2004 BMO (round 1)

Just finished it before lunch.
I mean "finished it" not really finished it...
anyone sits it today as well? what do u find about the questions?

Didn't get the last one, waffled a lot all way along from question 1, that's what i felt. the triangle bit is pretty easy though...

Reply 1

SsEe

13

Yeah...
I did a load of stuff on Q1. Came down to solving some horrible equation for two perfect squares and stopped there.

Tried Q2 for about 15 minutes but couldn't see myself making much progress.

The coloured numbers one was strange. Couldn't get anywhere with that.

For a while the 7 primes one looked like it was making progress and stuff but when it came down to it, I wouldn't have a way of proving that my answer was the smallest possible so stopped.

I spent most of my time on the last question. I was attempting a method of contradiction but I was finding it difficult to find exactly what I was looking for. I knew what I needed and with more time I think I could've fully solved it.

In the end I had a pile of paper with fragments of answers for four of the questions and didn't feel I would get anywhere with them so chose not to submit them.
My biggest mistake was spreading my effort around all of the questions. I think I should've attemted full answers to Q1 and Q5 and not even bothered with the others.
Another big problem is that I went in without any practice on that style of question. With the SMC I'd practiced lots of past papers and was comfortable answering at that level. BMO1 was a lot different.
Finally. The fact that I suck at maths probably didn't help things :smile:

.

Reply 2

n_e_m

I did it this morning as well...
I got 1,2,3,7 for question 1, anyone got the same?
agree that question 2 is pretty easy...
umn i did sth with Q3 and Q5 as well but not really got the answer...
couldnt understand Q4...
so in total, i just have 2 question done! hix...

Reply 3

Gemini

Me too!

I found Question 1 pretty easy, though I made a silly mistake! nem - I also got 1,2,5,7, and the other person taking it with me got the same. So I think you're alright.

Had no idea how to do Q2, ended up with a massive diagram with lines, circles and squiggles everywhere. It was one of those questions where you think that you're getting somewhere - and then work out 2x=2x! :biggrin:

Didn't like the look of Q3, so I ignored it :redface:

Q4) I got 1, 37, 67, 97, 127, 157, 187 but my proof was a bit iffy, so I don't think I got the marks for it.

Had no idea what the back to front E meant, so I ignored Q5 as well.

Don't you find that the time goes really quickly? I thought 3.5hrs would take ages, but I could've used another hour at least.

Reply 4

Spy_Lord

7

Mweh - its seems that everyone's done Question 1 correctly! Well I did it pretty solidly I think, I got n=1,2,3,7, and I came up with a decent proof for there being no other solutions.

I also did Question 3, and I got the correct answer (almost 100% sure) as n=11, but my explanation was fairly . . waffly.

I tried some of Q4, wrote some fragmented bits down, like showing that difference must be even and that the first of the seven primes must be greater than or equal to 7, but I didnt get a solution.

Q2, wrong diagram, I realised later. Q5 I would have had a go at, but ran outta time.

So yeah, 2 qs really. Thought it was quite fun :biggrin:

.

Reply 5

alwaysrebecca

OP

Spy_Lord

Mweh - its seems that everyone's done Question 1 correctly! Well I did it pretty solidly I think, I got n=1,2,3,7, and I came up with a decent proof for there being no other solutions.

I also did Question 3, and I got the correct answer (almost 100% sure) as n=11, but my explanation was fairly . . waffly.

I tried some of Q4, wrote some fragmented bits down, like showing that difference must be even and that the first of the seven primes must be greater than or equal to 7, but I didnt get a solution.

Q2, wrong diagram, I realised later. Q5 I would have had a go at, but ran outta time.

So yeah, 2 qs really. Thought it was quite fun :biggrin:

.

......for question 3, i got 22.....twice as much as 11...?
and for q4, i got a huge number 1061 or something, i can't remember..but i found the smallest d is 2*3*5*7=210.....

Reply 6

Spy_Lord

7

Well the reason I'm fairly sure that q3 gives 11 is that the teacher who was in the room has been doing the BMO questions for years, and he said the answer was 11. But don't worry about it, he himself said that the explanation counts for more, and you can pick up a lot of marks if you do that correctly. For example, one of my friends got 10, but he did all the working almost exactly like mine, but made a little error, so he'll probably get a decent mark for that.

Q4 . . no clue, really. I think you're right about the difference being 210, because I remember my friend talking about how the difference must be a multiple of 70, so could be right, that.

