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Algebra

Blazing goIITian

 Joined: 27 Dec 2010 Post: 1508
3 Mar 2012 01:25:05 IST
1 People liked this
1
626
rmo question.
Engineering Entrance , JEE Main , JEE Main & Advanced , Mathematics , Algebra

find all integers a,b,c such that

(x+1991)(x+a) + 1 can be written as (x+b)(x+c).

Blazing goIITian

Joined: 19 Jan 2008
Posts: 1139
3 Mar 2012 09:14:42 IST
1 people liked this

$\hspace{-16} Here \mathbf{(x+1991).(x+a)+1=0\Leftrightarrow (x+1991).(x+a)=-1}\\\\ So \mathbf{\left|(x+1991)-(x+a)\right|=2\Leftrightarrow \left|1991-a\right|=2}\\\\ So \mathbf{(1991-a)=\pm 2\Leftrightarrow a=1991\mp 2=1989\;,1993}\\\\ So \mathbf{a=\left\{1989,1993\right\}}\\\\ So \mathbf{(x+1991).(x+a)=-1=(x+b).(x+c)}\\\\ So \mathbf{(x+1991).(x+1989,1993)=-1=(x+b).(x+c)}\\\\ So \mathbf{b=\left\{1991,1989\right\}} and \mathbf{c=\left\{1989,1991\right\}}$

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