilotos Burgos 12. Russel J. Galanido 13. Caed Mark Medul Mendoza 14. Marvin Cato 15. Sumet Ketsri 16. Nixon Balandra 17. Lilanie Monique Torilla 18. Mark Elis Espiridion 19. Jacob Sabido 20. Christian Daang 21. Emmanuel David 22. Poetri Sonya Tarabunga 23. Roenz Joshlee Timbol 24. Ibrahim Demir 25. Christian Paul Patawaran 26. Amirul Faiz Abdul Muthalib 27. Norwyn Nicholson Kah 28. Chris Norman Algo 29. Daniel James Molina 30. Joem Canciller 31. Jhepoy Dizon 32. Ralph Macarasig 33. John Marco Latagan 34. Dreimuru Tempest 35. Kurara Chibana 36. Alea Astrea 37. Kumar Ayush 38. Lim Jing Ren 39. Lenard Guillermo 40. Mark Alvero 41. Radu Bogo 42. Randy Orton 43. Willie Revillame Wowowin 44. Mertkan Simsek 9. Problem: 11C8 , proposed by Ahmet Arduc, Tip: Key Fact(s): 29D9 9 is the maximum number of cubes that are needed to sum to any positive integer. If a,b,c>0 and a+b+c=1, what is the smallest value of \frac{1}{a}+\frac{9}{b}+\frac{16}{c}\text{ ?} Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Edge Ramos 3. Joseph Rodelas 4. Isaiah James de Dios Maling 5. Joselito Torculas 6. Rindell Mabunga 7. Roenz Joshlee Timbol 8. Marvin Cato 9. Nheil Ignacio 10. Melga Sonio 11. Russel J. Galanido 12. Sumet Ketsri 13. Caed Mark Medul Mendoza 14. Daniel James Molina 15. Nixon Balandra 16. Lilanie Monique Torilla 17. Poetri Sonya Tarabunga 18. Grant Lewis Bulaong 19. Richard Phillip Dimaala Fernandez 20. Christian Paul Patawaran 21. Ibrahim Demir 22. Amirul Faiz Abdul Muthalib 23. Norwyn Nicholson Kah 24. Ralph Macarasig 25. Lenard Guillermo 26. Kurara Chibana 27. Kumar Ayush 28. James Ericson 29. Jacob Sabido 30. Stefano Ongari 31. Randy Orton 32. Willie Revillame Wowowin 10. Problem: AC82 , proposed by Ahmet Arduc 10 is the only triangular number which is also the sum of 2 consecutive square odd numbers. Find the value of the following expression.$$1\cdot2+3\cdot4+5\cdot6+\cdots+2017\cdot2018$$Ah Math • Correct answers have been submitted by: 1. Joseph Rodelas 2. Jeffrey Robles 3. Edge Ramos 4. Isaiah James de Dios Maling 5. John Albert A. Reyes 6. Joselito Torculas 7. Nheil Ignacio 8. Rindell Mabunga 9. Marvin Cato 10. Russel J. Galanido 11. Chris Norman Algo 12. Angelu G. Leynes 13. Melga Sonio 14. Sumet Ketsri 15. Nixon Balandra 16. Caed Mark Medul Mendoza 17. Lilanie Monique Torilla 18. Richard Phillip Dimaala Fernandez 19. Jacob Sabido 20. Mark Elis Espiridion 21. Emmanuel David 22. Poetri Sonya Tarabunga 23. Christian Paul Patawaran 24. Ibrahim Demir 25. Amirul Faiz Abdul Muthalib 26. Daniel James Molina 27. Joem Canciller 28. Norwyn Nicholson Kah 29. Jhepoy Dizon 30. Fred Gutierrez 31. Dreimuru Tempest 32. Roenz Joshlee Timbol 33. Kurara Chibana 34. Lim Jing Ren 35. Kumar Ayush 36. Lenard Guillermo 37. Radu Bogo 38. Barry Villanueva 39. Randy Orton 40. Willie Revillame Wowowin 41. Mertkan Simsek 11. Problem: C1B5 , proposed by Ahmet Arduc 11 is the only palindromic prime with an even number of digits. What is the sum of the expression 2+3+5+8+13+\cdots+17711\text{ ?} Ah Math • Correct answers have been submitted by: 1. John Gamal Aziz Attia 2. Isaiah James de Dios Maling 3. Chris Norman Algo 4. Jeffrey Robles 5. Joselito Torculas 6. Melek Cimen 7. Rindell Mabunga 8. Angelu G. Leynes 9. Melga Sonio 10. Russel J. Galanido 11. John Albert A. Reyes 12. Nheil Ignacio 13. Adrian Pilotos Burgos 14. Caed Mark Medul Mendoza 15. Marvin Cato 16. Sumet Ketsri 17. Daniel James Molina 18. Lilanie Monique Torilla 19. Nixon Balandra 20. Jacob Sabido 21. Mark Elis Espiridion 22. Richard Phillip Dimaala Fernandez 23. Christian Paul Patawaran 24. Roenz Joshlee Timbol 25. Ibrahim Demir 26. Amirul Faiz Abdul Muthalib 27. Norwyn Nicholson Kah 28. Joem Canciller 29. Ralph Macarasig 30. Jhepoy Dizon 31. John Marco Latagan 32. Kurara Chibana 33. Lim Jing Ren 34. Kumar Ayush 35. Gluttony 36. Alea Astrea 37. Randy Orton 38. Willie Revillame Wowowin 39. Mertkan Simsek 40. Arjun Singh Rajawat 41. Radu Bogo 12. Problem: 24B8 , proposed by Ahmet Arduc 12 is the largest known even number expressible as the sum of two primes in one way. In the following sequence, find the units digit of the 2017th term$$1,3,4,7,11,18, 29,47,...$$Ah Math • Correct answers have been submitted by: 1. Isaiah James de Dios Maling 2. Jeffrey Robles 3. Richard Phillip Dimaala Fernandez 4. Joseph Rodelas 5. Joselito Torculas 6. Adrian Pilotos Burgos 7. Rindell Mabunga 8. Angelu G. Leynes 9. Nheil Ignacio 10. Russel J. Galanido 11. Melga Sonio 12. John Albert A. Reyes 13. Caed Mark Medul Mendoza 14. Marvin Cato 15. Sumet Ketsri 16. Nixon Balandra 17. Daniel James Molina 18. Lilanie Monique Torilla 19. Jacob Sabido 20. Emmanuel David 21. Poetri Sonya Tarabunga 22. Roenz Joshlee Timbol 23. Christian Daang 24. Christian Paul Patawaran 25. Ibrahim Demir 26. Amirul Faiz Abdul Muthalib 27. Norwyn Nicholson Kah 28. Joem Canciller 29. Fred Gutierrez 30. Ralph Macarasig 31. Jhepoy Dizon 32. Lim Jing Ren 33. Lenard Guillermo 34. Kurara Chibana 35. James Ericson 36. Alea Astrea 37. Kumar Ayush 38. Reymark Togno 39. Gluttony 40. Randy Orton 41. Willie Revillame Wowowin 42. Radu Bogo 43. Keedgwh 44. Arjun Singh Rajawat 45. Mertkan Simsek 13. Problem: 6392 , proposed by Ahmet Arduc 13 is the number of Archimedean solids. Find the 2017th term of the sequence 1,2,1,1,3,1,1,1,4,1,1,1,1,5,... Ah Math • Correct answers have been submitted by: 1. Sumet Ketsri 2. Rindell Mabunga 3. Russel J. Galanido 4. Joselito Torculas 5. Nheil Ignacio 6. Richard Phillip Dimaala Fernandez 7. Marvin Cato 8. Jeffrey Robles 9. Angelu G. Leynes 10. Caed Mark Medul Mendoza 11. Isaiah James de Dios Maling 12. Nixon Balandra 13. Daniel James Molina 14. Lilanie Monique Torilla 15. Jacob Sabido 16. Mark Elis Espiridion 17. Christian Daang 18. John Albert A. Reyes 19. Emmanuel David 20. Poetri Sonya Tarabunga 21. Christian Paul Patawaran 22. Roenz Joshlee Timbol 23. Ibrahim Demir 24. Amirul Faiz Abdul Muthalib 25. Joem Canciller 26. Norwyn Nicholson Kah 27. Fred Gutierrez 28. Ralph Macarasig 29. Jhepoy Dizon 30. Lim Jing Ren 31. John Marco Latagan 32. Hanelet Santos 33. Mark Alvero 34. Kurara Chibana 35. Alea Astrea 36. Lenard Guillermo 37. Gluttony 38. Reymark Togno 39. Radu Bogo 40. Stefano Ongari 41. Randy Orton 42. Willie Revillame Wowowin 43. Arjun Singh Rajawat 44. Monu Baba Rura Sirsa Up 45. Mertkan Simsek 14. Problem: 615D , proposed by Ahmet Arduc 14 is the smallest even number n with no solutions to φ(m) = n. Evaluate the following expression:$$1\cdot 2\cdot 3+2\cdot 3\cdot 4+3\cdot 4\cdot 5 \,+...