Australian Mathematics Competition! (1 Viewer)

[AMCtryhard]

hey all BOS members in this forum, it is this time of the year when we and I have to sit for the Australian Maths Comp. Getting participations everytime I have entered has made me feel soooooo bad and I have lost any motivation in my maths study. I have just got a practice Int. Yr 9-10 paper from my teacher as I am sitting the test tomorrow and was wondering if you could assist me in some of these really hard problems. By the way, I honestly think Westpac is utterly mean for dishing out these maths comps which are supposed to be like a year or 2 years above what level of maths you are doing..... (7 questions here, sorry so bad guys, there's like 30 but I;ve only done like 15-20 )

If you could solve any 1 or more of them with some sort of solution (answers are excellent since your doing the work), I'd be super grateful. Besides, I;m trying to scrape a Credit this time.

1. Each of Andrew, Bill, Clair, Daniel and Eva either always lies or is always truthful, and they know which each of them is.

Andrew says that Bill is a liar.
Bill says that Clair is a liar.
Clair says that Daniel is a liar.
Daniel says that Eva is a liar.

The largest possible number of liars among them can be
(A) 1
(b) 2
(C) 3
(D) 4
(E) 5

2. On her birthday in 2007, Rachel's age is equal to twice the sum of the digits of the year in which she was born. How many possible years are there in which she could have been born?
(A) 1
(B) 2
(C) 3
(D) 4
(E) 5
3. There arefour lifts in a building. Each makes three stops, which do not have to be on consecutive floors or include the ground floor. For any two floors, there is at least one lift which stops on both of them. What is the maximum amount of floors that this building can have?
(A) 4
(B) 5
(C) 6
(D) 7
(E) 12
4. A bee can fly or walk only in a straight line between any two corners on the inside of a cubic box of edge length 1. The bee managed to move so that it visited every corner of the box without passing through the smae point twice in the air or on the wall of the box. The largest possible length of such a path is:
(A) 2 + 5(root 2)
(B) 1 + 6(root 2)
(C) 7(root 2)
(D) (root 3) + 6(root 2)
(E) 4(root 3) + 3(root 2)

5. What is the smallest amount of odd numbers in the range 1...2006 such that, no matter how these numbers are chosen, there will always be two which add to 2008?

6. A lucky number is a positive integer which is 19 times the sum of its digits. How many different lucky numbers are there?

7. On my calculator screen the number 2659 can be read upside down as 6592. The digits that can be read upside down are 0,1,2,5,6,8,9 and are read as 0,1,2,5,9,8,6 respectively. Starting with 1, the fifth number that can be read upside down is 8 and the fifteenth is 21. What are the last three digits of the 2007th number that can be read upside down?



Thanks if you can, AMCtryhard!

 

[Poad]

Oh god, I hate these things so much, they're more tests of patterns/logic problem solving rather than pure maths. I'll be doing it tomorrow too. :\

 

[addikaye03]

i remember Q 4 from last year, grrrr

 

[buchanan]

Your Q1-7 are respectively Q20, 21, 24, 25, 26, 28, 30 in the 2007 Int. AMC.

Answers are:

20. C
21. D
24. B
25. D
26. 503
28. 11
30. 898

Here are the official solutions for these questions:

 

[hon1hon2hon3]

I can only come up with an answer for question 1 . dont even know did i used to right method .

but basically , there only two options turth or lies , and if u look close , it starts off with andrew . . . lets say andrew is "A" .

A says B tells lie.
B says C tells lie.
C says D tells lie.
D says E tells lie .

Let assume A tells lie , then B would be telling truth . and If B is telling the truth then C is lying . soo the order will be "lie, truth, lie , truth , lie"

now Assume A tells truth , soo B would be telling lies and C would be telling truth. soo now the order will be "turth , lie , turth , lie , turth "

And we need the biggest posible amount of liers , which is the first order we got , and is 3 people.




For the bee playing in a cube question , i only used logic to slove it . We all know diagonals are longer then walking along the side of the cube. and the longest possible path its root 3 . " eg, From the front bottom left corner to the back top right corner" ( If u know what i mean . . ) , and this longest path could be only taken once , because u need to play pass the centre of the cube . . and u cant do it more then once, becuase you cant play pass the same point more then once. . . Soo concluded up all that . diagonal is longer then the side , and only 1 root 3 can be taken. the answer is D.

This is the way how i do them , feel like sharing to others . Peace ~

 

[bored of sc]

These questions are ridiculously hard.

 

[lolokay]

well for the bee question you would have to be sure that you can do D if you were using logic alone. I just made the longest one I could think of straight away, which was greater than B, and since C and E can't be done I concluded D.

for questions numbered:
1. can't have 2 liars in a row (except for Daniel, Eva)
2. I just basically counted the ones that could be done (skipping the ones that obiouvsly couldn't be done) so you get 1959, 1965, 1971, 2001
3. there can only be 1 floor which does not have a lift stop twice at it. each lift can only stop twice at one floor, therefore the answer is number of lifts + 1 = 5
5. 2008 = 1003+1005. the number of odd numbers up to 1005 is 1006/2 = 503
6. I basically just went through all the multiples of 19 up to 19*21
7. you know the nth last digit repeats every 7^n numbers, lasting for 7^(n-1), you just see how many times 7^n goes into 2007 and figure out the digits from that

 

[AMCtryhard]

thank you so much buchanan for the solutions, I'm going to try and read and understand them. if I get a credit, I'll be sure to thank you all who have helped me but I wish I could do more +) Good night guys and I wish you the best of luck in your AMC.

 

[conics2008]

rubbish.

 

[dux&src]

Heys
i remember these questions from last year i think, some of them look familiar.

I don't even know how i get distinctions and high distinctions. Argh tomorrow AMC..

i don't feel confident.
But When i have to perform then yes i'll answer these questions. but atm i can't be bothered.

Good luck guys.

 

[Boxxxhead]

Haha, I miss these competitions.

The marks in these competitions are pretty freaking low, so it's not too hard to get Credit or higher. I got 63/120 in the Year 12 one, and got top 5% in NSW.

 

[Undermyskin]

You're kidding! Um, maybe it was more difficult 2 years ago...

Anyone knows what was the mark of those who got distinction last year?

 

[Boxxxhead]

No idea about last year.

In 2006, it was hard and the average results were very, very low.

 

[cheney31]

i got HD in this twice yay

it was like in year 9 and 10 though.