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# 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.
• 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

There are 52 problems and 46 key facts, waiting for math enthusiasts!

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Dear Math Enthusiasts,
1) Up now on, you will see some direct connections between challenging math problems and key facts: Tips! We'll try to add more and more tips day by day to help you to solve the problems. Please do not hesitate to send us an email that gives new connection ideas between the problems and the key facts.
2) We put the badge form inside an accordion menu. This will allow you to see the answer form directly.
3) When you answer any problem correctly, you'll get a message which gives the code of the problem. This way you'll be able to prove which problem you correctly solved when you promote the screenshot!
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Hope you will find these updates useful ☺
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1. Problem: 595A , proposed by Ahmet Arduc, Tip: Key Fact(s): 969B
$$\small{A=\frac{1}{1\times2\times3}+\frac{2}{3\times4\times5}+\frac{3}{5\times6\times7}+\cdots+\frac{2016}{4031\times4032\times4033}}$$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

2. Problem: B28E , proposed by Ahmet Arduc
Find the smallest natural number $n$ such that the last five digits of the product of $\\$ n$$\times935\times972\times975 are all zero. Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Joseph Rodelas 3. Yavuz Selim Koseoglu 4. John Gamal Aziz Attia 5. Mahmut Cemrek 6. Βαρελάς Γεώρ𝛾ιος 7. Isaiah James de Dios Maling 8. Adrian Pilotos Burgos 9. Joselito Torculas 10. Nheil Ignacio 11. Jacob Sabido 12. Gerald M. Pascua 13. Russel J. Galanido 14. John Albert A. Reyes 15. Roenz Joshlee Timbol 16. Marvin Cato 17. Rindell Mabunga 18. Chris Norman Algo 19. Melga Sonio 20. Mark Elis Espiridion 21. Angelu G. Leynes 22. Caed Mark Medul Mendoza 23. Sumet Ketsri 24. Richard Phillip Dimaala Fernandez 25. Daniel James Molina 26. Nixon Balandra 27. Poetri Sonya Tarabunga 28. Lilanie Monique Torilla 29. Christian Daang 30. Emmanuel David 31. Christian Paul Patawaran 32. Jake Gacuan 33. Ibrahim Demir 34. Amirul Faiz Abdul Muthalib 35. Norwyn Nicholson Kah 36. Joem Canciller 37. Ralph Macarasig 38. Jhepoy Dizon 39. Lim Jing Ren 40. John Marco Latagan 41. Dan Lang 42. Lenard Guillermo 43. Hanelet Santos 44. Dreimuru Tempest 45. Dreimuru Tempest 46. Srinivas Kanigiri 47. Kurara Chibana 48. Alea Astrea 49. Kumar Ayush 50. Reymark Togno 51. Gluttony 3. Problem: A275 , proposed by Ahmet Arduc, Tip: Key Fact(s): 969B$$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 4. Problem: 2BE2 , proposed by Ahmet Arduc 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 5. Problem: 218D , proposed by Ahmet Arduc 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 6. Problem: A13E , proposed by Ahmet Arduc 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{ ?}}$$Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Βαρελάς Γεώρ𝛾ιος 3. Isaiah James de Dios Maling 4. Joselito Torculas 5. Marvin Cato 6. Russel J. Galanido 7. Melga Sonio 8. Nheil Ignacio 9. Sumet Ketsri 10. Lilanie Monique Torilla 11. Nixon Balandra 12. Caed Mark Medul Mendoza 13. Poetri Sonya Tarabunga 14. Rindell Mabunga 15. Richard Phillip Dimaala Fernandez 16. Christian Paul Patawaran 17. Amirul Faiz Abdul Muthalib 18. Daniel James Molina 19. Norwyn Nicholson Kah 20. Jhepoy Dizon 21. Lenard Guillermo 22. Kurara Chibana 23. Afshiram Muhammed 7. Problem: 52D2 , proposed by Ahmet Arduc Find the value of$$\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}}$$Ah Math • Correct answers have been submitted by: 1. Melek Cimen 2. Joseph Rodelas 3. Yavuz Selim Koseoglu 4. Mahmut Cemrek 5. Jeffrey Robles 6. Βαρελάς Γεώρ𝛾ιος 7. Isaiah James de Dios Maling 8. John Albert A. Reyes 9. Nheil Ignacio 10. Adrian Pilotos Burgos 11. Joselito Torculas 12. Rindell Mabunga 13. Roenz Joshlee Timbol 14. Marvin Cato 15. Russel J. Galanido 16. Angelu G. Leynes 17. Melga Sonio 18. Caed Mark Medul Mendoza 19. Sumet