+\,31\cdot 32\cdot 33$$Ah Math • Correct answers have been submitted by: 1. Russel J. Galanido 2. Rindell Mabunga 3. Richard Phillip Dimaala Fernandez 4. Joselito Torculas 5. Jeffrey Robles 6. Isaiah James de Dios Maling 7. Caed Mark Medul Mendoza 8. Nheil Ignacio 9. John Albert A. Reyes 10. Lilanie Monique Torilla 11. Nixon Balandra 12. Jacob Sabido 13. Mark Elis Espiridion 14. Chris Norman Algo 15. Marvin Cato 16. Poetri Sonya Tarabunga 17. Christian Paul Patawaran 18. Christian Daang 19. Ibrahim Demir 20. Amirul Faiz Abdul Muthalib 21. Roenz Joshlee Timbol 22. Daniel James Molina 23. Joem Canciller 24. Norwyn Nicholson Kah 25. Ralph Macarasig 26. Jhepoy Dizon 27. Sumet Ketsri 28. John Marco Latagan 29. Fred Gutierrez 30. Kurara Chibana 31. Lim Jing Ren 32. James Ericson 33. Alea Astrea 34. Kumar Ayush 35. Lenard Guillermo 36. Radu Bogo 37. Randy Orton 38. Willie Revillame Wowowin 39. Mertkan Simsek 40. Arjun Singh Rajawat 15. Problem: A836 , proposed by Ahmet Arduc 15 is the number of 3-digit palindromic primes. Let A=2016+2017. How many proper irreducible positive fractions are there whose denominator is A? Ah Math • Correct answers have been submitted by: 1. John Gamal Aziz Attia 2. Jeffrey Robles 3. Joselito Torculas 4. Russel J. Galanido 5. Jacob Sabido 6. Marvin Cato 7. Nixon Balandra 8. Christian Daang 9. John Albert A. Reyes 10. Poetri Sonya Tarabunga 11. Richard Phillip Dimaala Fernandez 12. Rindell Mabunga 13. Kimi No Nawa 14. Caed Mark Medul Mendoza 15. Lilanie Monique Torilla 16. Roenz Joshlee Timbol 17. Isaiah James de Dios Maling 18. Christian Paul Patawaran 19. Ibrahim Demir 20. Amirul Faiz Abdul Muthalib 21. Norwyn Nicholson Kah 22. Daniel James Molina 23. John Rocel Perez 24. Ralph Macarasig 25. Jhepoy Dizon 26. Sumet Ketsri 27. Kurara Chibana 28. Lim Jing Ren 29. James Ericson 30. Alea Astrea 31. Kumar Ayush 32. Stefano Ongari 33. Mertkan Simsek 16. Problem: 7B83 , proposed by Ahmet Arduc 2017 is palindromic in base 31: 232. Let a+b+c=0. find the value of$$a\cdot\left(\frac{1}{b}+\frac{1}{c}+1\right)+b\cdot\left(\frac{1}{c}+\frac{1}{a}+1\right)+c\cdot\left(\frac{1}{a}+\frac{1}{b}+1\right)+2017$$Ah Math • Correct answers have been submitted by: 1. Richard Phillip Dimaala Fernandez 2. Joselito Torculas 3. Nixon Balandra 4. Jeffrey Robles 5. Jacob Sabido 6. Marvin Cato 7. Nheil Ignacio 8. Russel J. Galanido 9. Caed Mark Medul Mendoza 10. Lilanie Monique Torilla 11. Rindell Mabunga 12. Daniel James Molina 13. Roenz Joshlee Timbol 14. Christian Daang 15. Isaiah James de Dios Maling 16. Christian Paul Patawaran 17. Kimi No Nawa 18. Ibrahim Demir 19. Amirul Faiz Abdul Muthalib 20. Norwyn Nicholson Kah 21. Joem Canciller 22. John Lester Tan 23. John Rocel Perez 24. Ralph Macarasig 25. Jhepoy Dizon 26. Lim Jing Ren 27. Sumet Ketsri 28. Sigmund Dela Cruz 29. Srinivas Kanigiri 30. Kurara Chibana 31. James Ericson 32. Alea Astrea 33. Kumar Ayush 34. Radu Bogo 35. Stefano Ongari 36. Randy Orton 37. Willie Revillame Wowowin 38. Evan Gruda 39. Mertkan Simsek 17. Problem: E328 , proposed by Ahmet Arduc 17 is the number of wallpaper groups. In the matrix given below, if there are x numbers above 2017 and y numbers to the left of 2017, what is the sum of x and y ?$$\matrix{1 & 4 & 5 & 16 & 17 & \cdots \cr 2 & 3 & 6 & 15 & 18 & \cdots \cr 9 & 8 & 7 & 14 & 19 & \cdots \cr 10 & 11 & 12 & 13 & 20 & \cdots \cr 25 & 24 & 23 & 22 & 21 & \cdots \cr 26 & \cdots & \cdots & \cdots & \cdots & \cdots}$$Ah Math • Correct answers have been submitted by: 1. Kimi No Nawa 2. Caed Mark Medul Mendoza 3. Lilanie Monique Torilla 4. Nixon Balandra 5. Joselito Torculas 6. Marvin Cato 7. Rindell Mabunga 8. Richard Phillip Dimaala Fernandez 9. Russel J. Galanido 10. Jacob Sabido 11. Isaiah James de Dios Maling 12. Christian Daang 13. Chris Norman Algo 14. Amirul Faiz Abdul Muthalib 15. Daniel James Molina 16. Christian Paul Patawaran 17. Joem Canciller 18. Norwyn Nicholson Kah 19. Jeffrey Robles 20. Jhepoy Dizon 21. Lim Jing Ren 22. Hanelet Santos 23. Sumet Ketsri 24. Roenz Joshlee Timbol 25. Kurara Chibana 26. James Ericson 27. Stefano Ongari 28. Randy Orton 29. Willie Revillame Wowowin 30. Radu Bogo 31. Mertkan Simsek 18. Problem: B613 , proposed by Ahmet Arduc 1971 is the birth year of A. Arduc. Find the remainder when 1^{1971}+2^{1971}+3^{1971}+...+2016^{1971} is divided by 2017 ? Ah Math • Correct answers have been submitted by: 1. Marvin Cato 2. Joselito Torculas 3. Rindell Mabunga 4. Jacob Sabido 5. Russel J. Galanido 6. Richard Phillip Dimaala Fernandez 7. Nixon Balandra 8. Roenz Joshlee Timbol 9. Isaiah James de Dios Maling 10. Grant Lewis Bulaong 11. Kimi No Nawa 12. Lilanie Monique Torilla 13. Caed Mark Medul Mendoza 14. Christian Paul Patawaran 15. Jeffrey Robles 16. Ibrahim Demir 17. Christian Daang 18. Amirul Faiz Abdul Muthalib 19. Norwyn Nicholson Kah 20. Chris Norman Algo 21. Daniel James Molina 22. John Lester Tan 23. Ralph Macarasig 24. Jhepoy Dizon 25. Sumet Ketsri 26. Nheil Ignacio 27. Sigmund Dela Cruz 28. Srinivas Kanigiri 29. Kurara Chibana 30. Alea Astrea 31. Lim Jing Ren 32. Kumar Ayush 33. Randy Orton 34. Willie Revillame Wowowin 35. Radu Bogo 36. Mertkan Simsek 37. Keedgwh 38. Arjun Singh Rajawat 39. Monu Baba Rura Sirsa Up 19. Problem: EE2C , proposed by Ahmet Arduc 351 is the sum of five consecutive prime numbers: 61 + 67 + 71 + 73 + 79. By using only letters of English alphabet, label one marble 'A', two marbles 'B', three marbles 'C',..., twenty-six marbles 'Z'. Put these$$1+2+3+\cdots+26=351$$labeled marbles in a bag. Marbles are then drawn from the bag at random without replacement. What is the minimum number of marbles that must be drawn to guarantee drawing at least ten marbles with the same label? Ah Math • Correct answers have been submitted by: 1. Joselito Torculas 2. Rindell Mabunga 3. Nixon Balandra 4. Russel J. Galanido 5. Richard Phillip Dimaala Fernandez 6. Marvin Cato 7. Jacob Sabido 8. Kimi No Nawa 9. Caed Mark Medul Mendoza 10. Lilanie Monique Torilla 11. Jeffrey Robles 12. Sumet Ketsri 13. Isaiah James de Dios Maling 14. Christian Paul Patawaran 15. Christian Daang 16. Amirul Faiz Abdul Muthalib 17. Norwyn Nicholson Kah 18. Daniel James Molina 19. Joem Canciller 20. Ralph Macarasig 21. Jhepoy Dizon 22. Lim Jing Ren 23. Mark Alvero 24. Roenz Joshlee Timbol 25. Kurara Chibana 26. Kumar Ayush 27. Reymark Togno 28. Gluttony 29. Alea Astrea 30. Radu Bogo 31. Mertkan Simsek 20. Problem: E4C6 , proposed by Ahmet Arduc, Tip: Key Fact(s): 8632 1017 is the smallest number whose square contains 7 different digits. Find the remainder when 1001\times1002\times1003\times...\times2017 is divided by 1017!. Ah Math • Correct answers have been submitted by: 1. Kimi No Nawa 2. Caed Mark Medul Mendoza 3. Lilanie Monique Torilla 4. Joselito Torculas 5. Russel J. Galanido 6. Jacob Sabido 7. Rindell Mabunga 8. Nixon Balandra 9. Richard Phillip Dimaala Fernandez 10. Isaiah James de Dios Maling 11. Christian Paul Patawaran 12. Ibrahim Demir 13. Roenz Joshlee Timbol 14. Amirul Faiz Abdul Muthalib 15. Daniel James Molina 16. Norwyn Nicholson Kah 17. Jeffrey Robles 18. Jhepoy Dizon 19. Ralph Macarasig 20. Sumet Ketsri 21. Marvin Cato 22. Sigmund Dela Cruz 23. Kurara Chibana 24. Alea Astrea 25. Lim Jing Ren 26. Christian Daang 27. Kumar Ayush 28. Lenard Guillermo 29. Radu Bogo 30. Randy Orton 31. Willie Revillame Wowowin 32. Mertkan Simsek 33. Abhishek Singh 34. Arjun Singh Rajawat 35. Monu Baba Rura Sirsa Up 21. Problem: ABBA , proposed by Ahmet Arduc 21 is the smallest number of distinct squares needed to tile a square. How many pairs of distinct integers between 1 and 2017 inclusively have their products as multiple of 6? Ah Math • Correct answers have been submitted by: 1. Russel J. Galanido 2. Isaiah James de Dios Maling 3. Rindell Mabunga 4. Caed Mark Medul Mendoza 5. Amirul Faiz Abdul Muthalib 6. Lilanie Monique Torilla 7. Nixon Balandra 8. Richard Phillip Dimaala Fernandez 9. Daniel James Molina 10. Christian Paul Patawaran 11. Norwyn Nicholson Kah 12. Jeffrey Robles 13. Joselito Torculas 14. Jhepoy Dizon 15. Lim Jing Ren 16. Marvin Cato 17. Sumet Ketsri 18. Ikemen 22. Problem: DAB1 , proposed by Ahmet Arduc 1971 is the birth year of A. Arduc. An integer x plus 1971 is the square of a positive integer, and x minus 46 is the square of another positive integer. Find the value of x. Ah Math • Correct answers have been submitted by: 1. Rindell Mabunga 2. Russel J. Galanido 3. Roenz Joshlee Timbol 4. Amirul Faiz Abdul Muthalib 5. Caed Mark Medul Mendoza 6. Lilanie Monique Torilla 7. Isaiah James de Dios Maling 8. Nixon Balandra 9. Richard Phillip Dimaala Fernandez 10. Christian Paul Patawaran 11. Daniel James Molina 12. Joem Canciller 13. Norwyn Nicholson Kah 14. Joselito Torculas 15. Jeffrey Robles 16. Ralph Macarasig 17. Jhepoy Dizon 18. Lim Jing Ren 19. Sumet Ketsri 20. Marvin Cato 21. Hanelet Santos 22. Srinivas Kanigiri 23. Kurara Chibana 24. James Ericson 25. Alea Astrea 26. Kumar Ayush 27. Lenard Guillermo 28. Gluttony 29. Jacob Sabido 30. Radu Bogo 31. Mark Alvero 32. Stefano Ongari 33. Randy Orton 34. Willie Revillame Wowowin 35. Mertkan Simsek 23. Problem: D36C , proposed by Ahmet Arduc 23 is the smallest odd prime which is not a twin prime. Let x be a real number and$$A=\sqrt{x^2-26x+170}+\sqrt{x^2-52x+1700}.