[tex]\displaystyle\bf\\\frac{x}{y}+\frac{x+1}{y+1}+\frac{x+2}{y+2}+...+\frac{x+2009}{y+2009}=2010\\\\\\\frac{x+0}{y+0}+\frac{x+1}{y+1}+\frac{x+2}{y+2}+...+\frac{x+2009}{y+2009}=2010\\\\\text{\bf De la 0 la 2009 sunt 2010 fractii.}\\\\\text{\bf Egalitatea este adevarata doar daca fractiile sunt echiunitare:}\\\\\frac{x}{y}+\frac{x+1}{y+1}+\frac{x+2}{y+2}+...+\frac{x+2009}{y+2009}=\\\\=\underbrace{\bf1+1+1+...+1}_{2010~de~1}=2010\cdot1=2010\\\\\implies \boxed{\bf~x=y}\\\\\implies~(x-y)(x+y)=0[/tex]
