UKMT Maths Senior Challenge - how to prepare? - Page 4

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tomdebs

i saw this question on an SMC past paper.hoping one of you 'geniouses' can tell me how its done.

which one of the following is a prime.

a.1000^2+111^2 b.555^2+666^2 c.2000^2-999^2 d.1001^2+1002^2 e.1001^2+1003^2

Quickly glancing at the question, you can see that two options can immediately be ruled out: b and c.

c --> the difference of two squares means is equals (2000+999)*(2000-999)

b --> 555^2 and 666^2 share a common factor of 111^2, so when you add them the total will also have this factor of 111

Instead of working out D and E, it might be easier to do so algebraically, and doing this eliminates e immediately.

e --> let 1001 = n, so the sum becomes n^2 + (n+2)^2. Multiplied out and simplified this becomes n^2 + n^2 + 4n + 4, or 2n^2 + 4n + 4, or 2(n^2 + 2n + 2). This factorisation shows it can't be prime.

With the rest, er, I'll get back to you :p:

Reply 61

Robbie10538

It's (a), 1000^2 + 111^2

b is ruled out by seeing a common factor of 111
c is ruled out using difference of two squares formula
d and e ruled out by looking at last digits, 5 and 0 respectively

:biggrin:

edit: sorry aeiou81!

Reply 62

aeiou81

Robbie10538

edit: sorry aeiou81!

about d: OMFG :facepalm:

now I feel stupid :o:

But yes, to the person who posed the question: you just have to look for little things like ending in 5 or divisible by 2 and whether there are any factors or ways of factorising it :smile:

Reply 63

xinxin614

anyone have the solution for nov 2007 paper pls?

Reply 64

tomdebs

thanks aeiou for the pointers. prime numbers just freak me out sometimes

Reply 65

Robbie10538

xinxin614

anyone have the solution for nov 2007 paper pls?

CBDCEEDADDBCCDAAECBBBABED

I might be able to help with ones you're unsure of

Reply 66

Adam92

Solutions to some of the papers can be found here: http://www.wpr3.co.uk/UKMT/teachers/

Reply 67

tomdebs

im stuck on this question:
find the sum to infinity of the convergent series
1/2 + 1/4 + 2/8 + 3/16 + 5/32 + 8/64 + 13/128 + 21/256 +34/512 +.....?
its the last question so its supposed to be very difficult.please if you have any ideas post them.

Reply 68

aeiou81

tomdebs

Think about how you can break it down maybe? 2/8 simplifies to 1/4, 3/16 = 1/8 + 1/16, 5/32 = 1/8 + 1/32...

Reply 69

Simba

tomdebs

Well, let S = 1/2 + 1/4 + 2/8 + 3/16 + ...

Now, note that S/2 = 1/4 + 1/8 + 2/16 + 3/32 + ...

Adding these gives 3S/2 = 1/2 + (2/4 + 3/8 + 5/16 + ...)

But now note that (2/4 + 3/8 + 5/16 + ...) = 2*(2/8 + 3/16 + 5/32 + ...) = 2(S - 1/2 - 1/4).

Therefore 3S/2 = 1/2 + 2(S - 1/2 - 1/4).

And so S = 2 :smile:

Reply 70

tomdebs

thanks simba that was very good to say the least.can this technique work every time?

Reply 71

Simba

tomdebs

thanks simba that was very good to say the least.can this technique work every time?

Errr, it's probably worth spending a little time looking for a similar trick, yes. Stuff like that comes up semi-frequently in infinite series and other areas of maths.

Reply 72

Small123

miml

When I did it I just took the exams without revising (it is possible to get a Gold this way). I wouldn't take it too seriously, just turn up and see what you can do.

However, there are books publish by UKMT that would help. I have some now, and they would have been really useful when I was doing maths challenges. I also would recommend the Art of Problem solving wiki... focus on learning a few techniques (modular arithmetic is invaluable for almost all the number theory questions). Also it would be a good idea to do some geometry (I absolutely hate it, but geometry tends to be ignored after primary school).

What's that?

Reply 73

miml

Small123

What's that?

http://www.artofproblemsolving.com/Wiki/index.php/Main_Page

Reply 74

Robbie10538

NOoooo
my school didnt enter me for it. They said they would try to enter but it is a bit too soon.

Oh well :frown:

Reply 75

refref

awww

I better check tomorrow actually :s-smilie:

we usually have a maths lesson in the morning, but this year we dont.

Reply 76

Small123

Robbie10538

NOoooo
my school didnt enter me for it. They said they would try to enter but it is a bit too soon.

Oh well :frown:

Are you sure you're too late? I signed up for it yesterday.

Reply 77

tiny hobbit

Robbie10538

NOoooo
my school didnt enter me for it. They said they would try to enter but it is a bit too soon.

Oh well :frown:

Does that mean that they haven't entered anyone?

School's don't send names in, they just decide how many papers to order. If your school hasn't got any papers, is there a school nearby that has a spare - you could go there and do it.