Page 2 of 14 FirstFirst 123412 ... LastLast
Results 26 to 50 of 343

Thread: Australian Mathematics Competition

  1. #26
    ... shaon0's Avatar
    Join Date
    Mar 2008
    HSC
    2009
    Gender
    Male
    Location
    Guess
    Posts
    2,059
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by d3st1nyLiang View Post
    is it a)?

    A lucky number is a positive integer which is 19 times the sum of its digits. How
    many different lucky numbers are there?
    I don't know if my methods right:
    By using the conditions in the questions. It's easy to deduce that the number isn't a 1,2 or 4 digit number. So it's a 3 digit no.
    Let the digits be as follow: 100 digits in hundreds place, 10 digits in 10s place and 10 digits in 1's place.

    Alternatively, I think you can just go:
    Z*19=ABC where {Z,A,B,C E Z}
    So say for: a no. which has digits summing to 6 is:
    6*19=114

    Thus, Z is in between 5 and 16
    So 10 numbers.
    Last edited by shaon0; 2 Aug 2009 at 3:14 PM.

  2. #27
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by shaon0 View Post
    I don't know if my methods right:
    By using the conditions in the questions. It's easy to deduce that the number isn't a 1,2 or 4 digit number. So it's a 3 digit no.
    Let the digits be as follow: 100 digits in hundreds place, 10 digits in 10s place and 10 digits in 1's place.

    Alternatively, I think you can just go:
    Z*19=ABC where {Z,A,B,C E Z}
    So say for: a no. which has digits summing to 6 is:
    6*19=114

    Thus, Z is in between 5 and 16
    So 10 numbers.
    Close, the actual answer is 11 :|

    Here is the solution:

    As shaon showed, the lucky number must be a 3 digit number.

    Suppose the number is abc.

    Then , we have:



    ie



    Note that as b and c are the ten's digit and unit's digit respectively, then



    So the maximum is and therefore the maximum a is





    So we must only check for

    For a = 1, we have
    For a = 2, we have
    For a = 3, we have , and there are no other solutions.

    Hence there are exactly 11 lucky numbers, namely,


    We should post up easier questions first just to get the marathon going =)

  3. #28
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    PROBLEM 3

    Tina has a large number of 1 × 2 × 6 rectangular blocks. She wants
    to make a solid cube out of the blocks. What is the smallest number
    of blocks she needs?

    (A) 6 (B) 12 (C) 18 (D) 36 (E) 144

  4. #29
    hsc zomg
    Join Date
    Jan 2008
    HSC
    2010
    Gender
    Female
    Posts
    464
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    or is it (c)?
    HSC 2010



  5. #30
    ... shaon0's Avatar
    Join Date
    Mar 2008
    HSC
    2009
    Gender
    Male
    Location
    Guess
    Posts
    2,059
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by kurt.physics View Post
    Close, the actual answer is 11 :|

    Here is the solution:

    As shaon showed, the lucky number must be a 3 digit number.

    Suppose the number is abc.

    Then , we have:



    ie



    Note that as b and c are the ten's digit and unit's digit respectively, then



    So the maximum is and therefore the maximum a is





    So we must only check for

    For a = 1, we have
    For a = 2, we have
    For a = 3, we have , and there are no other solutions.

    Hence there are exactly 11 lucky numbers, namely,


    We should post up easier questions first just to get the marathon going =)
    Yeah it is 11. lol. I counted my solutions in between 5 and 16 incorrectly.

  6. #31
    New Member
    Join Date
    Aug 2009
    HSC
    N/A
    Gender
    Male
    Posts
    1
    Rep Power
    4

    Re: Australian Maths Competition

    Quote Originally Posted by kurt.physics View Post
    Here is 2008 q + a and 2006 answers.
    Hi Kurt,

    Thanks for these papers and answers for AMC. For some years there was just the answers and not teh solution methods as such. Can you send these also if you have, say for 2008, 2006,2005,2004 and 2003? I am taking part inthe JUnior level. Thanks.

  7. #32
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by nerdsforever View Post
    or is it (c)?
    Yes, the answer is (c)

    Here my solution:










  8. #33
    Retiring Aquawhite's Avatar
    Join Date
    Jul 2008
    HSC
    2010
    Uni Grad
    2013
    Gender
    Male
    Location
    Gold Coast
    Posts
    4,962
    Rep Power
    8

    Re: Australian Maths Competition

    I think I am meant to be in this >_>... I'm really not in the mood for doing this since the Trials are just around the corner and all I've focused on is the syllabus and not really extra curricular (interesting maths ...).