$$What is the square of the minimum value of A ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Daniel James Molina 4. Richard Phillip Dimaala Fernandez 5. Rindell Mabunga 6. Norwyn Nicholson Kah 7. John Lester Tan 8. Joselito Torculas 9. Jeffrey Robles 10. Ralph Macarasig 11. Jhepoy Dizon 12. Kimi No Nawa 13. Caed Mark Medul Mendoza 14. Lilanie Monique Torilla 15. Nixon Balandra 16. Russel J. Galanido 17. Jacob Sabido 18. Marvin Cato 19. Sumet Ketsri 20. Sigmund Dela Cruz 21. Christian Daang 22. Kurara Chibana 23. Joem Canciller 24. Gluttony 25. Randy Orton 26. Willie Revillame Wowowin 27. Mertkan Simsek 24. Problem: D7C4 , proposed by Ahmet Arduc The product of 4 consecutive numbers n(n+1)(n+2)(n+3) is divisible by 24. How many x values are there which makes \sqrt{2017-\sqrt{x}} an integer? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Norwyn Nicholson Kah 3. Kimi No Nawa 4. Isaiah James de Dios Maling 5. Joselito Torculas 6. Daniel James Molina 7. Jeffrey Robles 8. Richard Phillip Dimaala Fernandez 9. Ralph Macarasig 10. Rindell Mabunga 11. Caed Mark Medul Mendoza 12. Lilanie Monique Torilla 13. Jhepoy Dizon 14. Nixon Balandra 15. Russel J. Galanido 16. Lim Jing Ren 17. Sumet Ketsri 18. Marvin Cato 19. Mark Alvero 20. John Albert A. Reyes 21. Roenz Joshlee Timbol 22. Smahi Abdeslem 23. Srinivas Kanigiri 24. Kurara Chibana 25. James Ericson 26. Afshiram Muhammed 27. Alea Astrea 28. Jacob Sabido 29. Kumar Ayush 30. Lenard Guillermo 31. Serdal Aslantas 32. Radu Bogo 33. Mertkan Simsek 25. Problem: A4C5 , proposed by Ahmet Arduc, Tip: Key Fact(s): 7695 25 is the smallest square number that can be written as a sum of 2 consecutive squares.$$A=\left(1-\frac{1}{2^2}\right)\cdot\left(1-\frac{1}{3^2}\right)\cdot...\cdot\left(1-\frac{1}{2017^2}\right)$$If A is written in its simplest form as \frac{a}{b}, what is the sum of a and b ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Joselito Torculas 3. Jeffrey Robles 4. Norwyn Nicholson Kah 5. Richard Phillip Dimaala Fernandez 6. Sumet Ketsri 7. Ralph Macarasig 8. Kimi No Nawa 9. Rindell Mabunga 10. Caed Mark Medul Mendoza 11. Lilanie Monique Torilla 12. Nixon Balandra 13. Isaiah James de Dios Maling 14. Russel J. Galanido 15. Jhepoy Dizon 16. Lim Jing Ren 17. Daniel James Molina 18. Jacob Sabido 19. Marvin Cato 20. Fred Gutierrez 21. Nheil Ignacio 22. John Albert A. Reyes 23. Srinivas Kanigiri 24. Chris Norman Algo 25. Kurara Chibana 26. Kumar Ayush 27. Joem Canciller 28. Gluttony 29. Reymark Togno 30. Roenz Joshlee Timbol 31. Mark Alvero 32. Radu Bogo 33. Randy Orton 34. Willie Revillame Wowowin 35. Mertkan Simsek 36. Christian Daang 26. Problem: A378 , proposed by Ahmet Arduc 1953 is a Kaprekar constant in base 2. Find the sum of all real numbers x for which$$1953^x+1954^x+...+1984^x=1985^x+1986^x+...+2015^x.$$Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Joselito Torculas 4. Jhepoy Dizon 5. Jeffrey Robles 6. Ralph Macarasig 7. Norwyn Nicholson Kah 8. Daniel James Molina 9. Richard Phillip Dimaala Fernandez 10. Russel J. Galanido 11. Kimi No Nawa 12. Caed Mark Medul Mendoza 13. Lilanie Monique Torilla 14. Marvin Cato 15. Nixon Balandra 16. Rindell Mabunga 17. Kurara Chibana 18. Sumet Ketsri 19. Kumar Ayush 20. Lenard Guillermo 21. Randy Orton 22. Willie Revillame Wowowin 27. Problem: 3DC5 , proposed by Ahmet Arduc 27 is the largest integer which is the sum of the digits of its cube. Find the coefficient of x^2 when$$\left(1+x\right)\left(1+2x\right)\left(1+4x\right)\cdots\left(1+2^9x\right)$$is expanded ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Richard Phillip Dimaala Fernandez 4. Joselito Torculas 5. Russel J. Galanido 6. Sumet Ketsri 7. Caed Mark Medul Mendoza 8. Lilanie Monique Torilla 9. Daniel James Molina 10. Marvin Cato 11. Hanelet Santos 12. Jacob Sabido 13. Rindell Mabunga 14. Nixon Balandra 15. Jeffrey Robles 16. Sigmund Dela Cruz 17. Kurara Chibana 18. James Ericson 19. Alea Astrea 20. Randy Orton 21. Willie Revillame Wowowin 22. Keedgwh 23. Arjun Singh Rajawat 28. Problem: 9E31 , proposed by Ahmet Arduc, Tip: Key Fact(s): 7695 30 is the 4th pyramidal number. For how many rational numbers between 0 and 1, written as a fraction in its lowest terms, the product of its numerator and denominator will be 30!. Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Russel J. Galanido 3. Marvin Cato 4. Daniel James Molina 5. Joselito Torculas 6. Isaiah James de Dios Maling 7. Richard Phillip Dimaala Fernandez 8. Kimi No Nawa 9. Caed Mark Medul Mendoza 10. Rindell Mabunga 11. Nixon Balandra 12. Sumet Ketsri 13. Roenz Joshlee Timbol 14. Smahi Abdeslem 15. Kurara Chibana 16. Lenard Guillermo 17. Lilanie Monique Torilla 29. Problem: EEAA , proposed by Ahmet Arduc 255 equals 11111111 in base 2. The increasing sequence 1, 3, 4, 9, 10, 12, 13,... consists of all positive integers which are powers of 3 or sums of distinct powers of 3. What is the 255th term of this sequence? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Russel J. Galanido 3. Sumet Ketsri 4. Jacob Sabido 5. Joselito Torculas 6. Marvin Cato 7. Richard Phillip Dimaala Fernandez 8. Isaiah James de Dios Maling 9. Daniel James Molina 10. Caed Mark Medul Mendoza 11. Nixon Balandra 12. Rindell Mabunga 13. Jeffrey Robles 14. Dreimuru Tempest 15. Roenz Joshlee Timbol 16. Kurara Chibana 17. Lim Jing Ren 18. Kumar Ayush 19. Stefano Ongari 20. Randy Orton 21. Willie Revillame Wowowin 22. Lilanie Monique Torilla 30. Problem: 442B , proposed by Ahmet Arduc 30 is the largest number such that every smaller coprime to it is prime. What is the sum of the first 2017 terms of the given sequence?$$\style{color:red}1,1,\style{color:red}1,1,2,\style{color:red}1,1,2,3,\style{color:red}1,1,2,3,4,\style{color:red}1,1,2,3,4,5,\style{color:red}1,...