Reply 7

n_e_m

Q3 i agree with 11, at first i found 7 but then i checked it again i realized that 11 must be the smallest value
Q4 my friend got 203 but he said his method wasnt clear...
his list is 23 53 83 113 143 173 203
anyone know when will we get the result?

Reply 8

alwaysrebecca

OP

umm...i still dont get the 11. If the set is{1 2 3 4 5 6 7 8 9 10 11}, we could color the first 5 numbers red supposably, and the last 6 blue. and in this case, we couldn't find such x, y, z, w of the same colour that x+y+z=w, could we?

Reply 9

alwaysrebecca

OP

and 203 is a prime? isn't it 7*29?..

Reply 10

beauford

alwaysrebecca

umm...i still dont get the 11. If the set is{1 2 3 4 5 6 7 8 9 10 11}, we could color the first 5 numbers red supposably, and the last 6 blue. and in this case, we couldn't find such x, y, z, w of the same colour that x+y+z=w, could we?

1 is red. 3 is red. 1+1+1=3.

Reply 11

Camford

12

Can anyone put the questions up? I want to have a look.

Reply 12

n_e_m

1. Each of Paul and jenny has a whole number of pounds.
> he says to her: ''if u give me £3 i will have n times as much as u
> she says to him: '' if u dive me £n i will have 3 times as much as u.
>
> given that all these statements are true and that n is a positive
> integer, what are the possible values of n?
>
>
> 2 Let ABC be an acute angled and let D, E be the feet of the
> perpendiculars from A, B to BC, CA respectively. Let P be the point
where
> the line AD meets the semicircle constructed outwardly on BC, and Q
be
> the point where the line BE meets the semicirlce constructed
outwardly on
> AC. prove that CP= CQ
>
>
> 3 Determine the least natural number n for which the following result
> holds: no matter how the element of the set {1,2,3,...n} are coloured
> blue or red, there are integers x,y,z,w in the set (not neccesary
> distinct) of the same colour such that x+y+z=w
>
>
> 4 Determine the least possible value of the largest term in an
> arithmetic progression of seven distinct primes
>
>
> 5
> Let S be a set of rational numbers with the following properties:
> i, 1/2 € S
> ii, if x€S then both 1/(1+x) and x/(1+x) € S
> prove that S contains all rational numbers in the interval 0<x<1

Reply 13

alwaysrebecca

OP

ok..i didn't see the word "distinct".... :mad:

Reply 14

Gemini

alwaysrebecca

......for question 3, i got 22.....twice as much as 11...?
and for q4, i got a huge number 1061 or something, i can't remember..but i found the smallest d is 2*3*5*7=210.....

What did you get for a?

Reply 15

alwaysrebecca

OP

a? u mean the first term of the progression? i picked 11, but i know i got it wrong now..i checked my anwser and found 851=37*23.....sign....

Reply 16

Gemini

I know

I checked my answers and found one was wrong - (I've got a table of all the primes up to 5000)

I've been trying to figure out exactly what the answer was...starting to think its a trick question :biggrin:

Reply 17

alwaysrebecca

OP

could tell me the anwser if u find one from the table....questions on primes are all "trick" questions to me....what the hell am i suppoed to know about primes!! they extend in such irregular fashion, even the most sophisticated mathematicians on earth are struggling with them obviously.....

Reply 18

beauford

Suppose the common difference is not a multiple of a prime p, with p<=7. Then p will divide one of the terms of the series. If the first term of the series is p, then if p<=5, the (p+1)th term will also be divisible by p and be not equal to p, and hence composite. Thus, 2,3 and 5 all divide the common difference. Further, either the first term is 7, or 7 divides the common difference. In the latter case, we have a difference being a multiple of 210, so the highest term is at least 6*210 > 1200

Otherwise, we take first term 7, and difference a multiple of 2,3 and 5, so of 30.
If d=30, we get 187 is composite.
If d=60 187 again
d=90 187 again
d=120 , 247 is divisible by 13
d=150 gives us an ok sequence and we're there. Answer: 907

Reply 19

alwaysrebecca

OP

beauford

Further, either the first term is 7, or 7 divides the common difference. In the latter case, we have a difference being a multiple of 210, so the highest term is at least 6*210 > 1200

Had same sort of thinking for most parts...igorned the fact that the first term could be 7!!..i feel so stupid..... thanks very much for your reply..i feel so happy now..

2004 BMO (round 1)

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