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Ecuatie in N

 
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alex2008
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Joined: 19 Oct 2008
Posts: 489
Location: Tulcea

PostPosted: Sat Jan 10, 2009 11:32 pm    Post subject: Ecuatie in N Reply with quote

Rezolvati in \mathb{N} ecuatia : \frac{1}{x+2006}+\frac{2007}{y+2007}=1 , daca x-y\in \mathb{N}* .
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mihai miculita
Pitagora


Joined: 12 Nov 2007
Posts: 94
Location: Oradea, Romania

PostPosted: Thu Mar 26, 2009 5:46 pm    Post subject: Reply with quote

1). \left{ \begin{array} {c} x=2 \\ y=1 \end{array}  \Rightarrow \frac{1}{x+2006}+\frac{2007}{y+2007}=\frac{1}{2+2006}+\frac{2007}{1+2007}=\frac{1}{2008}+\frac{2007}{2008}=1
2).  y<1\Rightarrow y=0 \Rightarrow \frac{1}{x+2006}+\frac{2007}{y+2007}=\frac{1}{x+2006}+\frac{2007}{2007}>\frac{1}{x+2006}+1>1\Rightarrow  \frac{1}{x+2006}+\frac{2007}{y+2007}>1.
3). \left{ \begin{array} {c} x>2 \\ y>1 \end{array}  \Rightarrow \frac{1}{x+2006}+\frac{2007}{y+2007}<\frac{1}{2+2006}+\frac{2007}{1+2007}=\frac{1}{2008}+\frac{2007}{2008}=1\Rightarrow  \frac{1}{x+2006}+\frac{2007}{y+2007}<1.
\mbox{Unica solutie este: } (x;y)=(2;1).
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