$$Ah Math • Correct answers have been submitted by: 1. Joselito Torculas 2. Sumet Ketsri 3. Amirul Faiz Abdul Muthalib 4. Marvin Cato 5. Nixon Balandra 6. Caed Mark Medul Mendoza 7. Isaiah James de Dios Maling 8. Richard Phillip Dimaala Fernandez 9. Daniel James Molina 10. Jeffrey Robles 11. Jacob Sabido 12. Dreimuru Tempest 13. Russel J. Galanido 14. Kurara Chibana 15. Lim Jing Ren 16. Rindell Mabunga 17. Roenz Joshlee Timbol 18. Stefano Ongari 19. Randy Orton 20. Willie Revillame Wowowin 21. Lilanie Monique Torilla 22. Mertkan Simsek 31. Problem: B854 , proposed by Ahmet Arduc, Tip: Key Fact(s): A031878. The sum of the first 31 odd primes is a perfect square. Let S be a square. There are five distinct circles in the plane of S which have a diameter both of whose endpoints are vertices of S. Let T be a 2017-sided regular polygon. How many distinct circles in the plane of T have a diameter both of whose endpoints are vertices of T ? Ah Math • Correct answers have been submitted by: 1. Dreimuru Tempest 2. Amirul Faiz Abdul Muthalib 3. Nixon Balandra 4. Richard Phillip Dimaala Fernandez 5. Joselito Torculas 6. Marvin Cato 7. Isaiah James de Dios Maling 8. Sumet Ketsri 9. Kimi No Nawa 10. Caed Mark Medul Mendoza 11. Kurara Chibana 12. Russel J. Galanido 13. Rindell Mabunga 14. Lim Jing Ren 15. Jeffrey Robles 16. Randy Orton 17. Willie Revillame Wowowin 18. Lilanie Monique Torilla 32. Problem: 95E3 , proposed by Ahmet Arduc 32 the largest known power with all decimal digits being prime. 8 points on a circle are numbered 0, 1, 2,..., 6, and 7 in counter clockwise order. A ladybug moves in a counter-clockwise direction from one point to another, starting from point 0, 1 point in its first move, 2 points in its second move, 3 points in its third move, and so on. Thus, it will be on point 1 after its first move, on point 3 after its second move, on point 6 after its third move, and so on. On which point, will it be after its 2017th move? Ah Math • Correct answers have been submitted by: 1. Joselito Torculas 2. Richard Phillip Dimaala Fernandez 3. Nixon Balandra 4. Jacob Sabido 5. Amirul Faiz Abdul Muthalib 6. Daniel James Molina 7. Isaiah James de Dios Maling 8. Marvin Cato 9. Sumet Ketsri 10. Roenz Joshlee Timbol 11. Kimi No Nawa 12. Caed Mark Medul Mendoza 13. Chris Norman Algo 14. Russel J. Galanido 15. Kurara Chibana 16. Rindell Mabunga 17. Lim Jing Ren 18. Jeffrey Robles 19. Joem Canciller 20. Gluttony 21. Stefano Ongari 22. Mark Alvero 23. Randy Orton 24. Willie Revillame Wowowin 25. Lilanie Monique Torilla 26. Keedgwh 27. Arjun Singh Rajawat 33. Problem: 3E1C , proposed by Ahmet Arduc 33 is the largest number that is not a sum of distinct triangular numbers. What is the sum of the first digits (not the units digits) of all powers of 2, from 2^0 to 2^{2017} inclusive ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Joselito Torculas 3. Isaiah James de Dios Maling 4. Richard Phillip Dimaala Fernandez 5. Marvin Cato 6. Nixon Balandra 7. Jacob Sabido 8. Russel J. Galanido 9. Kurara Chibana 10. Caed Mark Medul Mendoza 11. Rindell Mabunga 12. Lim Jing Ren 13. Lilanie Monique Torilla 14. Mertkan Simsek 34. Problem: 1715 , proposed by Ahmet Arduc 1910 is the birth year of Cahit Arf. Let O(n) denote the sum of the odd digits of n. For example, O(1910)=1+9+1=11. What is the sum of$$O(1)+O(2)+O(3)+...+O(2017)$$Ah Math • Correct answers have been submitted by: 1. Ikemen 2. Smahi Abdeslem 3. Russel J. Galanido 4. Sumet Ketsri 5. Nixon Balandra 6. Richard Phillip Dimaala Fernandez 7. Joselito Torculas 8. Marvin Cato 9. Isaiah James de Dios Maling 10. Amirul Faiz Abdul Muthalib 11. Kimi No Nawa 12. Caed Mark Medul Mendoza 13. Kurara Chibana 14. Rindell Mabunga 15. Roenz Joshlee Timbol 16. Jeffrey Robles 17. Lilanie Monique Torilla 18. Mertkan Simsek 35. Problem: C877 , proposed by Ahmet Arduc 35 is the number of hexominoes. The sequence u_1, u_2, u_3,... satisfies u_1=1, u_{2017}=2017, and, for all n\ge3, u_n is the average (arithmetic mean) of the first n-1 terms. What is the sum of the first three terms? Ah Math • Correct answers have been submitted by: 1. Dreimuru Tempest 2. Andrew Chiu 3. Marvin Cato 4. Ikemen 5. Richard Phillip Dimaala Fernandez 6. Isaiah James de Dios Maling 7. Joselito Torculas 8. Jacob Sabido 9. Russel J. Galanido 10. Nixon Balandra 11. Sumet Ketsri 12. Amirul Faiz Abdul Muthalib 13. Kurara Chibana 14. Rindell Mabunga 15. Kimi No Nawa 16. Caed Mark Medul Mendoza 17. Alea Astrea 18. Lim Jing Ren 19. Kumar Ayush 20. Jeffrey Robles 21. Joem Canciller 22. Gluttony 23. Randy Orton 24. Willie Revillame Wowowin 25. Lilanie Monique Torilla 36. Problem: C2BC , proposed by Ahmet Arduc 36 is the smallest number which is the sum of pairs of distinct odd primes in four ways. A piece of graph paper is folded once so that (2, 1) is matched with (0, 5). If (a, b) is matched with itself where a+b=2017, what is the value of a ? Ah Math • Correct answers have been submitted by: 1. Ikemen 2. Isaiah James de Dios Maling 3. Jacob Sabido 4. Caed Mark Medul Mendoza 5. Nixon Balandra 6. Amirul Faiz Abdul Muthalib 7. Richard Phillip Dimaala Fernandez 8. Marvin Cato 9. Andrew Chiu 10. Joselito Torculas 11. Russel J. Galanido 12. Sumet Ketsri 13. Kurara Chibana 14. Alea Astrea 15. Rindell Mabunga 16. Jeffrey Robles 17. Kumar Ayush 18. Joem Canciller 19. Gluttony 20. Mark Alvero 21. Randy Orton 22. Willie Revillame Wowowin 23. Lilanie Monique Torilla 24. Mertkan Simsek 37. Problem: E4AE , proposed by Ahmet Arduc 37 is the maximum number of 5th powers needed to sum to any number. For the sets of consecutive integers$$\{1\}, \{2, 3\}, \{4, 5, 6\}, \{7, 8, 9, 10\},...$$let S_n be the sum of the elements in the nth set. What is the remainder when S_{2017} is divided by 9 ? Ah Math • Correct answers have been submitted by: 1. Ikemen 2. Richard Phillip Dimaala Fernandez 3. Kurara Chibana 4. Isaiah James de Dios Maling 5. Russel J. Galanido 6. Marvin Cato 7. Amirul Faiz Abdul Muthalib 8. Joselito Torculas 9. Smahi Abdeslem 10. Jacob Sabido 11. Alea Astrea 12. Nixon Balandra 13. Sumet Ketsri 14. Lim Jing Ren 15. Rindell Mabunga 16. Kimi No Nawa 17. Caed Mark Medul Mendoza 18. Jeffrey Robles 19. Lenard Guillermo 20. Joem Canciller 21. Gluttony 22. Stefano Ongari 23. Randy Orton 24. Willie Revillame Wowowin 25. Lilanie Monique Torilla 26. Keedgwh 27. Arjun Singh Rajawat 28. Monu Baba Rura Sirsa Up 29. Mertkan Simsek 38. Problem: 57BD , proposed by Ahmet Arduc 38 is the largest known even number which can be represented as sum of two distinct primes in only one way. What is the remainder when the sum of the first 2017 terms of the sequence$$1,\hspace{0.2cm}1+2,\hspace{0.2cm}1+2+2^2,...,\hspace{0.2cm}1+2+2^2+...+2^{n-1},...$$is divided by 9 ? Ah Math • Correct answers have been submitted by: 1. Sumet Ketsri 2. James Ericson 3. Andrew Chiu 4. Joselito Torculas 5. Richard Phillip Dimaala Fernandez 6. Caed Mark Medul Mendoza 7. Isaiah James de Dios Maling 8. Nixon Balandra 9. Jeffrey Robles 10. Marvin Cato 11. Amirul Faiz Abdul Muthalib 12. Lim Jing Ren 13. Jacob Sabido 14. Rindell Mabunga 15. Alea Astrea 16. Kurara Chibana 17. Russel J. Galanido 18. Roenz Joshlee Timbol 19. Mark Alvero 20. Randy Orton 21. Radu Bogo 22. Willie Revillame Wowowin 23. Lilanie Monique Torilla 24. Keedgwh 25. Arjun Singh Rajawat 39. Problem: 3785 , proposed by Ahmet Arduc 39 is the smallest number which has 3 different partitions into 3 parts with the same product. 428 has the property that its square appears as two consecutive integers:$$428^2=183184$$What is the sum of all 3-digit numbers with this property? