Nice Maths Puzzle

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JamesCFraser
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Nice Maths Puzzle

Postby JamesCFraser » Sat Dec 01, 2007 1:05 am UTC

Evaluate the following efficiently and without the use of a calculator:

(14 + 20074 + 20084)/(12 + 20072 + 20082)

I'd be interested to see different methods. Though, I am pleased with the method I came up with.
Spoiler:
My solution:

(14 + 20074 + 20084)/(12 + 20072 + 20082)

Expressed in base 2007 is:
(14 + 104 + 114)/(12 + 102 + 112)
=(1 + 10000 + 14641*)/(1 + 100 + 121*)

*NB: Powers of 11 are quickly evaluated using Pascal's triangle {this can be proven by expanding (1 + 10)n}

=24642/222
=12321/111
=111*

*NB: numbers of the form 1234...4321 are (11...11)2 {this can be proven by expanding (1 + 10 + 100 + ... + 10n)2}

Expressed in denary is:
=20072 + 2007 + 1

Now a simple long multiplication and division yields:
(14 + 20074 + 20084)/(12 + 20072 + 20082) = 4030057

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Mathmagic
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Re: Nice Maths Puzzle

Postby Mathmagic » Sat Dec 01, 2007 6:42 am UTC

Ummm, I guess there's no point in making a solution thread:

Spoiler:
To start: let x = 2007

-> (1 + x4 + (x + 1)4)/(1 + x2 + (x + 1)2)

Expanding this out (after some algebra that I won't bother showing):

-> (2x4 + 4x3 + 6x2 + 4x + 2)/(2x2 + 2x + 2)

Taking out the common factor of 2, then recognizing that x4 + 2x3 + 3x2 + 2x + 1 is just (x2 + x + 1)2 (Because (x0 + x1 +...+ xn-1 + xn)2 displays this kind of symmetry (which could in theory, be proven using the multiplication of series, but I don't really feel like it):

-> (x2 + x + 1)2/(x2 + x + 1)

-> (x2 + x + 1)

Replacing x with 2007:

-> 20072 + 2007 + 1

Which can be easily calculated with basic multiplication.
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Macbi
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Re: Nice Maths Puzzle

Postby Macbi » Sat Dec 01, 2007 12:02 pm UTC

BMO question one!
(I also got 2,3 and 4. questions 5 and 6 didn't look possible though)
My working went:
Spoiler:
2x4+4x3+6x2+4x+2=2x4+4x3+6x2+4x+2
=>
1+x4+(x4+4x3+6x2+4x+1)=2(x4+2x3+3x2+2x+1)
=>
1+x4+(x+1)4=2(x2+x+1)2
=>
(1+x4+(x+1)4)/2(x2+x+1)=(x2+x+1)
=>
(1+x4+(x+1)4)/(1+x2+(x+1)2)=(x2+x+1)
...
(14 + 20074 + 20084)/(12 + 20072 + 20082)=20072+2007+1=4030057
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SimonM
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Re: Nice Maths Puzzle

Postby SimonM » Sat Dec 01, 2007 10:21 pm UTC

Macbi wrote:BMO question one!
(I also got 2,3 and 4. questions 5 and 6 didn't look possible though)
My working went:


Question 6 was easy enough. I found the two geometry questions a bitch. I also messed up question 2, but hopefully I'll still get in the 0+ for that anyway
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Macbi
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Re: Nice Maths Puzzle

Postby Macbi » Sun Dec 02, 2007 9:53 am UTC

Was it strong induction?
I tried, but couldn't get the odd numbers to work.
    Indigo is a lie.
    Which idiot decided that websites can't go within 4cm of the edge of the screen?
    There should be a null word, for the question "Is anybody there?" and to see if microphones are on.

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SimonM
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Re: Nice Maths Puzzle

Postby SimonM » Sun Dec 02, 2007 1:52 pm UTC

Macbi wrote:Was it strong induction?
I tried, but couldn't get the odd numbers to work.


It was in indeed. MSN?
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fordred
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Re: Nice Maths Puzzle

Postby fordred » Wed Dec 05, 2007 11:04 pm UTC

That's the answer I got :D
I also solved question 2, but spent to much time on question 3, and ended up doing it completely wrong.

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SimonM
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Re: Nice Maths Puzzle

Postby SimonM » Sat Dec 08, 2007 11:08 am UTC

fordred wrote:That's the answer I got :D
I also solved question 2, but spent to much time on question 3, and ended up doing it completely wrong.


Ah... Geometry, questions 4 and 6 were so good as well....
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