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# 第五届全国大学生数学竞赛试题(专数)

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1036464679@qq.com

2013c10??26F

K.2?R2tü????r????C1,C2???uP:.ò??C2÷C1????±(?w?)E??±,ù?,C2t??P:?‘XC2??\$???\$?.PΓ?P:??\$?;,-?,??%9?.y??C?P??D? ?(?:)???%????,ù???R.Pγ:R2∪{∞}???C???üC?,§òQ∈R2\{P}N?→??→?¤???PQt??:Q??,÷vPQ·PQ??=R2,|y:γ(Γ)?????.

K.??n?????B(t)ún×1Y??b(t)?O?

?

?B(t)=(bij(t)),b(t)=??b1(t)?...??bn(t)?

ù￥bij(t),bi(t)t?'ut??￠Xê?‘a.Pd(t)=detB(t),di(t)?^b(t)“OB(t)??1i????¤????n??Y????1??a.ed(t)k￠?t0|??

B(t0)X=b(t0)

¤?'uX???N?5?§|,áy2:d(t),d1(t),···,dn(t)7kgê≥1??ú?a.K.??f(x)∈C2[0,a]÷vf(0)=1,f??(0)=0,…

0<f(x)<x,(x∈(0,a))

-xn+1=f(xn),x1∈(0,a).

(1)|y{xn}???|4?.

(2)áˉ{nxn}′????`2nd§e??§|?T?.

K.??a>1,?êf:(0,+∞)→(0,+∞)??.|y?3au??????ê??{xn}|??

f??(xn)<f(axn),n∈N+.

K.??f:[?1,1]→R?ó?ê,f3[0,1]t4O,q??g′[?,1,1]t??à?ê,|y:

2??1

fgdx≥??1

fdx??1

gdx.

?1?1?1

1

K.PEij?(i,j) ??1ù{ ??0??n?????.4

Γr={X∈Rn×n|rankX=r}(r=0,1,···,n)

…4φ:Rn×n→Rn×n??¤N??,=÷v

φ(AB)=φ(A)φ(B),?A,B∈Rn×n

y2:

(1)?A,B∈Γrk

rankφ(A)=rankφ(B).

(2)eφ(0)=0,…?3,???1??Y??W|??φ(W)=0,K7?3?_???R|??

φ(Eij)=REijR?1

é??Eijt¤á.

2