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Math Help - Australian Maths Competition

  1. #1
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    Australian Maths Competition

    Hey,

    I'm in Year 10, does anybody know the answers to the last 5 questions?

    All I know:
    The question that asks "how many numbers under 2010 have 3 factors" is 14.

    Also, it would be nice if you can post what the question was, only if you can though

    Cheers,

    EDIT:

    So far we have:

    Question 26. how many numbers under 2010 have 3 factors
    A. 14

    Question 27. Two 10*18*L blocks are placed on either side of a cylinder of length L to stop it from rolling. One block has 10*L face on the ground while the other block has an 18*L face on the ground. The block on the left sticks out 4 units more than the one on the right.
    What is the radius of the cylinder?
    A. 25

    Question 28. A 3-digit number is subtracted from a 4-digit number and the result is a 3-digit number.
    □□□□-□□□=□□□
    The 10digits are all different.
    What is the smallest possible result?
    A. 246

    Question 29. I have a list of thirty numbers where the first number is 1, the last number is 30 and each of the other numbers is on more than the average of its two neighbours. what is the largest number in the list?
    A. 226

    Question 30. There are so many towns on the island of Tetra, all connected by roads. Reach town has three roads leading to three other different towns: one red road, one yellow road and one blue road, where no two roads meet other than at towns. If you start from any town and travel along red and yellow roads alternately (RYRY...) you will get back to your starting town after having travelled over six different roads. In fact RYRYRY will always get you back to where you started. In the same way, going along yellow and blue roads alternately will always get you back to the starting point after travelling along six different roads (YBYBYB). On the other hand, going along red and blue roads alternately will always get you back to the starting point after travelling along four different roads (RBRB). How many towns are there on Tetra?
    A. 24
    Last edited by Username; August 7th 2010 at 09:57 PM.
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  2. #2
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    I'd be happy to attempt them (I teach Year 10 myself) but I don't have a copy of the questions. If you post them I'm happy to help you answer them.

    Oh and Mods, the Australian Mathematics Competition has already passed for this year, the OP is just asking for help with questions he/she did not understand when competing.
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  3. #3
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    Quote Originally Posted by Username View Post
    Also, it would be nice if you can post what the question was, only if you can though
    For tips on typesetting, you can see the LaTeX Help Subforum. For diagrams it might be easiest to use a scanner. LaTeX is a great tool to have available, if you don't already know how to use it.
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  4. #4
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    Question 27

    Two 10*18*L blocks are placed on either side of a cylinder of length L to stop it from rolling. One block has 10*L face on the ground while the other block has an 18*L face on the ground. The block on the left sticks out 4 units more than the one on the right.
    Australian Maths Competition-27.jpg
    What is the radius of the cylinder?

    That is question 27
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  5. #5
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    Question 28
    A 3-digit number is subtracted from a 4-digit number and the result is a 3-digit number.
    □□□□-□□□=□□□
    The 10digits are all different.
    What is the smallest possible result?
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  6. #6
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    Quote Originally Posted by chessweicong View Post
    Two 10*18*L blocks are placed on either side of a cylinder of length L to stop it from rolling. One block has 10*L face on the ground while the other block has an 18*L face on the ground. The block on the left sticks out 4 units more than the one on the right.
    Click image for larger version. 

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ID:	18456
    What is the radius of the cylinder?

    That is question 27
    1. Have a look at my sketch.

    2. According to the text you know:

    18+x-(10+y)=4

    3. Use Pythagorean theorem to calculate x or y, respectively:

    (r-10)^2+x^2=r^2 or (r-18)^2+y^2=r^2

    4. Solve the system of equations for (x, y, r):

    \left|\begin{array}{rcl}18+x-(10+y)&=&4 \\ (r-10)^2+x^2&=&r^2 \\ (r-18)^2+y^2&=&r^2\end{array}\right.

    5. I've got (x, y, r) = (20, 24, 25)
    Attached Thumbnails Attached Thumbnails Australian Maths Competition-zylinderbremse.png  
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  7. #7
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    Quote Originally Posted by chessweicong View Post
    Two 10*18*L blocks are placed on either side of a cylinder of length L to stop it from rolling. One block has 10*L face on the ground while the other block has an 18*L face on the ground. The block on the left sticks out 4 units more than the one on the right.
    Click image for larger version. 

Name:	27.jpg 
Views:	47 
Size:	3.7 KB 
ID:	18456
    What is the radius of the cylinder?

    That is question 27


    If one block sticks out by 4 more than the other,
    then the taller one would be exactly in the centre of the shorter one,
    if they could overlap,
    which we can do in a sketch.

    When we do this, we see that ... in going from the corner of one block to the corner of the other
    is a movement of 4 horizontally and 8 vertically.

    The centre of the circle lies on the bisector of that line segment.
    This point is 14 units above the floor.
    In moving along the bisector, we go 2 units horizontally and 1 unit vertically.

    We can keep moving along the bisector until our height above the floor
    equals the distance from the block edge.

    Then, using Pythagoras' theorem, we need to find the equal squares as follows..

    6^2+2^2 and 16^2

    7^2+4^2 and 17^2

    8^2+6^2 and 18^2

    9^2+8^2 and 19^2

    10^2+10^2 and 20^2

    11^2+12^2 and 21^2

    12^2+14^2 and 22^2

    13^2+16^2 and 23^2

    14^2+18^2 and 24^2

    15^2+20^2 and 25^2

    The final sum of squares equals the 3rd square, so r=25 units.
    This does assume the radius is a whole number.
    Attached Thumbnails Attached Thumbnails Australian Maths Competition-cylinder-centre.jpg  
    Last edited by Archie Meade; August 6th 2010 at 11:39 AM.
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  8. #8
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    How do you do Q28?

    These AMC questions are insanely hard :/
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  9. #9
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    Quote Originally Posted by chessweicong View Post
    Question 28
    A 3-digit number is subtracted from a 4-digit number and the result is a 3-digit number.
    □□□□-□□□=□□□
    The 10digits are all different.
    What is the smallest possible result?
    This question just takes some inspired guesswork...

    I assume that each number can't start with 0, so...

    First, a 4 digit number minus a 3 digit number gives you a 3 digit number - it should be pretty obvious that the 4 digit number starts with 1.

    If a 1 is gone, then the next smallest digit is 2, so I would have the answer start with 2...

    So far that is

    1\,\circ\,\circ\,\circ - \circ\,\circ\,\circ = 2\,\circ\,\circ...

    Then the next thing I would think about is "What is the smallest possible number I can make with the remaining digits for the answer?"

    Do you want to try it from here?
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  10. #10
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    I think it's 420?
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  11. #11
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    Does 420 start with a 2?
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  12. #12
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    Oh crap! I forgot about that =/

    I guess it will be...

    10**-***=2**
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  13. #13
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    I'm stuck o.o
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  14. #14
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    Another reason it can't be 420 is because that would make the 1's digit of the other two numbers be the same.

    I'll give another digit and a hint:
     1\,0\,\square\,\square\:-\:7\,\square\,\square\:=\:2\,\square\,\square

    Hint: the 4 digit is divisible by 9.
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  15. #15
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    Quote Originally Posted by eumyang View Post
    Hint: the 4 digit is divisible by 9.
    What?? :O

    I don't get what you mean by that.

    EDIT:
    Is the answer 1053-764=289
    ??
    Last edited by Username; August 7th 2010 at 03:15 AM.
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