初中數(shù)學奧賽

1.

f(s)=|s^2+2s-2009|/2,其中s為奇數(shù).

f(s)=|(s+1)^2-2010|/2.

44^2<2010<46^2,而f(43)=37,f(45)=53

==>

當s=43時,f(s)取最小值=37



2.

(x(k+1))^2=(xk+1)^2

==>

(x0)^2+(x1)^2+(x2)^2+…+(x(2008))^2=

=(x0)^2+(x0+1)^2+(x1+1)^2+…+(x(2007)+1)^2=

=[(x0)^2+(x1)^2+…+(x(2007))^2]+2009+

+2[x0+x1+…+x(2007)]

==>

x0+x1+…+x2008=

=[(x(2008))^2+2*x(2008)-2009]/2

==>

|x0+x1+…+x(2008)|=|(x(2008))^2+2*x(2008)-2009|/2=

=f(x(2008))

根據(jù)1.得:

|x0+x1+…+x(2008)|≥37.





3.

構造一個數(shù)列,使x(2008)=43.



取x1=2,x2=3,...,x(981)=982,

x(982)=-983,x(983)=-982,x(984)=-981,..,x(1965)=0,

x(1966)=1,x(1967)=2,..,x(2008)=43

根據(jù)2.得:

|x0+x1+…+x(2008)|=f(x(2008))=

=f(43)=37.

==>

|x0+x1+x2+…x(2008)︱的最小值為37.