Ketsri 20. Daniel James Molina 21. Nixon Balandra 22. Lilanie Monique Torilla 23. Mark Elis Espiridion 24. Christian Daang 25. Emmanuel David 26. Jacob Sabido 27. Poetri Sonya Tarabunga 28. Richard Phillip Dimaala Fernandez 29. Grant Lewis Bulaong 30. Ibrahim Demir 31. Amirul Faiz Abdul Muthalib 32. Norwyn Nicholson Kah 33. Christian Paul Patawaran 34. Joem Canciller 35. Ralph Macarasig 36. Jhepoy Dizon 37. Lim Jing Ren 38. Lenard Guillermo 39. Mark Alvero 40. Dreimuru Tempest 41. Kurara Chibana 42. James Ericson 43. Alea Astrea 44. Reymark Togno 45. Gluttony 8. Problem: 79B2 , proposed by Ahmet Arduc If n is a positive even multiple of 5 and$$\small{8^2+10^2+12^2+18^2+20^2+22^2+\cdots+(n-2)^2+n^2+(n+2)^2}$$is divisible by 9, what is the minumum value of n ? Ah Math • Correct answers have been submitted by: 1. Yavuz Selim Koseoglu 2. Mahmut Cemrek 3. Jeffrey Robles 4. Nheil Ignacio 5. Isaiah James de Dios Maling 6. Joselito Torculas 7. Rindell Mabunga 8. Richard Phillip Dimaala Fernandez 9. Melga Sonio 10. Angelu G. Leynes 11. Adrian Pilotos Burgos 12. Russel J. Galanido 13. Caed Mark Medul Mendoza 14. Marvin Cato 15. Sumet Ketsri 16. Nixon Balandra 17. Lilanie Monique Torilla 18. Mark Elis Espiridion 19. Jacob Sabido 20. Christian Daang 21. Emmanuel David 22. Poetri Sonya Tarabunga 23. Roenz Joshlee Timbol 24. Ibrahim Demir 25. Christian Paul Patawaran 26. Amirul Faiz Abdul Muthalib 27. Norwyn Nicholson Kah 28. Chris Norman Algo 29. Daniel James Molina 30. Joem Canciller 31. Jhepoy Dizon 32. Ralph Macarasig 33. John Marco Latagan 34. Dreimuru Tempest 35. Kurara Chibana 36. Alea Astrea 37. Kumar Ayush 38. Lim Jing Ren 39. Lenard Guillermo 9. Problem: 11C8 , proposed by Ahmet Arduc, Tip: Key Fact(s): 2169 If a,b,c>0 and a+b+c=1, what is the smallest value of \frac{1}{a}+\frac{9}{b}+\frac{16}{c}\text{ ?} Ah Math • Correct answers have been submitted by: 1. Jeffrey Robles 2. Edge Ramos 3. Joseph Rodelas 4. Isaiah James de Dios Maling 5. Joselito Torculas 6. Rindell Mabunga 7. Roenz Joshlee Timbol 8. Marvin Cato 9. Nheil Ignacio 10. Melga Sonio 11. Russel J. Galanido 12. Sumet Ketsri 13. Caed Mark Medul Mendoza 14. Daniel James Molina 15. Nixon Balandra 16. Lilanie Monique Torilla 17. Poetri Sonya Tarabunga 18. Grant Lewis Bulaong 19. Richard Phillip Dimaala Fernandez 20. Christian Paul Patawaran 21. Ibrahim Demir 22. Amirul Faiz Abdul Muthalib 23. Norwyn Nicholson Kah 24. Ralph Macarasig 25. Lenard Guillermo 26. Kurara Chibana 27. Kumar Ayush 28. James Ericson 10. Problem: AC82 , proposed by Ahmet Arduc Find the value of the following expression.$$1\cdot2+3\cdot4+5\cdot6+\cdots+2017\cdot2018$$Ah Math • Correct answers have been submitted by: 1. Joseph Rodelas 2. Jeffrey Robles 3. Edge Ramos 4. Isaiah James de Dios Maling 5. John Albert A. Reyes 6. Joselito Torculas 7. Nheil Ignacio 8. Rindell Mabunga 9. Marvin Cato 10. Russel J. Galanido 11. Chris Norman Algo 12. Angelu G. Leynes 13. Melga Sonio 14. Sumet Ketsri 15. Nixon Balandra 16. Caed Mark Medul Mendoza 17. Lilanie Monique Torilla 18. Richard Phillip Dimaala Fernandez 19. Jacob Sabido 20. Mark Elis Espiridion 21. Emmanuel David 22. Poetri Sonya Tarabunga 23. Christian Paul Patawaran 24. Ibrahim Demir 25. Amirul Faiz Abdul Muthalib 26. Daniel James Molina 27. Joem Canciller 28. Norwyn Nicholson Kah 29. Jhepoy Dizon 30. Fred Gutierrez 31. Dreimuru Tempest 32. Roenz Joshlee Timbol 33. Kurara Chibana 34. Lim Jing Ren 35. Kumar Ayush 36. Lenard Guillermo 11. Problem: C1B5 , proposed by Ahmet Arduc What is the sum of the expression 2+3+5+8+13+\cdots+17711\text{ ?