    Ima probably end up with just a participation or a credit if I'm lucky probably. Not that I care XD
    #overandout

  9. #34
    everything's fuzzy.. 9876543210's Avatar
    Join Date
    Nov 2008
    HSC
    2009
    Gender
    Female
    Location
    innisfree
    Posts
    54
    Rep Power
    5

    Re: Australian Maths Competition

    thx kurt!
    nic =]

  10. #35
    hsc zomg
    Join Date
    Jan 2008
    HSC
    2010
    Gender
    Female
    Posts
    464
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Cheers kurt! you're awesome. what do you usually get in the maths comp?

    Here's a new question.

    A sequence {u1, u2, . . . , un} of real numbers is defined by




    for n≥ 3.


    What is u2008?
    (A) −√2
    (B) 2008(√2 − 2008pi)
    (C) 1003√2 − 1004pi
    (D) pi
    (E)√2
    HSC 2010



  11. #36
    New Member
    Join Date
    Dec 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    16
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Is it E) ?

  12. #37
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by nerdsforever View Post
    Cheers kurt! you're awesome. what do you usually get in the maths comp?

    Here's a new question.

    A sequence {u1, u2, . . . , un} of real numbers is defined by





    for n≥ 3.


    What is u2008?
    (A) −√2
    (B) 2008(√2 − 2008pi)
    (C) 1003√2 − 1004pi
    (D) pi
    (E)√2


















    We see the pattern emerging s.t.







    So consider the pattern in modulo 6.

    The 2008th term will be equivalent a specific residue modulo 6 and hence we can determine which term the 2008th term is equivalent to




  13. #38
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Problem 5

    The sum of three numbers is 4, the sum of their squares is 10 and the sum of their
    cubes is 22. What is the sum of their fourth powers?

  14. #39
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Australian Maths Competition

    I re-found the website which has the papers =)

    AMC papers (this is in Chinese ;P)

    AMC papers (in English)

    (for the second link you have to click on the "Downloads" tab to the left)
    Last edited by kurt.physics; 4 Aug 2009 at 7:31 AM.


  15. #40
    New Member
    Join Date
    Aug 2009
    HSC
    2012
    Gender
    Female
    Posts
    5
    Rep Power
    4

    Australian Mathematics Competition

    Hi everyone~
    i was just wondering if anyone had the AMT solutions for year 2002 - 2006 (intermediate division)???

  16. #41
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Australian Mathematics Competition

    Quote Originally Posted by appl3licious View Post
    Hi everyone~
    i was just wondering if anyone had the AMT solutions for year 2002 - 2006 (intermediate division)???
    Here they are =)
    Last edited by kurt.physics; 4 Aug 2009 at 7:29 AM.

  17. #42
    Senior Member
    Join Date
    Jun 2007
    HSC
    N/A
    Gender
    Undisclosed
    Posts
    847
    Rep Power
    6

    Re: Australian Maths Competition

    I have posted the solutions here

    Solutions

  18. #43
    New Member
    Join Date
    Aug 2009
    HSC
    2012
    Gender
    Female
    Posts
    5
    Rep Power
    4

    Re: Australian Mathematics Competition

    Quote Originally Posted by kurt.physics View Post
    Here they are =)
    thank you for sharing~ u really saved me ^^

  19. #44
    Executive Member
    Join Date
    May 2008
    HSC
    2010
    Gender
    Male
    Posts
    1,470
    Rep Power
    5

    Re: Westpac Maths Comp marathon

    A train travelling at constant speed takes a quarter of a minute to pass a signpost
    and takes three-quarters of a minute to pass completely through a tunnel which is
    600m in length. The speed of the train, in kilometres per hour, is
    (A) 50 (B) 56 (C) 64 (D) 72 (E) 80

    I really dont get this question, even though it is extremely simple. Shouldnt it be 48 km/h?

    BTW no idea to the above.
    Subjects 2009:
    English Advanced
    Mathematics Extension 1
    Mathematics Extension 2
    Biology
    Chemistry
    Physics

    Swans Fan Forever.