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Andrew Chiu 3. Nixon Balandra 4. Caed Mark Medul Mendoza 5. Marvin Cato 6. Joselito Torculas 7. Rindell Mabunga 8. Isaiah James de Dios Maling 9. Richard Phillip Dimaala Fernandez 10. Kurara Chibana 11. Russel J. Galanido 12. James Ericson 13. Alea Astrea 14. Jacob Sabido 15. Sumet Ketsri 16. Lim Jing Ren 17. Jeffrey Robles 18. Randy Orton 19. Willie Revillame Wowowin 20. Keedgwh 21. Arjun Singh Rajawat 40. Problem: 2783 , proposed by Ahmet Arduc 40 is a pentagonal pyramidal number. What is the greatest number less than 2017, when divided by the sum of its digits gives the greatest remainder? Ah Math • Correct answers have been submitted by: 1. Nixon Balandra 2. Isaiah James de Dios Maling 3. Marvin Cato 4. Andrew Chiu 5. Amirul Faiz Abdul Muthalib 6. Joselito Torculas 7. Rindell Mabunga 8. Caed Mark Medul Mendoza 9. Lilanie Monique Torilla 10. Richard Phillip Dimaala Fernandez 11. Kurara Chibana 12. Russel J. Galanido 13. James Ericson 14. Sumet Ketsri 15. Alea Astrea 16. Jacob Sabido 17. Jeffrey Robles 18. Joem Canciller 19. Gluttony 20. Randy Orton 21. Willie Revillame Wowowin 22. Mertkan Simsek 41. Problem: 398B , proposed by Ahmet Arduc 41 is the lowest number whose cube is the sum of 3 cube numbers in 2 different ways. The numbers$$1,\frac{1}{2},\frac{1}{3},\frac{1}{4},...,\frac{1}{2016},\frac{1}{2017}$$are all written on a board. Two numbers x and y are selected randomly from the list, deleted, and replaced by the single number x+y+x\cdot y. This is done repeatedly until one number is left. What is the last number remaining on the board? Ah Math • Correct answers have been submitted by: 1. Isaiah James de Dios Maling 2. Nixon Balandra 3. Kurara Chibana 4. Russel J. Galanido 5. Rindell Mabunga 6. Marvin Cato 7. James Ericson 8. Andrew Chiu 9. Ikemen 10. Caed Mark Medul Mendoza 11. Sumet Ketsri 12. Joselito Torculas 13. Alea Astrea 14. Richard Phillip Dimaala Fernandez 15. Jacob Sabido 16. Amirul Faiz Abdul Muthalib 17. Lim Jing Ren 18. Jeffrey Robles 19. Lenard Guillermo 20. Joem Canciller 21. Gluttony 22. Stefano Ongari 23. Mark Alvero 24. Randy Orton 25. Willie Revillame Wowowin 42. Problem: DE1D , proposed by Ahmet Arduc 42 is the he smallest perfect square that is the mean of two cubed twin primes. Let ABC be a triangle. E \in [BC]. If |AB|=32, |BE|=16, |EC|=48, and m(BCA)=24^{\circ}, what is the measure of the angle EAB ? Ah Math • Correct answers have been submitted by: 1. Sumet Ketsri 2. Kurara Chibana 3. Russel J. Galanido 4. Marvin Cato 5. Rindell Mabunga 6. Ikemen 7. Richard Phillip Dimaala Fernandez 8. Isaiah James de Dios Maling 9. Nixon Balandra 10. Kimi No Nawa 11. Amirul Faiz Abdul Muthalib 12. Caed Mark Medul Mendoza 13. Jeffrey Robles 14. Lilanie Monique Torilla 15. Joem Canciller 16. Gluttony 17. Andrew Chiu 18. Roenz Joshlee Timbol 19. Randy Orton 20. Willie Revillame Wowowin 21. Mertkan Simsek 43. Problem: B87D , proposed by Ahmet Arduc 43 is the smallest non-palindromic prime which on subtracting its reverse gives a perfect square. Let ABC be a triangle. Let D be a point in the interior of ABC, such that, |AD|=|DB|. If m(DBA)=11^{\circ}, m(DBC)=38^{\circ}, and m(DAC)=19^{\circ}, what is the measure of the angle m(DCA) ? Ah Math • Correct answers have been submitted by: 1. Kurara Chibana 2. Russel J. Galanido 3. Amirul Faiz Abdul Muthalib 4. Kimi No Nawa 5. Caed Mark Medul Mendoza 6. Marvin Cato 7. Rindell Mabunga 8. Sumet Ketsri 9. Jeffrey Robles 10. Lim Jing Ren 11. Lilanie Monique Torilla 12. Andrew Chiu 13. Isaiah James de Dios Maling 14. Randy Orton 15. Willie Revillame Wowowin 16. Mertkan Simsek 44. Problem: C426 , proposed by Amirul Faiz Abdul Muthalib 44 is the number of derangements of 5 objects. Let P(x) be a polynomial where deg(P(x))=2015. If P(0)=1, P(1)=2, P(2)=3, ..., P(2013)=2014, and P(2014)=2015, but P(2015)=2017, what is the value of P(2017) ? Ah Math • Correct answers have been submitted by: 1. Christian Daang 2. Sumet Ketsri 3. Kumar Ayush 4. Rindell Mabunga 5. Kurara Chibana 6. Russel J. Galanido 7. Marvin Cato 8. Jeffrey Robles 9. Nixon Balandra 10. Isaiah James de Dios Maling 11. Richard Phillip Dimaala Fernandez 12. Kimi No Nawa 13. Caed Mark Medul Mendoza 14. Lilanie Monique Torilla 15. Roenz Joshlee Timbol 16. Randy Orton 17. Willie Revillame Wowowin 45. Problem: 143B , proposed by Amirul Faiz Abdul Muthalib 45 is both a Kaprekar and a triangular number. Ahmet added up all the odd integers from 1 to a certain number on his calculator and obtained a sum of 2017. Unfortunately, he mistakenly entered one of the numbers twice. What is the number that he added twice? Ah Math • Correct answers have been submitted by: 1. Richard Phillip Dimaala Fernandez 2. Kumar Ayush 3. Nixon Balandra 4. Kurara Chibana 5. Russel J. Galanido 6. Caed Mark Medul Mendoza 7. Marvin Cato 8. Reymark Togno 9. Lim Jing Ren 10. Rindell Mabunga 11. Jeffrey Robles 12. Sumet Ketsri 13. Isaiah James de Dios Maling 14. Lilanie Monique Torilla 15. Lenard Guillermo 16. Jacob Sabido 17. Joem Canciller 18. Gluttony 19. Andrew Chiu 20. Roenz Joshlee Timbol 21. Mark Alvero 22. Randy Orton 23. Willie Revillame Wowowin 24. Mertkan Simsek 46. Problem: 7964 , proposed by Amirul Faiz Abdul Muthalib, Tip: Key Fact(s): 7695 46 is the number of chromosomes all human beings have in every cell of their body. What is the smallest positive integer n \neq 2017 such that the fraction$$\frac{n-2017}{20n+17}$$is NOT in its simplest form (reducible fraction)? Ah Math • Correct answers have been submitted by: 1. Nixon Balandra 2. Marvin Cato 3. Sumet Ketsri 4. Rindell Mabunga 5. Kurara Chibana 6. Russel J. Galanido 7. Lim Jing Ren 8. Caed Mark Medul Mendoza 9. Andrew Chiu 10. Isaiah James de Dios Maling 11. Jeffrey Robles 12. Roenz Joshlee Timbol 13. Randy Orton 14. Willie Revillame Wowowin 47. Problem: 76AC , proposed by Amirul Faiz Abdul Muthalib 47 is the largest number of cubes that cannot tile a cube. Four consecutive integers are greater than 2017. The smallest is divisible by 5, the second is divisible by 7, the third is divisible by 9, and the largest is divisible by 11. Find the sum of the smallest such consecutive integers. Ah Math • Correct answers have been submitted by: 1. Isaiah James de Dios Maling 2. Rindell Mabunga 3. Kurara Chibana 4. Russel J. Galanido 5. Lim Jing Ren 6. Caed Mark Medul Mendoza 7. Lilanie Monique Torilla 8. Marvin Cato 9. Sumet Ketsri 10. Jeffrey Robles 11. Mark Alvero 12. Roenz Joshlee Timbol 13. Randy Orton 14. Willie Revillame Wowowin 15. Mertkan Simsek 48. Problem: D256 , proposed by Ahmet Arduc 1971 is the birth year of A. Arduc. ﻿The factorial base of numeration is defined as a_1+a_2⋅2!+a_3⋅3!+...+a_n⋅n! where a_1,a_2,...,a_n are integer coefficients such that 0≤a_k≤k. Thus, 1971 can be written in factorial base of numeration as$$1+1⋅2!+0⋅3!+2⋅4!+4⋅5!+2⋅6!$$where the sum of the coefficients is 10. What is the sum of all coefficients used in factorial base of numeration to write all numbers from 1 to 2017? Ah Math • Correct answers have been submitted by: 1. Kurara Chibana 2. Russel J. Galanido 3. Amirul Faiz Abdul Muthalib 4. Marvin Cato 5. Kimi No Nawa 6. Caed Mark Medul Mendoza 7. Rindell Mabunga 8. Lilanie Monique Torilla 9. Sumet Ketsri 49. Problem: 8D49 , proposed by Ahmet Arduc, Tip: Key Fact(s): 7695 49 is the smallest number with exactly 8 representations as a sum of three distinct primes. Given$$A=\frac{1}{2017}+\frac{2}{2017^2}+\frac{3}{2017^3}+...