} Ah Math • Correct answers have been submitted by: 1. John Gamal Aziz Attia 2. Isaiah James de Dios Maling 3. Chris Norman Algo 4. Jeffrey Robles 5. Joselito Torculas 6. Melek Cimen 7. Rindell Mabunga 8. Angelu G. Leynes 9. Melga Sonio 10. Russel J. Galanido 11. John Albert A. Reyes 12. Nheil Ignacio 13. Adrian Pilotos Burgos 14. Caed Mark Medul Mendoza 15. Marvin Cato 16. Sumet Ketsri 17. Daniel James Molina 18. Lilanie Monique Torilla 19. Nixon Balandra 20. Jacob Sabido 21. Mark Elis Espiridion 22. Richard Phillip Dimaala Fernandez 23. Christian Paul Patawaran 24. Roenz Joshlee Timbol 25. Ibrahim Demir 26. Amirul Faiz Abdul Muthalib 27. Norwyn Nicholson Kah 28. Joem Canciller 29. Ralph Macarasig 30. Jhepoy Dizon 31. John Marco Latagan 32. Kurara Chibana 33. Lim Jing Ren 34. Kumar Ayush 35. Gluttony 36. Alea Astrea 12. Problem: 24B8 , proposed by Ahmet Arduc In the following sequence, find the units digit of the 2017th term$$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 13. Problem: 6392 , proposed by Ahmet Arduc 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 14. Problem: 615D , proposed by Ahmet Arduc Evaluate the following expression: 1\times2\times3+2\times3\times4+3\times4\times5+...+31\times32\times33 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 15. Problem: A836 , proposed by Ahmet Arduc 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 16. Problem: 7B83 , proposed by Ahmet Arduc 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$$Ah Math • Correct answers have been submitted by: 1. Richard Phillip Dimaala Fernandez 2. Joselito Torculas 3. Nixon Balandra 4. Jeffrey Robles 5. Jacob Sabido 6. Marvin Cato 7. Nheil Ignacio 8. Russel J. Galanido 9. Caed Mark Medul Mendoza 10. Lilanie Monique Torilla 11. Rindell Mabunga 12. Daniel James Molina 13. Roenz Joshlee Timbol 14. Christian Daang 15. Isaiah James de Dios Maling 16. Christian Paul Patawaran 17. Kimi No Nawa 18. Ibrahim Demir 19. Amirul Faiz Abdul Muthalib 20. Norwyn Nicholson Kah 21. Joem Canciller 22. John Lester Tan 23. John Rocel Perez 24. Ralph Macarasig 25. Jhepoy Dizon 26. Lim Jing Ren 27. Sumet Ketsri 28. Sigmund Dela Cruz 29. Srinivas Kanigiri 30. Kurara Chibana 31. James Ericson 32. Alea Astrea 33. Kumar Ayush 17. Problem: E328 , proposed by Ahmet Arduc In the matrix given below, if there are x numbers above 2017 and y numbers to the left of 2017, what is the sum of x and y ?$$\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}$$Ah Math • Correct answers have been submitted by: 1. Kimi No Nawa 2. Caed Mark Medul Mendoza 3. Lilanie Monique Torilla 4. Nixon Balandra 5. Joselito Torculas 6. Marvin Cato 7. Rindell Mabunga 8. Richard Phillip Dimaala Fernandez 9. Russel J. Galanido 10. Jacob Sabido 11. Isaiah James de Dios Maling 12. Christian Daang 13. Chris Norman Algo 14. Amirul Faiz Abdul Muthalib 15. Daniel James Molina 16. Christian Paul Patawaran 17. Joem Canciller 18. Norwyn Nicholson Kah 19. Jeffrey Robles 20. Jhepoy Dizon 21. Lim Jing Ren 22. Hanelet Santos 23. Sumet Ketsri 24. Roenz Joshlee Timbol 25. Kurara Chibana 26. James Ericson 18. Problem: B613 , proposed by Ahmet Arduc Find the remainder when 1^{1971}+2^{1971}+3^{1971}+...+2016^{1971} is divided by 2017 ? Ah Math • Correct answers have been submitted by: 1. Marvin Cato 2. Joselito Torculas 3. Rindell Mabunga 4. Jacob Sabido 5. Russel J. Galanido 6. Richard Phillip Dimaala Fernandez 7. Nixon Balandra 8. Roenz Joshlee Timbol 9. Isaiah James de Dios Maling 10. Grant Lewis Bulaong 11. Kimi No Nawa 12. Lilanie Monique Torilla 13. Caed Mark Medul Mendoza 14. Christian Paul Patawaran 15. Jeffrey Robles 16. Ibrahim Demir 17. Christian Daang 18. Amirul Faiz Abdul Muthalib 19. Norwyn Nicholson Kah 20. Chris Norman Algo 21. Daniel James Molina 22. John Lester Tan 23. Ralph Macarasig 24. Jhepoy Dizon 25. Sumet Ketsri 26. Nheil Ignacio 27. Sigmund Dela Cruz 28. Srinivas Kanigiri 29. Kurara Chibana 30. Alea Astrea 31. Lim Jing Ren 32. Kumar Ayush 19. Problem: EE2C , proposed by Ahmet Arduc By using only letters of English alphabet, label one marble 'A', two marbles 'B', three marbles 'C',..., twenty-six marbles 'Z'. Put these$$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 20. Problem: E4C6 , proposed by Ahmet Arduc, Tip: Key Fact(s): A86B 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 21. Problem: ABBA , proposed by Ahmet Arduc 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 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 23. Problem: D36C , proposed by Ahmet Arduc Let x be a real number and$$A=\sqrt{x^2-26x+170}+\sqrt{x^2-52x+1700}.