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Home » High School Contests & Programs » American Mathematics Competitions
 
My Fifth Practice Senior Meet #6
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darkquantum
Yang-Mills Theory
Yang-Mills Theory

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#1
My Fifth Practice Senior Meet #6
Given that 1/x-1/y=1/97. Compute all possible ordered pairs (x,y)
'
x,y are positive integers

PostPosted: Mon Oct 06, 2003 5:49 am  Back to top 
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darkquantum
Yang-Mills Theory
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#2
any takers?

PostPosted: Tue Oct 07, 2003 7:41 pm  Back to top 
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JBL
Birch & Swinnerton Dyer
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#3
I like prime numbers.
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Joel
Hi Deeps! <3

PostPosted: Thu Oct 09, 2003 1:01 pm  Back to top 
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ComplexZeta
Birch & Swinnerton Dyer
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#4
This problem involves Simon's favorite factoring trick. That shouldn't be too much of a hint, should it?
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\mathbb{C}\zeta
Simon Rubinstein-Salzedo

PostPosted: Thu Oct 09, 2003 2:43 pm  Back to top 
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Tare
Navier-Stokes Equations
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#5
I thought you were Simon...
Anyways, Click to reveal hidden content
1/x-1/y=1/97 Arrow y/xy-x/xy=1/97 Arrow (y-x)/xy=1/97 Arrow y-x=1 so y=x+1. Substituting that for xy=97 gives us x(x+1)=97 Arrow x:^2:+x-97=0 Arrow x=(-1:pm::sqrt:1+388)/2...it doesn't look right...am I right so far?

PostPosted: Thu Oct 09, 2003 2:50 pm  Back to top 
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JBL
Birch & Swinnerton Dyer
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#6
(y-x)/xy=1/97 --> y-x=1
is a rather problematic statement -- what if y-x = 2 and xy = 2*97.

Keep hacking at it.
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Joel
Hi Deeps! <3

PostPosted: Thu Oct 09, 2003 3:03 pm  Back to top 
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Tare
Navier-Stokes Equations
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#7
OK then, trying again...Click to reveal hidden content
so I got it right till I said (y-x)/xy=1/97 Arrow y-x=1 and xy=97. So cross multiplying (y-x)/xy=1/97 gives me 97(y-x)=xy Arrow 97y-97x=xy Arrow 97x+xy-97y=0...now what? I don't see any way to factor this...now's a good time for that factoring trick Simon Smile


PostPosted: Thu Oct 09, 2003 3:49 pm  Back to top 
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Osiris
Yang-Mills Theory
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#8
Click to reveal hidden content
x < y.

1 / x - 1 / y = 1 / 97
(y - x) / xy = 1 / 97
97 (y - x) = xy
xy + 97x - 97y = 0
(x - 97)(y + 97) = - 972

97 is prime, so the only possibility is

x - 97 = - 1, y + 97 = 972 => x = 96, y = 9312.
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Tom Yue

PostPosted: Thu Oct 09, 2003 4:13 pm  Back to top 
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TheSouthpaw
Riemann Hypothesis
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#9
DUDE! IT WAS STARING ME RIGHT IN THE FACE! I CANT BELIEVE I COULDNT FACTOR THAT! GR!

PostPosted: Thu Oct 09, 2003 4:20 pm  Back to top 
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ComplexZeta
Birch & Swinnerton Dyer
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#10
I am Simon. Simon sometimes likes to speak in the third person. Simon's favorite factoring trick is:

(x-1)(y-1)=xy-x-y+1, or, more generally,
(x-k)(y-k)=xy-kx-ky+k^2.

You'd be amazed how many problems you can solve by simply remembering that you can factor xy-x-y+1 (or xy+x+y+1).
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\mathbb{C}\zeta
Simon Rubinstein-Salzedo

PostPosted: Thu Oct 09, 2003 4:26 pm  Back to top 
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Tare
Navier-Stokes Equations
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#11
I feel there's a lot more possible answers than that...
From Click to reveal hidden content
(x - 97)(y + 97) = - 97:^2:
I got the following four cases:
1.) Click to reveal hidden content
x-97=1, y+97=-97:^2: Arrow x=98, y=-9506

2.) Click to reveal hidden content
x-97=-1, y+97=97:^2: Arrow x=96, y=9506

3.) Click to reveal hidden content
x-97=97:^2:, y+97=-1 Arrow x=9312, y=-98

4.) Click to reveal hidden content
x-97=-97:^2:, y+97=1 Arrow x=-9312, y=-96


*sigh* need some well-deserved sleep...can't think straight, I feel the same way as TheSouthpaw.

OK Simon, Tare's perfectly fine with that Smile
Last edited by Tare on Thu Oct 09, 2003 5:08 pm; edited 1 time in total 
PostPosted: Thu Oct 09, 2003 5:03 pm  Back to top 
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Osiris
Yang-Mills Theory
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#12
The question stated that x and y are positive; and 972 - 97 is not equal to 9506. Wink
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Tom Yue

PostPosted: Thu Oct 09, 2003 5:07 pm  Back to top 
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Tare
Navier-Stokes Equations
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#13
That's it, bedtime! Smile

PostPosted: Thu Oct 09, 2003 5:09 pm  Back to top 
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darkquantum
Yang-Mills Theory
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#14
yuppers
theres only one solution

PostPosted: Fri Oct 10, 2003 3:02 pm  Back to top 
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darkquantum
Yang-Mills Theory
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#15
yuppers
theres only one solution

PostPosted: Fri Oct 10, 2003 3:03 pm  Back to top 
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