$$whose n th term is \large{\frac{n}{2017^n}}. If A is written in its simplest form as \large{\frac{a}{b}}, what is the sum of a and b ? Ah Math • Correct answers have been submitted by: 1. Isaiah James de Dios Maling 2. Amirul Faiz Abdul Muthalib 3. Kurara Chibana 4. Russel J. Galanido 5. Marvin Cato 6. Kimi No Nawa 7. Caed Mark Medul Mendoza 8. Sumet Ketsri 9. Rindell Mabunga 10. Joem Canciller 11. Daniel James Molina 12. Lilanie Monique Torilla 13. Jacob Sabido 14. Andrew Chiu 15. Nixon Balandra 16. Jeffrey Robles 17. Roenz Joshlee Timbol 18. Randy Orton 19. Willie Revillame Wowowin 50. Problem: 7E39 , proposed by Amirul Faiz Abdul Muthalib 50 is the smallest number that can be written as the sum of 2 squares in 2 distinct ways. Given two sequences {{a}_{n}} and {{b}_{n}} with initial terms a_1 and b_1 such that {{a}_{1}}={{b}_{1}}=2017,${{a}_{2}}=2!\,+0!\,+1!\,+7!=5044,\text{ }{{a}_{3}}=5!\,+0!\,+4!\,+4!=169,\,\,\ldots$ which every term of {{a}_{n}}is the sum of the factorial of the digits of the preceding term, ${{b}_{2}}={{2}^{2}}+{{0}^{2}}+{{1}^{2}}+{{7}^{2}}=54,\text{ }{{b}_{3}}={{5}^{2}}+{{4}^{2}}=41,\text{ }{{b}_{4}}={{4}^{2}}+{{1}^{2}}=17,\ldots$which every term of {{b}_{n}} is the sum of the squares of the digits of the preceding term. What is the value of {{a}_{2017}}+{{b}_{2017}} ? Ah Math • Correct answers have been submitted by: 1. Kurara Chibana 2. Russel J. Galanido 3. Caed Mark Medul Mendoza 4. Lilanie Monique Torilla 5. Marvin Cato 6. Sumet Ketsri 7. Andrew Chiu 8. Nixon Balandra 9. Isaiah James de Dios Maling 10. Rindell Mabunga 11. Jeffrey Robles 12. Mark Alvero 13. Randy Orton 14. Willie Revillame Wowowin 51. Problem: 4E84 , proposed by Ahmet Arduc 51 is the 6th Motzkin number. The 51 stars of the US flag are numbered from 1 to 51, from top right corner to lower left corner. 13 of them are picked at random. Among those selected, if the probability that the third smallest is numbered as 7 is an irreducible fraction \large{\frac{a}{b}}, what is the sum of a and b ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Sumet Ketsri 3. Kurara Chibana 4. Russel J. Galanido 5. Kimi No Nawa 6. Caed Mark Medul Mendoza 7. Isaiah James de Dios Maling 8. Jeffrey Robles 9. Rindell Mabunga 52. Problem: EE37 , proposed by Amirul Faiz Abdul Muthalib 52 is the 5th Bell number. How many ways are there to choose 4 distinct integers, \{a,\,b,\,c,\,d\}, from the set N=\{1,\,2,\,3,\,...\,, 2016,\,2017\} such that \{a,\,b,\,c,\,d\} form an increasing geometric sequence with a common ratio of positive integer? Ah Math • Correct answers have been submitted by: 1. Sumet Ketsri 2. Kurara Chibana 3. Russel J. Galanido 4. Kimi No Nawa 5. Caed Mark Medul Mendoza 6. Isaiah James de Dios Maling 7. Lilanie Monique Torilla 8. Jeffrey Robles 9. Roenz Joshlee Timbol 10. Mark Alvero 11. Rindell Mabunga 12. Radu Bogo 13. Reymark Togno 53. Problem: 2D73 , proposed by Ahmet Arduc 64 is the smallest number with 7 divisors. Let S(n) and P(n) denote the sum and the product, respectively, of the digits of the integer n. For example, S(64)=6+4=10 and P(64)=6 \cdot 4=24. Let N be the sum of S(N) and P(N). What is the sum of all two-digit N numbers? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Kimi No Nawa 4. Lilanie Monique Torilla 5. Caed Mark Medul Mendoza 6. Kurara Chibana 7. Russel J. Galanido 8. Jeffrey Robles 9. Reymark Togno 10. Rindell Mabunga 11. Roenz Joshlee Timbol 54. Problem: E584 , proposed by Amirul Faiz Abdul Muthalib 54 is the smallest number that can be written as the sum of 3 squares in 3 ways. A polynomial P(x) of degree 2016 satisfies$$P(0)=0,\,\,P(1)=\frac{1}{2},\,\,P(2)=\frac{2}{3},\,\,P(3)=\frac{3}{4},\,\,\ldots ,\,\,P(2016)=\frac{2016}{2017} .$$If P(2017) form an irreducible fraction of \large{\frac{a}{b}}, what is a+b? Ah Math 55. Problem: 86C7 , proposed by Ahmet Arduc 55 is the largest triangular number in the Fibonacci sequence. Consider the sequence of numbers$$1,\,2,\,3,\,5,\,8,\,3,\,1,\,...$$For n>2, the nth term of the sequence is the units digit of the sum of the two previous terms. What is the sum of the first 2017 terms? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Kurara Chibana 4. Russel J. Galanido 5. Caed Mark Medul Mendoza 6. Jeffrey Robles 7. James Ericson 8. Reymark Togno 9. Sumet Ketsri 10. Nixon Balandra 11. Jacob Sabido 12. Stefano Ongari 13. Randy Orton 14. Willie Revillame Wowowin 15. Rindell Mabunga 16. Mertkan Simsek 56. Problem: DAA3 , proposed by Ahmet Arduc co-writer: Mehmet Emin Arduc. After a certain dilation, if point (86,286) maps onto point (103,343), point (106,126) maps onto point (127,151), and point (a,b) maps onto point (2017,2017), what is the sum of a and b ? Ah Math 57. Problem: 7B63 , proposed by Ahmet Arduc 57 because this it the 57th problem :-) The sum of 57 consecutive positive integers is a perfect square. What is the smallest possible value of this sum? Ah Math • Correct answers have been submitted by: 1. Caed Mark Medul Mendoza 2. Kimi No Nawa 3. Amirul Faiz Abdul Muthalib 4. James Ericson 5. Jacob Sabido 6. Roenz Joshlee Timbol 7. Kurara Chibana 8. Russel J. Galanido 9. Sumet Ketsri 10. Jeffrey Robles 11. Dan Lang 12. Rindell Mabunga 13. Reymark Togno 14. Randy Orton 15. Radu Bogo 16. Willie Revillame Wowowin 17. Lilanie Monique Torilla 18. Mertkan Simsek 58. Problem: 5484 , proposed by Ahmet Arduc 18 is the only number that is twice the sum of its digits.$$A=2!\cdot4!\cdot6!\cdot8!\cdot10!\cdot12!\cdot14!\cdot16!\cdot18!$$How many perfect squares are divisors of A? Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Ikemen 3. Kimi No Nawa 4. Caed Mark Medul Mendoza 5. Amirul Faiz Abdul Muthalib 6. Kurara Chibana 7. Russel J. Galanido 8. Rindell Mabunga 9. Stefano Ongari 10. Lilanie Monique Torilla 11. Randy Orton 12. Willie Revillame Wowowin 59. Problem: B18D , proposed by Amirul Faiz Abdul Muthalib Watch first S2E2 of NUMB3RS (the television crime drama): Better or Worse. The irreducible fractions between 0 and 1 are listed in ascending order, with denominators that are at most 2017. What is the sum of a, b, c, and d, if \large{\frac{a}{b}} and \large{\frac{c}{d}} are two adjacent fractions of \large{\frac{17}{20}} ? Ah Math 60. Problem: B57E , proposed by Ahmet Arduc A Platonic Solid: Cube. Numbers from the set \{1, 289, 577, 865, 1153, 1441, 1729, 2017\} are placed at the vertices of a cube in such a manner that the sum of the four numbers on each face is the same. What is this common sum? Ah Math • Correct answers have been submitted by: 1. Kurara Chibana 2. Russel J. Galanido 3. Randy Orton 4. Sumet Ketsri 5. Willie Revillame Wowowin 6. Kimi No Nawa 7. Caed Mark Medul Mendoza 8. Lilanie Monique Torilla 9. Rindell Mabunga 10. Jeffrey Robles 11. Amirul Faiz Abdul Muthalib 12. Mertkan Simsek 61. Problem: B375 , proposed by Ahmet Arduc 61 is the smallest prime whose reversal (16) is a square.$$A=1!+2!+3!\,+ \,...\, +\,2017!$$What is the sum of the units and the tens digit of A ? Ah Math • Correct answers have been submitted by: 1. Randy Orton 2. Willie Revillame Wowowin 3. Kimi No Nawa 4. Caed Mark Medul Mendoza 5. Lilanie Monique Torilla 6. Roenz Joshlee Timbol 7. Kurara Chibana 8. Russel J. Galanido 9. Rindell Mabunga 10. Christian Daang 11. Amirul Faiz Abdul Muthalib 12. Ikemen 13. Keedgwh 14. Arjun Singh Rajawat 62. Problem: A3BD , proposed by Ahmet Arduc 62 is the smallest number that can be written as the sum of 3 distinct squares in 2 ways. The first term of a sequence is 2017. Each succeeding term is the sum of the squares of the digits of the previous term. What is the 2017th term of the sequence? Ah Math • Correct answers have been submitted by: 1. Randy Orton 2. Willie Revillame Wowowin 3. Kurara Chibana 4. Russel J. Galanido 5. Caed Mark Medul Mendoza 6. Lilanie Monique Torilla 7. Roenz Joshlee Timbol 8. Jeffrey Robles 9. Amirul Faiz Abdul Muthalib 10. Mertkan Simsek 63. Problem: 513D , proposed by Ahmet Arduc 63 is the number of partially ordered sets of 5 elements.