$$What is the square of the minimum value of A ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Daniel James Molina 4. Richard Phillip Dimaala Fernandez 5. Rindell Mabunga 6. Norwyn Nicholson Kah 7. John Lester Tan 8. Joselito Torculas 9. Jeffrey Robles 10. Ralph Macarasig 11. Jhepoy Dizon 12. Kimi No Nawa 13. Caed Mark Medul Mendoza 14. Lilanie Monique Torilla 15. Nixon Balandra 16. Russel J. Galanido 17. Jacob Sabido 18. Marvin Cato 19. Sumet Ketsri 20. Sigmund Dela Cruz 21. Christian Daang 22. Kurara Chibana 23. Joem Canciller 24. Gluttony 24. Problem: D7C4 , proposed by Ahmet Arduc How many x values are there which makes \sqrt{2017-\sqrt{x}} an integer? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Norwyn Nicholson Kah 3. Kimi No Nawa 4. Isaiah James de Dios Maling 5. Joselito Torculas 6. Daniel James Molina 7. Jeffrey Robles 8. Richard Phillip Dimaala Fernandez 9. Ralph Macarasig 10. Rindell Mabunga 11. Caed Mark Medul Mendoza 12. Lilanie Monique Torilla 13. Jhepoy Dizon 14. Nixon Balandra 15. Russel J. Galanido 16. Lim Jing Ren 17. Sumet Ketsri 18. Marvin Cato 19. Mark Alvero 20. John Albert A. Reyes 21. Roenz Joshlee Timbol 22. Smahi Abdeslem 23. Srinivas Kanigiri 24. Kurara Chibana 25. James Ericson 26. Afshiram Muhammed 27. Alea Astrea 28. Jacob Sabido 29. Kumar Ayush 30. Lenard Guillermo 25. Problem: A4C5 , proposed by Ahmet Arduc, Tip: Key Fact(s): 969B$$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)$$If A is written in its simplest form as \frac{a}{b}, what is the sum of a and b ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Joselito Torculas 3. Jeffrey Robles 4. Norwyn Nicholson Kah 5. Richard Phillip Dimaala Fernandez 6. Sumet Ketsri 7. Ralph Macarasig 8. Kimi No Nawa 9. Rindell Mabunga 10. Caed Mark Medul Mendoza 11. Lilanie Monique Torilla 12. Nixon Balandra 13. Isaiah James de Dios Maling 14. Russel J. Galanido 15. Jhepoy Dizon 16. Lim Jing Ren 17. Daniel James Molina 18. Jacob Sabido 19. Marvin Cato 20. Fred Gutierrez 21. Nheil Ignacio 22. John Albert A. Reyes 23. Srinivas Kanigiri 24. Chris Norman Algo 25. Kurara Chibana 26. Kumar Ayush 27. Joem Canciller 28. Gluttony 29. Reymark Togno 26. Problem: A378 , proposed by Ahmet Arduc Find the sum of all real numbers x for which$$1953^x+1954^x+...+1984^x=1985^x+1986^x+...+2015^x.$$Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Joselito Torculas 4. Jhepoy Dizon 5. Jeffrey Robles 6. Ralph Macarasig 7. Norwyn Nicholson Kah 8. Daniel James Molina 9. Richard Phillip Dimaala Fernandez 10. Russel J. Galanido 11. Kimi No Nawa 12. Caed Mark Medul Mendoza 13. Lilanie Monique Torilla 14. Marvin Cato 15. Nixon Balandra 16. Rindell Mabunga 17. Kurara Chibana 18. Sumet Ketsri 19. Kumar Ayush 20. Lenard Guillermo 27. Problem: 3DC5 , proposed by Ahmet Arduc Find the coefficient of x^2 when$$\left(1+x\right)\left(1+2x\right)\left(1+4x\right)\cdots\left(1+2^9x\right)$$is expanded ? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Isaiah James de Dios Maling 3. Richard Phillip Dimaala Fernandez 4. Joselito Torculas 5. Russel J. Galanido 6. Sumet Ketsri 7. Caed Mark Medul Mendoza 8. Lilanie Monique Torilla 9. Daniel James Molina 10. Marvin Cato 11. Hanelet Santos 12. Jacob Sabido 13. Rindell Mabunga 14. Nixon Balandra 15. Jeffrey Robles 16. Sigmund Dela Cruz 17. Kurara Chibana 18. James Ericson 19. Alea Astrea 28. Problem: 9E31 , proposed by Ahmet Arduc, Tip: Key Fact(s): 969B For how many rational numbers between 0 and 1, written as a fraction in its lowest terms, the product of its numerator and denominator will be 30!. Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Russel J. Galanido 3. Marvin Cato 4. Daniel James Molina 5. Joselito Torculas 6. Isaiah James de Dios Maling 7. Richard Phillip Dimaala Fernandez 8. Kimi No Nawa 9. Caed Mark Medul Mendoza 10. Rindell Mabunga 11. Nixon Balandra 12. Sumet Ketsri 13. Roenz Joshlee Timbol 14. Smahi Abdeslem 15. Kurara Chibana 16. Lenard Guillermo 29. Problem: EEAA , proposed by Ahmet Arduc The increasing sequence 1, 3, 4, 9, 10, 12, 13,... consists of all positive integers which are powers of 3 or sums of distinct powers of 3. What is the 255th term of this sequence? