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Algebra

Blazing goIITian

Joined:	26 Mar 2009
Post:	1428

27 Sep 2009 09:52:19 IST

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387

integral solutions

None

Find all integers a,b,c, and d satisfying:

1<= a<=b<=c<=d
ab+cd = a+b+c+d+3

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Comments (4)

akki ~~ unlucky forever ~~

Blazing goIITian

Joined: 11 May 2008

Posts: 491

27 Sep 2009 21:19:32 IST

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let

a=1+w

b=1+x

c=1+y

d=1+z

so we have,

(1+w+x+wx)+(1+y+z+yz) = 7+x+y+z+w

wx+yz = 5 . . . . . . . (i)

with the constraint 0<=w<=x<=y<=z with all w , x , y , z belongs to integers

we have

wx + yz = 5+0

= 4+1

= 3+2

= 2+3

= 1+4

= 0+5

ignoring first three cases on basis of constraint 0<=w<=x<=y<=z

we have only 3 possible sets of solution,

w x y z

1 1 2 2

1 1 1 4

0 0 1 5

=> (a,b,c,d) = = (2,2,2,5) , (2,2,3,3) and (1,1,2,6) are only possible solution

Aakash Anuj

Blazing goIITian

Joined: 26 Mar 2009

Posts: 1428

27 Sep 2009 21:56:37 IST

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Very good...a solution indeed....thanx

lerrel pinto

Hot goIITian

Joined: 25 Sep 2009

Posts: 156

27 Sep 2009 22:09:53 IST

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yes thats the right answer.........nice question

Aakash Anuj

Blazing goIITian

Joined: 26 Mar 2009

Posts: 1428

27 Sep 2009 22:30:58 IST

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ya this question was of the regional mathematics olympiad

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