$$3^{280} \lt n^{140} \lt (k\cdot n)^{70}$$If there are 2017 positive integers n that satisfies the given inequality, what is the positive value of k \,? Ah Math • Correct answers have been submitted by: 1. Randy Orton 2. Willie Revillame Wowowin 3. Kurara Chibana 4. Russel J. Galanido 5. Kimi No Nawa 6. Caed Mark Medul Mendoza 7. Lilanie Monique Torilla 8. Jeffrey Robles 9. Amirul Faiz Abdul Muthalib 64. Problem: D8E1 , proposed by Ahmet Arduc, Tip: Key Fact(s): B13A 64 is the smallest number with 7 divisors. A lattice point is a point in the plane with integer coordinates. How many lattice points are on the line segment whose endpoints are (45, 64) and (1971, 2017), both ends excluded? Ah Math • Correct answers have been submitted by: 1. Caed Mark Medul Mendoza 2. Randy Orton 3. Willie Revillame Wowowin 4. Lilanie Monique Torilla 5. Jeffrey Robles 6. Amirul Faiz Abdul Muthalib 7. Mertkan Simsek 65. Problem: 89E9 , proposed by Ahmet Arduc 65 is the lowest integer that becomes square if its reverse is either added to or subtracted from it. A 65×65×65 woden cube is formed by gluing together 65^3 unit cubes. What is the least number of unit cubes that cannot be seen from a single point? Ah Math • Correct answers have been submitted by: 1. Randy Orton 2. Willie Revillame Wowowin 3. Kimi No Nawa 4. Caed Mark Medul Mendoza 5. Lilanie Monique Torilla 6. Radu Bogo 7. Amirul Faiz Abdul Muthalib 8. Mertkan Simsek 66. Problem: 792C , proposed by Amirul Faiz Abdul Muthalib 66 is a triangular palindromic number. Given that z is a complex number such that \large{z+\frac{1}{z}}=\normalsize{2\cos17}^{\circ}. Find the least integer that is greater than$${\left( {{z}^{2017}}+\frac{1}{{{z}^{2017}}} \right)}^{-1}$$Ah Math 67. Problem: 1D15 , proposed by Ahmet Arduc 67 is the smallest number which is both palindromic in base 5 and in base 6. A two-digit number sequence is as follows:$$13, 92, 78, 12, 43, 72, a, b, c, ...$$What is the sum of a, b, and c of this sequence? Ah Math 68. Problem: BB2C , proposed by Amirul Faiz Abdul Muthalib 68 is the smallest composite number that becomes prime by turning it upside down. Amirul and Ahmet play a game. Each player, in turn, has to name a natural number that is less than but at least half the previous number. The player who names the number 1 loses. If Amirul starts by naming 2017, what is the next number that Ahmet should choose to ensure that he will win at the end? Ah Math 69. Problem: C294 , proposed by Ahmet Arduc 69 is a value of n where n² and n³ together contain each digit once. A bag contains 69 marbles, numbered with the first 69 prime numbers. 5 marbles are drawn simultaneously at random. Among those selected, if the probability that the sum of the numbers on the marbles drawn is even is an irreducible fraction \large{\frac{a}{b}}, what is the sum of a and b ? Ah Math 70. Problem: D8E5 , proposed by Ahmet Arduc, Tip: Key Fact(s): C181 70 is the smallest weird number. Find the least positive integer that has exactly 70 positive integer divisors. Ah Math 71. Problem: 99E4 , proposed by Ahmet Arduc 71 divides the sum of the primes less than it. The four points A(2, 2), B(10, 1), C(2017, t), and D(1, 7) lie in the coordinate plane. If P(7, 3) is the point to get the minimum possible value of PA+PB+PC+PD over all points on the plane, what is the value of t ? Ah Math 72. Problem: D64B , proposed by Ahmet Arduc 72 is the maximum number of spheres that can touch another sphere in a lattice packing in 6 dimensions. How many positive perfect squares less than 10^8 are multiples of 72? Ah Math 73. Problem: 62C9 , proposed by Ihsan Yucel 73 is the smallest multi-digit number which is one less than twice its reverse. What is the maximum number of points of intersection of 73 coplanar squares if none of the sides of any two squares have the same slope? Ah Math 74. Problem: 5AD1 , proposed by Ihsan Yucel 74 is the number of non-Hamiltonian polyhedra with a minimum number of vertices. In a country, there are n farms and in each farm, there are at most 15 types of fruit trees. For any three of the farms, it's certain that two of them have the same type of fruit tree. What is the minimum number of n to guarantee that 69 different farms have the same type of fruit tree? Ah Math 75. Problem: 64DE , proposed by Ahmet Arduc 75 is the number of orderings of 4 objects with ties allowed. ABCD is a square with sides 75 cm. P is a point on BC such that PC=35 cm. If R is a variable point on the diagonal BD, find the least value of RC+RP ? Ah Math 76. Problem: 6A4D , proposed by Ahmet Arduc 76 is an automorphic number. Let a and b be two real numbers that satisfy a \cdot b=76. What is the minimum value of (a+b)^2 ? Ah Math 77. Problem: DAB4 , proposed by Ahmet Arduc 77 is the largest number that cannot be written as a sum of distinct numbers whose reciprocals add up to 1. For n \ge 2, what is the minumum value of the integer n that satisfies the inequality$$\left( 1 - \frac{1}{2^2} \right)\left( 1 - \frac{1}{3^2} \right) \cdot ... \cdot \left( 1 - \frac{1}{n^2} \right) < \frac{1017}{2017}$$Ah Math 78. Problem: 3B7C , proposed by Ihsan Yucel 78 is the lowest number which can be written as the sum of 4 distinct squares in 3 ways. Let P_1, P_2, P_3, ... , P_{78} be some coplanar points with k lines on the same plane. If exactly two line passes through each point and exactly three of these points lie on each line, what is the value of k ? Ah Math 79. Problem: 7611 , proposed by Ahmet Arduc 79 is the smallest prime whose sum of digits is a 4th power. What are the last 5 digits of the sum$$1+11+111\,+...+\,\underbrace{111...11}_{2017\text{ digits}}$$Ah Math 80. Problem: 5185 , proposed by Ahmet Arduc 80 is the smallest number with exactly 7 representations as a sum of three distinct primes. What is the sum of the terms of the 2017th set if the sets are given like$$\{1,2,3,4\}, \{5,6,7,8\}, \{9,1,0,1\}, \{1,1,2,1\},...$$Ah Math 81. Problem: 77ED , proposed by Ahmet Arduc 81 is the square of the sum of its digits; there are no other numbers with this property except 0 and 1. How many digits are there before the 81th 5 in the following number$$a0b00c000d0000e00000a000000b0000000c...$$if e=5 ? Ah Math 82. Problem: D467 , proposed by Ahmet Arduc 82 is the international telephone dialing code for Korea. Starting with 1, Russel J. Galanido lists the counting numbers in order but omits all those that use the digit 5. What is the 2017th number in the list? Ah Math 83. Problem: 5788 , proposed by Ahmet Arduc 83 is the sum of the first 3 primes ending with 1. Let f_1, f_2, f_3, ... be a sequence of integers satisfying f_{n-1}+f_n=2n for all n \ge 2. If f_1=83, what is the value of f_{2017} \, ? Ah Math 84. Problem: C2C9 , proposed by Ahmet Arduc 84 is the smallest integer with 8 different representations as a sum of 2 primes. Find the number of pairs of positive integers a and b such that a < b and$$\frac{1}{a}+\frac{1}{b}=\frac{1}{84}.