    I can do everythign through Him

    who gives me strength

    Philippians 4:13

  20. #45
    Executive Member
    Join Date
    May 2008
    HSC
    2010
    Gender
    Male
    Posts
    1,470
    Rep Power
    5

    Re: Westpac Maths Comp marathon

    How many different pairs of 2-digit numbers multiply to give a 3-digit number with
    all digits the same?
    (A) 5 (B) 6 (C) 7 (D) 8 (E) 9
    Subjects 2009:
    English Advanced
    Mathematics Extension 1
    Mathematics Extension 2
    Biology
    Chemistry
    Physics

    Swans Fan Forever.

    I can do everythign through Him

    who gives me strength

    Philippians 4:13

  21. #46
    Bored of study-freak's Avatar
    Join Date
    Feb 2008
    HSC
    2009
    Gender
    Male
    Posts
    1,145
    Rep Power
    5

    Re: Westpac Maths Comp marathon

    For the train Q, (D) is the answer.
    Remember that the train takes 45 seconds to travel the distance of 600m+its own length.

  22. #47
    Bored of study-freak's Avatar
    Join Date
    Feb 2008
    HSC
    2009
    Gender
    Male
    Posts
    1,145
    Rep Power
    5

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by Lukybear View Post
    How many different pairs of 2-digit numbers multiply to give a 3-digit number with
    all digits the same?
    (A) 5 (B) 6 (C) 7 (D) 8 (E) 9
    111=37x3
    222=37x6
    333=37x9
    444=37x12
    555=37x15
    666=37x18
    777=37x21
    888=37x24=74x12
    999=37x27
    (C) 7.

  23. #48
    Bored of study-freak's Avatar
    Join Date
    Feb 2008
    HSC
    2009
    Gender
    Male
    Posts
    1,145
    Rep Power
    5

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by kurt.physics View Post
    Problem 5



    The sum of three numbers is 4, the sum of their squares is 10 and the sum of their

    cubes is 22. What is the sum of their fourth powers?

  24. #49
    The A-Team addikaye03's Avatar
    Join Date
    Nov 2006
    HSC
    2008
    Gender
    Male
    Location
    Albury-Wodonga, NSW
    Posts
    1,275
    Rep Power
    6

    Re: Westpac Maths Comp marathon

    Quote Originally Posted by kurt.physics View Post
    Problem 5

    The sum of three numbers is 4, the sum of their squares is 10 and the sum of their
    cubes is 22. What is the sum of their fourth powers?
    Consider polynomial: P(x)=x^4-4x^3+8x^2+(8/3)x+a0

    S1=a+b+c=4
    --> S1-4=0

    S2=a^2+b^2+c^2=10
    -->S2-4(S1)+16=0

    S3=a^3+b^3+c^3=22
    --> S3-4(S2)+8(S1)+8=0

    S4=a^4+b^4+c^4=x
    --> S4-4(S3)+8(S2)+(8/3)S1+4(y)=0

    S4=88-80-32/3-4a0

    S4=-(8/3+4a0) Where a0 is the constant of P(x)

    Thought this method would work but i would need to know a0 for it to work...fml

    The method your looking for though i'll do tomoz if not answered

    Update: Above was the method i was referring to
    Last edited by addikaye03; 5 Aug 2009 at 10:18 AM.
    2010 : B Engineering (Civil or Electrical) @ Uni of Melbourne

    Week 14: (3 Sets x 6 Reps)| Bench Press: 100kg (1 rep: 120kg)| Squats: 150kg| Deadlift: 170kg|

  25. #50
    Executive Member
    Join Date
    May 2008
    HSC
    2010
    Gender
    Male
    Posts
    1,470
    Rep Power
    5

    Re: Westpac Maths Comp marathon

    *study freak:

    what rule is applied when you say:
    x^2+y^2+z^2 = (x+y+z)^2 - 2(xy+yz+xz)
    Subjects 2009:
    English Advanced
    Mathematics Extension 1
    Mathematics Extension 2
    Biology
    Chemistry
    Physics

    Swans Fan Forever.

    I can do everythign through Him

    who gives me strength

    Philippians 4:13

Page 2 of 14 FirstFirst 123412 ... LastLast

Thread Information

Users Browsing this Thread

There are currently 1 users browsing this thread. (0 members and 1 guests)

Posting Permissions

  • You may not post new threads
  • You may not post replies
  • You may not post attachments
  • You may not edit your posts
  •