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Russel J. Galanido 3. Sumet Ketsri 4. Jacob Sabido 5. Joselito Torculas 6. Marvin Cato 7. Richard Phillip Dimaala Fernandez 8. Isaiah James de Dios Maling 9. Daniel James Molina 10. Caed Mark Medul Mendoza 11. Nixon Balandra 12. Rindell Mabunga 13. Jeffrey Robles 14. Dreimuru Tempest 15. Roenz Joshlee Timbol 16. Kurara Chibana 17. Lim Jing Ren 18. Kumar Ayush 30. Problem: 442B , proposed by Ahmet Arduc What is the sum of the first 2017 terms of the given sequence?$$\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 31. Problem: B854 , proposed by Ahmet Arduc 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 32. Problem: 95E3 , proposed by Ahmet Arduc 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 33. Problem: 3E1C , proposed by Ahmet Arduc 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 34. Problem: 1715 , proposed by Ahmet Arduc Let O(n) denote the sum of the odd digits of n. For example, O(1872)=1+7=8. What is the sum of$$O(1)+O(2)+O(3)+...+O(2017)$$Ah Math • Correct answers have been submitted by: 1. Ikemen 2. Smahi Abdeslem 3. Russel J. Galanido 4. Sumet Ketsri 5. Nixon Balandra 6. Richard Phillip Dimaala Fernandez 7. Joselito Torculas 8. Marvin Cato 9. Isaiah James de Dios Maling 10. Amirul Faiz Abdul Muthalib 11. Kimi No Nawa 12. Caed Mark Medul Mendoza 13. Kurara Chibana 14. Rindell Mabunga 15. Roenz Joshlee Timbol 16. Jeffrey Robles 35. Problem: C877 , proposed by Ahmet Arduc The sequence u_1, u_2, u_3,... satisfies u_1=1, u_{2017}=2017, and, for all n\ge3, u_n is the average (arithmetic mean) of the first n-1 terms. What is the sum of the first three terms? Ah Math • Correct answers have been submitted by: 1. Dreimuru Tempest 2. Andrew Chiu 3. Marvin Cato 4. Ikemen 5. Richard Phillip Dimaala Fernandez 6. Isaiah James de Dios Maling 7. Joselito Torculas 8. Jacob Sabido 9. Russel J. Galanido 10. Nixon Balandra 11. Sumet Ketsri 12. Amirul Faiz Abdul Muthalib 13. Kurara Chibana 14. Rindell Mabunga 15. Kimi No Nawa 16. Caed Mark Medul Mendoza 17. Alea Astrea 18. Lim Jing Ren 19. Kumar Ayush 20. Jeffrey Robles 21. Joem Canciller 22. Gluttony 36. Problem: C2BC , proposed by Ahmet Arduc A piece of graph paper is folded once so that (2, 1) is matched with (0, 5). If (a, b) is matched with itself where a+b=2017, what is the value of a ? Ah Math • Correct answers have been submitted by: 1. Ikemen 2. Isaiah James de Dios Maling 3. Jacob Sabido 4. Caed Mark Medul Mendoza 5. Nixon Balandra 6. Amirul Faiz Abdul Muthalib 7. Richard Phillip Dimaala Fernandez 8. Marvin Cato 9. Andrew Chiu 10. Joselito Torculas 11. Russel J. Galanido 12. Sumet Ketsri 13. Kurara Chibana 14. Alea Astrea 15. Rindell Mabunga 16. Jeffrey Robles 17. Kumar Ayush 18. Joem Canciller 19. Gluttony 37. Problem: E4AE , proposed by Ahmet Arduc For the sets of consecutive integers$$\{1\}, \{2, 3\}, \{4, 5, 6\}, \{7, 8, 9, 10\},...$$let S_n be the sum of the elements in the nth set. What is the remainder when S_{2017} is divided by 9 ? Ah Math • Correct answers have been submitted by: 1. Ikemen 2. Richard Phillip Dimaala Fernandez 3. Kurara Chibana 4. Isaiah James de Dios Maling 5. Russel J. Galanido 6. Marvin Cato 7. Amirul Faiz Abdul Muthalib 8. Joselito Torculas 9. Smahi Abdeslem 10. Jacob Sabido 11. Alea Astrea 12. Nixon Balandra 13. Sumet Ketsri 14. Lim Jing Ren 15. Rindell Mabunga 16. Kimi No Nawa 17. Caed Mark Medul Mendoza 18. Jeffrey Robles 19. Lenard Guillermo 20. Joem Canciller 21. Gluttony 38. Problem: 57BD , proposed by Ahmet Arduc What is the remainder when the sum of the first 2017 terms of the sequence$$1,\hspace{0.2cm}1+2,\hspace{0.2cm}1+2+2^2,...,\hspace{0.2cm}1+2+2^2+...+2^{n-1},...