Answers to Senior Maths Challenge!!!

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Ralfskini
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#21
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#21
Where can we see the questions?
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Micky
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P.S. It's clear, though, that the answers to 15 and 25 are wrong, by the above.
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theone
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#23
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(Original post by Micky)
k, I've got a copy of the paper, here's the answers I get:

1 D
2 A
3 A
4 C
5 A
6 B
7 C
8 E
9 B
10 D
11 C
12 C
13 A
14 E
15 ?
16 A
17 C
18 E
19 E
20 B
21 B
22 ?
23 B
24 A
25 ? (probably D, though)

I'm afraid I haven't worked out 15, though it's not A or B (since obviously 1 and 2003squared are both factors, but I couldn't see a straightforward way to prove any of the others), and don't have a proof for 25, it's all very well citing 3 pairs as answers, but I wouldn't like to conjecture the answer was D without a formal proof (do you have one?).


Does anyone have a proof for 22, all I noticed was that D is the sum of all the numbers from 1 to 100, but couldn't see how to fit that to the question :?

I think everything that you've answered is right.

Here's the answers for the three questions (at least what I did):

15) Since 2003 is prime the only factors are 1, 2003, 2003^2, 2003^3.... 2003^2003. Of these, only 1002 can be epxressed in the form (2003^n)2 (i.e. n being between 0-1001).

22) Say 100 = a/2 + b + 2c.

C may take 51 values, from 0-50. After this, there are 100-2c+1 options for b (since b can be any value from 0 to 100-2c), then a/2 is fixed. Therefore we have an arithmetic series with 51 terms, first term 101 and common difference -2, which should add to 2601.

25) Rearranging we get 19y + 38x = 3xy, as such 57y + 114x = 9xy and factorising (3x-19)(3y-38) = 19.38. Since 19.38 = 19.19.2, and each of the two brackets is a +ive integer, it follows that 3x-19 must be a factor of 19.19.2, and is therefore 1,2,19,38, 19^2 or 19.19.2. Trying each possibility we get 3 possibilites for x and 3 corresponding values for y.

Somebody want to throw some rep my way for this? I only have 21 points
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Micky
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#24
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*applauds*

Nice work Just one question (probably has a simple answer which I'm missing, but anyway...)


(Original post by theone)

25) Rearranging we get 19y + 38x = 3xy, as such 57y + 114x = 9xy and factorising (3x-19)(3y-38) = 19.38. Since 19.38 = 19.19.2, and each of the two brackets is a +ive integer, it follows that 3x-19 must be a factor of 19.19.2, and is therefore 1,2,19,38, 19^2 or 19.19.2. Trying each possibility we get 3 possibilites for x and 3 corresponding values for y.
Why can't both brackets be negative? We know x and y are both positive integers, but that doesn't necessarily mean 3x-19<0
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Unregistered899
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Does Anyone Know If The Questions In The Senior Maths Challenge Are The Same Each Year???
Thanx
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theone
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(Original post by Micky)
*applauds*

Nice work Just one question (probably has a simple answer which I'm missing, but anyway...)

Why can't both brackets be negative? We know x and y are both positive integers, but that doesn't necessarily mean 3x-19<0
Because if 3x - 19 < 0 this implies x < 19/3 and y < 38/3. as such 1/x + 2/y > 1/(19/3) + 2/(38/3) = 3/19 + 6/38 = 12/38 = 6/19. So 1/x + 2/y > 6/19 > 3/19.
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Micky
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(Original post by Unregistered899)
Does Anyone Know If The Questions In The Senior Maths Challenge Are The Same Each Year???
Thanx
Did you really think they would be?! They have a certain style, but there's nothing that you can really 'learn' that will help drastically, it's mostly how quickly you can think mathematically, and there's a lot of 'thinking outside the box'.

edit: good point theone, that's what I get for not having a pen and paper handy!
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theone
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#28
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(Original post by Unregistered899)
Does Anyone Know If The Questions In The Senior Maths Challenge Are The Same Each Year???
Thanx
Obviously not, otherwise we'd all have looked at last years paper.
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Micky
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#29
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#29
Although sometimes it helps to know the factors of the years (they like questions that involve the year), though that wouldn't take long this year!

Olympiad round 1 is 3rd December this year isn't it?
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Ralfskini
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#30
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Does anybody know where I can get the questions to this years smc? I didn't find it too difficult and would like to have another look at them as I had to hand my question paper in. I only got 70 marks but arrived really late and everybody had been working for half an hour.
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theone
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(Original post by Micky)
Although sometimes it helps to know the factors of the years (they like questions that involve the year), though that wouldn't take long this year!

Olympiad round 1 is 3rd December this year isn't it?
It is indeed.
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Micky
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#32
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Great(!) Our sixth form panto is the night before - better work out my priorities before I get to the pub :|
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It'sPhil...
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(Original post by theone)
It is indeed.
what score do you think you'll need to a) get a gold cert and b) go through to the bmo round 1
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theone
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(Original post by It'sPhil...)
what score do you think you'll need to a) get a gold cert and b) go through to the bmo round 1
Judging on last year's, which was about the same difficulty, 70 for a gold, 82 for BMO, but obviously this may change...
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Micky
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(Original post by theone)
Judging on last year's, which was about the same difficulty, 70 for a gold, 82 for BMO, but obviously this may change...
The head of maths at school told me you needed over 105 to be put forward for BMO1, which I thought was a bit high! I think it's meant to be slightly less than 1000 that do BMO1, about 100 BMO2.

Oh, and Ralfskini, see if any of your maths teachers have copies left, the teachers at my school certainly did.
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Ralfskini
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(Original post by Micky)
The head of maths at school told me you needed over 105 to be put forward for BMO1, which I thought was a bit high! I think it's meant to be slightly less than 1000 that do BMO1, about 100 BMO2.

Oh, and Ralfskini, see if any of your maths teachers have copies left, the teachers at my school certainly did.
Thanks, I'll do that.
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theone
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#37
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(Original post by Micky)
The head of maths at school told me you needed over 105 to be put forward for BMO1, which I thought was a bit high! I think it's meant to be slightly less than 1000 that do BMO1, about 100 BMO2.

Oh, and Ralfskini, see if any of your maths teachers have copies left, the teachers at my school certainly did.
It is a not that high, it's normally 80s/low 90s. The BMO1 was done by 650 people last year (including me ) and BMO2 is for the top 100 scorers from BMO1, although some people (normally pupils in years 11 and below) can be invited even if they miss out on the top 100.
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Ralfskini
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(Original post by theone)
It is a not that high, it's normally 80s/low 90s. The BMO1 was done by 650 people last year (including me ) and BMO2 is for the top 100 scorers from BMO1, although some people (normally pupils in years 11 and below) can be invited even if they miss out on the top 100.
Is the BMO round much more difficult?
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theone
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(Original post by Ralfskini)
Is the BMO round much more difficult?
The BMO is unbelievably harder. There are 5 questions, all of which to be answered, in 3 1/2 hrs. Full solutions are required, and guesswork gets no marks. There's a link to past papers on http://www.bmoc.maths.org/ .
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Ralfskini
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(Original post by theone)
The BMO is unbelievably harder. There are 5 questions, all of which to be answered, in 3 1/2 hrs. Full solutions are required, and guesswork gets no marks. There's a link to past papers on http://www.bmoc.maths.org/ .
Cheers.
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