令x1=2000,y1=1/x1
则x1^2/(x1^2+1) +y1^2/(y1^2+1) =x1^2/(x1^2+1)+(1/x1)^2/((1/x1)^2+1)
=x1^2/(x1^2+1)+1/(x1^2+1)
=(x1^2+1)/(x1^2+1)
=1
所以
2000+1/2000+2001+1/2001+2002+1/2002+2003+1/2003+2004+1/2004+2005+1/2005=6
令x1=2000,y1=1/x1
则x1^2/(x1^2+1) +y1^2/(y1^2+1) =x1^2/(x1^2+1)+(1/x1)^2/((1/x1)^2+1)
=x1^2/(x1^2+1)+1/(x1^2+1)
=(x1^2+1)/(x1^2+1)
=1
所以
2000+1/2000+2001+1/2001+2002+1/2002+2003+1/2003+2004+1/2004+2005+1/2005=6