$$is divided by 9 ? Ah Math • Correct answers have been submitted by: 1. Sumet Ketsri 2. James Ericson 3. Andrew Chiu 4. Joselito Torculas 5. Richard Phillip Dimaala Fernandez 6. Caed Mark Medul Mendoza 7. Isaiah James de Dios Maling 8. Nixon Balandra 9. Jeffrey Robles 10. Marvin Cato 11. Amirul Faiz Abdul Muthalib 12. Lim Jing Ren 13. Jacob Sabido 14. Rindell Mabunga 15. Alea Astrea 16. Kurara Chibana 17. Russel J. Galanido 18. Roenz Joshlee Timbol 39. Problem: 3785 , proposed by Ahmet Arduc 428 has the property that its square appears as two consecutive integers:$$428^2=183184$$What is the sum of all 3-digit numbers with this property? Ah Math • Correct answers have been submitted by: 1. Amirul Faiz Abdul Muthalib 2. Andrew Chiu 3. Nixon Balandra 4. Caed Mark Medul Mendoza 5. Marvin Cato 6. Joselito Torculas 7. Rindell Mabunga 8. Isaiah James de Dios Maling 9. Richard Phillip Dimaala Fernandez 10. Kurara Chibana 11. Russel J. Galanido 12. James Ericson 13. Alea Astrea 14. Jacob Sabido 15. Sumet Ketsri 16. Lim Jing Ren 17. Jeffrey Robles 40. Problem: 2783 , proposed by Ahmet Arduc What is the greatest number less than 2017, when divided by the sum of its digits gives the greatest remainder? Ah Math • Correct answers have been submitted by: 1. Nixon Balandra 2. Isaiah James de Dios Maling 3. Marvin Cato 4. Andrew Chiu 5. Amirul Faiz Abdul Muthalib 6. Joselito Torculas 7. Rindell Mabunga 8. Caed Mark Medul Mendoza 9. Lilanie Monique Torilla 10. Richard Phillip Dimaala Fernandez 11. Kurara Chibana 12. Russel J. Galanido 13. James Ericson 14. Sumet Ketsri 15. Alea Astrea 16. Jacob Sabido 17. Jeffrey Robles 18. Joem Canciller 19. Gluttony 41. Problem: 398B , proposed by Ahmet Arduc The numbers$$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 42. Problem: DE1D , proposed by Ahmet Arduc 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 43. Problem: B87D , proposed by Ahmet Arduc 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 44. Problem: C426 , proposed by Amirul Faiz Abdul Muthalib 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 45. Problem: 143B , proposed by Amirul Faiz Abdul Muthalib 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 46. Problem: 7964 , proposed by Amirul Faiz Abdul Muthalib, Tip: Key Fact(s): 969B What is the smallest positive integer n \neq 2017 such that the fraction$$\frac{n-2017}{20n+17}$$is NOT in its simplest form (reducible fraction)? Ah Math • Correct answers have been submitted by: 1. Nixon Balandra 2. Marvin Cato 3. Sumet Ketsri 4. Rindell Mabunga 5. Kurara Chibana 6. Russel J. Galanido 7. Lim Jing Ren 8. Caed Mark Medul Mendoza 9. Andrew Chiu 47. Problem: 76AC , proposed by Amirul Faiz Abdul Muthalib Four consecutive integers are greater than 2017. The smallest is divisible by 5, the second is divisible by 7, the third is divisible by 9, and the largest is divisible by 11. Find the sum of the smallest such consecutive integers. Ah Math • Correct answers have been submitted by: 1. Isaiah James de Dios Maling 2. Rindell Mabunga 3. Kurara Chibana 4. Russel J. Galanido 5. Lim Jing Ren 6. Caed Mark Medul Mendoza 7. Lilanie Monique Torilla 8. Marvin Cato 9. Sumet Ketsri 48. Problem: D256 , proposed by Ahmet Arduc ﻿The factorial base of numeration is defined as a_1+a_2⋅2!+a_3⋅3!+...+a_n⋅n! where a_1,a_2,...,a_n are integer coefficients such that 0≤a_k≤k. Thus, 1971 can be written in factorial base of numeration as$$1+1⋅2!+0⋅3!+2⋅4!+4⋅5!+2⋅6!$$where the sum of the coefficients is 10. What is the sum of all coefficients used in factorial base of numeration to write all numbers from 1 to 2017? Ah Math • Correct answers have been submitted by: 1. Kurara Chibana 2. Russel J. Galanido 3. Amirul Faiz Abdul Muthalib 4. Marvin Cato 5. Kimi No Nawa 6. Caed Mark Medul Mendoza 7. Rindell Mabunga 8. Lilanie Monique Torilla 9. Sumet Ketsri 49. Problem: 8D49 , proposed by Ahmet Arduc, Tip: Key Fact(s): 5DB3 Given$$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 50. Problem: 7E39 , proposed by Amirul Faiz Abdul Muthalib 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 51. Problem: 4E84 , proposed by Ahmet Arduc 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 ☆★☆★☆★ Newest Problem! ★☆★☆★☆ 52. Problem: EE37 , proposed by Amirul Faiz Abdul Muthalib 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 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. Order Score # of Correct Answers Names Country 1.096.19 51Sumet Ketsri Thailand 2.085.51 49Kurara Chibana Japan 3.084.35 50Marvin Cato Philippines 4.080.13 50Russel J. Galanido South Korea 5.078.25 46Isaiah James de Dios Maling Philippines 6.078.25 46Amirul Faiz Abdul Muthalib Malaysia 7.072.57 43Jeffrey Robles Philippines 8.072.00 