# Australian Maths Competition

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• Aug 5th 2010, 10:31 PM
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

A. 24
• Aug 5th 2010, 11:29 PM
Prove It
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.
• Aug 5th 2010, 11:59 PM
undefined
Quote:

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.
• Aug 6th 2010, 06:32 AM
chessweicong
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?

That is question 27
• Aug 6th 2010, 06:38 AM
chessweicong
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?
• Aug 6th 2010, 11:10 AM
earboth
Quote:

Originally Posted by chessweicong
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?

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)
• Aug 6th 2010, 11:38 AM
Quote:

Originally Posted by chessweicong
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?

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.
• Aug 6th 2010, 09:30 PM
How do you do Q28?

These AMC questions are insanely hard :/
• Aug 6th 2010, 10:13 PM
Prove It
Quote:

Originally Posted by chessweicong
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...

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?
• Aug 6th 2010, 10:45 PM
I think it's 420?
• Aug 6th 2010, 11:02 PM
Prove It
• Aug 6th 2010, 11:11 PM
Oh crap! I forgot about that =/

I guess it will be...

10**-***=2**
• Aug 6th 2010, 11:27 PM
I'm stuck o.o
• Aug 7th 2010, 12:45 AM
eumyang
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.
• Aug 7th 2010, 03:57 AM
Quote:

Originally Posted by eumyang
Hint: the 4 digit is divisible by 9.

What?? :O

I don't get what you mean by that.

EDIT: