Ecuatie in N
Posted: Sat Jan 10, 2009 11:32 pm
Rezolvati in \( \mathb{N} \) ecuatia : \( \frac{1}{x+2006}+\frac{2007}{y+2007}=1 \) , daca \( x-y\in \mathb{N}* \) .
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). \( \left{ \begin{array} {c} x>2 \\ y>1 \end{array} \Rightarrow \frac{1}{x+2006}+\frac{2007}{y+2007}<\frac{1}{2+2006}+\frac{2007}{1+2007}=\frac{1}{2008}+\frac{2007}{2008}=1\Rightarrow \frac{1}{x+2006}+\frac{2007}{y+2007}<1. \)
\( \mbox{Unica solutie este: } (x;y)=(2;1). \)