47Nixon Balandra Philippines 9.069.23 36Lilanie Monique Torilla Philippines 10.068.68 50Caed Mark Medul Mendoza Philippines 11.061.03 44Richard Phillip Dimaala Fernandez Philippines 12.058.45 49Rindell Mabunga Philippines 13.054.79 35Jacob Sabido Philippines 14.052.99 41Joselito Torculas Philippines 15.050.86 36Lim Jing Ren Malaysia 16.050.00 26Alea Astrea Philippines 17.036.54 19Lenard Guillermo Philippines 18.034.97 20Gluttony Philippines 19.031.71 26Norwyn Nicholson Kah Philippines 20.031.63 25Jhepoy Dizon Philippines 21.029.84 32Daniel James Molina Philippines 22.028.49 20Ralph Macarasig Philippines 23.027.08 26Joem Canciller Philippines 24.026.92 14John Albert A. Reyes Philippines 25.025.26 17Nheil Ignacio Philippines 26.024.22 23Kumar Ayush India 27.023.16 17Ibrahim Demir Turkey 28.022.41 26Roenz Joshlee Timbol Philippines 29.022.16 22Christian Paul Patawaran Philippines 30.021.15 11Melga Sonio Philippines 31.021.15 11Angelu G. Leynes Philippines 32.020.31 13Poetri Sonya Tarabunga Indonesia 33.019.78 12Andrew Chiu Philippines 34.016.62 11Mark Elis Espiridion Philippines 35.016.41 16James Ericson Thailand 36.015.38 8Joseph Rodelas Philippines 37.014.54 11Chris Norman Algo Philippines 38.013.85 12Dreimuru Tempest Philippines 39.013.46 7Adrian Pilotos Burgos Philippines 40.011.78 7Ikemen Japan 41.011.72 16Christian Daang Philippines 42.011.54 6Yavuz Selim Koseoglu Turkey 43.011.54 6Mahmut Cemrek Turkey 44.011.54 6John Gamal Aziz Attia Egypt 45.009.42 7Srinivas Kanigiri India 46.009.42 7Emmanuel David Philippines 47.008.57 21Kimi No Nawa Japan 48.008.01 5Sigmund Dela Cruz Philippines 49.007.85 7Reymark Togno Philippines 50.007.69 6Mark Alvero Philippines 51.007.69 4Chayapol Thailand 52.007.69 4Βαρελάς Γεώρ𝛾ιος Greece 53.006.87 5John Lester Tan Philippines 54.006.01 5Hanelet Santos Philippines 55.006.01 5Grant Lewis Bulaong Philippines 56.005.77 3Gerald M. Pascua Philippines 57.005.34 5Jake Gacuan Philippines 58.004.95 6Fred Gutierrez Philippines 59.004.40 4Smahi Abdeslem Algeria 60.003.85 2Melek Cimen Turkey 61.003.85 2Edge Ramos Philippines 62.003.85 2Afshiram Muhammed Turkey 63.002.83 5John Marco Latagan Philippines 64.002.80 4Dan Lang Philippines 65.002.56 2Stefano Ongari Italy 66.001.92 1Suleyman Akarsu Turkey 67.001.92 1Serkan Callioglu Turkey 68.001.92 1Rosendo Parra Milian Peru 69.001.92 1Muhammed Aydogdu Turkey 70.001.92 1Mohamed Karamany Egypt 71.001.28 2John Rocel Perez Philippines 72.000.64 1John Patrick Philippines Key Fact A4A7 The product of a rational number and an irrational number is irrational. Ah Math Key Fact 969B For a fraction to be in lowest terms, its numerator and denominator must be relatively prime. Ah Math Key Fact 7B55 The square of any even integer is of form 4k. Ah Math Key Fact E856 The square of any odd integer is of form 4k+1. Ah Math Key Fact D7B8 The product of four consecutive natural numbers is never a perfect square. Ah Math Key Fact 7C45$$1001=7\cdot11\cdot13$$Ah Math Key Fact ECD9$$n\cdot n!=(n+1)!-n!$$Ah Math Key Fact 47EB$$\frac{1}{n\cdot(n+1)}=\frac{1}{n}-\frac{1}{n+1}$$Ah Math Key Fact 1529$$\frac{n}{(n+1)!}=\frac{1}{n!}-\frac{1}{(n+1)!}$$Ah Math Key Fact 72A4$$n^4+n^2+1=(n^2+1-n)\cdot(n^2+1+n)$$Ah Math Key Fact C4CC$$(n+1)^2-(n+1)+1=n^2+n+1$$Ah Math Key Fact B4C3 For a=k\cdot (2k+1) where k \in N,$$a^2+(a+1)^2+⋯+(a+k)^2\\=(a+k+1)^2+(a+k+2)^2+⋯+(a+2k)^2$$Ah Math Key Fact B65C$$\sum_{k=1}^n k \cdot (k+1) \cdot (k+2) = \frac{n\cdot(n+1)\cdot(n+2)\cdot(n+3)}{4}$$Ah Math Key Fact 7B38 \textbf{Sophie-Germain Identity}:$$\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}$$Ah Math Key Fact 7695 The square of any odd integer leaves remainder 1 upon division by 8. Ah Math Key Fact 965A When n is a positive even integer,$$n⋅(n+4)⋅(n+8)⋅(n+12)$$is divisible by 8. Ah Math Key Fact DDC5 n^3-n is always divisible by 6, where n \in Z. Ah Math Key Fact A86B The product of any n consecutive integer is always divisible by n! Ah Math Key Fact D477 If n is a positive integer,$$(n+1)\cdot(n+2)⋅…⋅(2n)$$is divisible by 2^n. Ah Math Key Fact ADBA The binomial expansion of (x+y)^n has n+1 terms. Ah Math Key Fact EADA The binomial expansion of$$(a_1+a_2+⋯+a_{r-1}+a_r )^n$$has \large {n+r-1 \choose r-1} terms. Ah Math Key Fact BAB3 The sum of all the elements of the subsets of the set {a_1,a_2,a_3,…,a_n } is$$2^{n-1}·(a_1+a_2+a_3+⋯+a_n )$$Ah Math Key Fact 2EBB There are$$\frac{n\cdot(n+1)\cdot(2n+1)}{6}$$squares on an n\times n chessboard. Ah Math Key Fact 13D3 The \textbf{Arithmetic Mean} (\textbf{AM}) of positive real numbers x_1, x_2, ...,x_n is defined as$$\frac{x_1 + x_2 + ... + x_n}{n}$$Ah Math Key Fact 9A34 The \textbf{Geometric Mean} (\textbf{GM}) of positive real numbers x_1, x_2, ...,x_n is defined as$$\root n \of {x_1 \cdot x_2 \cdot ...