$$Ah Math 85. Problem: 69D1 , proposed by Ahmet Arduc 85 is a centered triangular number. Let ABC be an equilateral triangle. Let D be a point in the interior of ABC, such that, |DA|=5 cm, |DB|=12 cm, and |DC|=13 cm. What is the measure of the angle m(ADB), in degrees? Ah Math 86. Problem: A795 , proposed by Ahmet Arduc 86 is the largest known n for which 2^n contain NO zeros. Let ABCD be a square. Let E be a point in the interior of ABCD, such that, |EB|=11 cm, |EC|=6 cm, and |ED|=7 cm. What is the measure of the angle m(CED), in degrees? Ah Math 87. Problem: 9896 , proposed by Ahmet Arduc 87 is the sum of the squares of the first 4 primes. If A=\underbrace{111\cdots11}_{2017\text{ digits}}, what is the sum of the digits of 2017 \cdot A \,? Ah Math 88. Problem: 5D33 , proposed by Ahmet Arduc 88 can be read the same upside down or when viewed in a mirror. Let A be the sum of all 4-digit numbers that can be formed by 2, 3, 5, and 9, and B be the sum of all 4-digit numbers that can be formed by 1, 4, 6, and 7. What is the sum of all positive prime factors of A-B, if digits are allowed to be repeated for all numbers? Ah Math 89. Problem: 1972 , proposed by Ahmet Arduc 89 is a prime and a Fibonacci number. Let ABC be a triangle. If h_a=10 units and h_b=12 units, what can be the maximum integer value of h_c? Ah Math ☆★☆★☆★ Newest Problem! ★☆★☆★☆ 90. Problem: A845 , proposed by Ahmet Arduc 90 is the smallest number having 6 representations as a sum of four positive squares. Caed Mark Medul Mendoza has put 90 unit cubes together side by side to get a long cuboid with dimensions of 1 unit x 1 unit x 90 units. The numbers on the faces of the nth cube are the successive integers from 6n-5 to 6n, where the opposite faces have the sum of 12n-5. What will be the greatest possible sum of the numbers of the visible 272 unit squares? Ah Math 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. Enter your E-mail and User Code to Log in. I need a user code I already have a user code  By clicking Send Form, you confirm that you have read and agree to our Terms and Conditions written under the "Info" tab. ## 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. Order Score # of Correct Answers Names Badges Country FSSFSP 1.076.60 77Amirul Faiz Abdul Muthalib110 Malaysia 2.072.09 86Caed Mark Medul Mendoza4101 Philippines 3.067.22 70Jeffrey Robles134 Philippines 4.066.67 60Lilanie Monique Torilla055 Philippines 5.062.24 57Sumet Ketsri558 Thailand 6.060.16 62Kurara Chibana40 Japan 7.060.00 54Willie Revillame Wowowin00 Philippines 8.059.59 63Russel J. Galanido130 South Korea 9.052.90 53Isaiah James de Dios Maling426 Philippines 10.048.73 50Marvin Cato130 Philippines 11.043.09 55Mertkan Simsek50 Turkey 12.041.97 48Nixon Balandra346 Philippines 13.041.54 54Randy Orton60 United States of America 14.041.24 60Rindell Mabunga131 Philippines 15.036.28 40Jacob Sabido05 Philippines 16.035.26 44Richard Phillip Dimaala Fernandez445 Philippines 17.030.62 41Joselito Torculas426 Philippines 18.029.39 36Lim Jing Ren00 Malaysia 19.028.89 26Alea Astrea00 Philippines 20.025.60 24Radu Bogo024 Romania 21.025.56 23Arjun Singh Rajawat04 India 22.021.11 19Lenard Guillermo02 Philippines 23.020.20 40Roenz Joshlee Timbol01 Philippines 24.020.20 20Gluttony00 Philippines 25.018.32 26Norwyn Nicholson Kah038 Philippines 26.018.27 25Jhepoy Dizon00 Philippines 27.017.24 32Daniel James Molina01 Philippines 28.016.46 20Ralph Macarasig00 Philippines 29.016.36 18Mark Alvero00 Philippines 30.015.65 26Joem Canciller10 Philippines 31.015.56 14John Albert A. Reyes06 Philippines 32.014.60 17Nheil Ignacio08 Philippines 33.013.99 23Kumar Ayush01 India 34.013.38 17Ibrahim Demir01 Turkey 35.012.80 22Christian Paul Patawaran01 Philippines 36.012.22 11Melga Sonio00 Philippines 37.012.22 11Angelu G. Leynes10 Philippines 38.011.74 13Poetri Sonya Tarabunga00 Indonesia 39.011.43 12Andrew Chiu00 Philippines 40.010.05 42Kimi No Nawa7166 Japan 41.010.00 18James Ericson10 Thailand 42.009.60 11Mark Elis Espiridion00 Philippines 43.009.46 20Christian Daang125 Philippines 44.008.89 8Monu Baba Rura Sirsa Up00 India 45.008.89 8Joseph Rodelas10 Philippines 46.008.40 11Chris Norman Algo06 Philippines 47.008.37 16Stefano Ongari00 Italy 48.008.00 12Reymark Togno00 Philippines 49.008.00 12Dreimuru Tempest20 Philippines 50.007.78 7Adrian Pilotos Burgos00 Philippines 51.006.67 6Yavuz Selim Koseoglu20 Turkey 52.006.67 6Mahmut Cemrek00 Turkey 53.006.67 6John Gamal Aziz Attia21 Egypt 54.006.17 10Ikemen30 Japan 55.005.44 7Srinivas Kanigiri00 India 56.005.44 7Emmanuel David00 Philippines 57.004.63 5Sigmund Dela Cruz05 Philippines 58.004.44 4Chayapol02 Thailand 59.004.44 4Βαρελάς Γεώρ𝛾ιος00 Greece 60.003.97 5John Lester Tan00 Philippines 61.003.47 5Hanelet Santos00 Philippines 62.003.47 5Grant Lewis Bulaong01 Philippines 63.003.33 3Gerald M. Pascua00 Philippines 64.003.33 3Barry Villanueva02 Philippines 65.003.09 5Jake Gacuan09 Philippines 66.002.96 4Mark Lawrence P Velasco00 Philippines 67.002.86 6Fred Gutierrez00 Philippines 68.002.54 4Smahi Abdeslem04 Algeria 69.002.31 5Dan Lang00 Philippines 70.002.22 2Melek Cimen10 Turkey 71.002.22 2Edge Ramos00 Philippines 72.002.22 2Afshiram Muhammed00 Turkey 73.002.22 16Keedgwh00 India 74.001.63 5John Marco Latagan00 Philippines 75.001.11 2Evan Gruda00 United States of America 76.001.11 1Suleyman Akarsu00 Turkey 77.001.11 1Serkan Callioglu00 Turkey 78.001.11 1Rosendo Parra Milian01 Peru 79.001.11 1Rdvnaksu11 Turkey 80.001.11 1Muhammed Aydogdu00 Turkey 81.001.11 1Mohamed Karamany10 Egypt 82.001.11 1Captain Magneto00 Germany 83.001.11 1Abhishek Singh00 India 84.000.89 2Serdal Aslantas01 Romania 85.000.74 2John Rocel Perez00 Philippines 86.000.37 1Mark Allen Facun00 Philippines 87.000.37 1John Patrick03 Philippines • Books, Websites, etc. ✔ Singapore Mathematical Olympiads-2005 ✔ Singapore Mathematical Olympiads-2006 ✔ Singapore Mathematical Olympiads-2007 ✔ Singapore Mathematical Olympiads-2008 ✔ Singapore Mathematical Olympiads-2009 ✔ Singapore Mathematical Olympiads-2010 ✔ Singapore Mathematical Olympiads-2011 ✔ Singapore Mathematical Olympiads-2012 ✔ Singapore Mathematical Olympiads-2013 ✔ 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 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$$Ah Math Key Fact 5DB3 \small{\textbf{Fibonacci Numbers}}:\\[0pt]\text{Sequence defined recursively by } F_1=F_2=1 \text{ and } F_{n+2}=F_{n+1}+F_n \\ \text{for all } n \in N. Ah Math Key Fact 47EB A finite series is given by all the terms of a finite sequence, added together. Ah Math Key Fact 13D3 An infinite series is given by all the terms of an infinite sequence, added together. Ah Math Key Fact EAA4 The nth partial sum of a series is the sum of the first n terms. Ah Math Key Fact 3332 The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series. Ah Math Key Fact 2B69 The sequence of partial sums of a series sometimes tends to a real limit. If this does not happen, we say that the series has no sum. Ah Math Key Fact C62B A series can have a sum only if the individual terms tend to zero. But there are some series (with individual terms tending to zero) that do not have sums. Ah Math Key Fact 1529 The sum of an infinite series of the form$$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

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