\cdot x_n}$$Ah Math Key Fact 89DD For any set of positive real numbers x_1,\ldots,x_n, the arithmetic mean is greater than or equal to the geometric mean. That is,$$\frac{x_1 + x_2 + ... + x_n}{n}\ge\root n \of {x_1 \cdot x_2 \cdot ...\cdot x_n}$$Ah Math Key Fact ABAD The \textbf{Harmonic Mean} (\textbf{HM}) of positive real numbers x_1, x_2, ...,x_n is defined to be$$\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$$Ah Math Key Fact 2169 For any set of positive real numbers x_1,\ldots,x_n, the arithmetic mean is greater than or equal to the harmonic mean. That is,$$\frac{x_1 + x_2 + ... + x_n}{n}\ge\frac{n}{\frac{1}{x_1}+\frac{1}{x_2}+...+\frac{1}{x_n}}$$Ah Math Key Fact 988C For positive real numbers a, b, \text{and } c,$$\frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge3$$. Ah Math Key Fact 3332 \textbf{Cauchy–Schwarz Inequality}:\\[0pt] \text{For all sequences of real numbers a_i and b_i, we have}$$\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$$Ah Math Key Fact 8C41 \textbf{ Chebyshev Inequality}:\\[5pt]\text{ Let }x_1, x_2,..., x_n \text{ and } y_1, y_2,..., y_n \text{ be two sequences of real }\\ \text{ numbers, such that } x_1 \le x_2 \le \cdots \le x_n \text{ and } y_1 \le y_2 \le \cdots \le y_n. \\ \text{ Then, }$$\frac{(x_1+x_2+...+x_n)(y_1+y_2+...+y_n)}{n} \le x_1y_1+x_2y_2+...+x_ny_n$$Ah Math Key Fact B13A Let y=k be any line which intersects y=ax^2+bx+c at two points, say P and Q, where a, b, c, k \in R and a \neq 0. Then the abscissa of the midpoint of the line segment PQ is the abscissa of the vertex of the parabola. Ah Math Key Fact 8B56 \textbf{De Moivre's Formula}: For any angle \alpha and for any integer n,$$(\text{cos } \alpha + i\cdot \text{sin } \alpha)^n=\text{cos } n\alpha + i\cdot \text{sin } n\alpha$$Ah Math Key Fact EAA4 \textbf{Fibonacci Numbers}:\\[0pt]\text{Sequence defined recursively by } F_1=F_2=1 \text{ and } F_{n+2}=F_{n+1}+F_n \\ \text{for all } n \in N. Ah Math Key Fact 2B69 A finite series is given by all the terms of a finite sequence, added together. Ah Math Key Fact 8A33 An infinite series is given by all the terms of an infinite sequence, added together. Ah Math Key Fact E6E7 The n-th partial sum of a series is the sum of the first n terms. Ah Math Key Fact C62B The sequence of partial sums of a series sometimes tends to a real limit. If this happens, we say that this limit is the sum of the series. Ah Math Key Fact 7446 The sequence of partial sums of a series sometimes tends to a real limit. If this does not happen, we say that the series has no sum. Ah Math Key Fact 8632 A series can have a sum only if the individual terms tend to zero. But there are some series (with individual terms tending to zero) that do not have sums. Ah Math Key Fact 5DB3 The sum of an infinite series of the form$$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 8146
Relative error = error / measurement Ah Math

Key Fact E1B9
The bisector of an angle of the triangle divides the opposite side into segments proportional to the other two sides. Ah Math

Key Fact 29D9
The sum of the exterior angles of any (convex) polygon is a constant, namely two straight angles (360$^{\circ}$). Ah Math

Key Fact BCC8
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 A376
Two equal circles in a plane cannot have only one common tangent. 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

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.

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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