2008年IMO中国国家集训队选拔考试

2008ABC,ABAC,BCE,AED(E).AEEF,CE=CF,CFBDG.:CF=FG.(){xn}x1=2,x2=12,xn+2=6xn+1-xn(n=1,2,).p,qxp.:q2,3,q2p-1.().:a1a2an,a1,a1+a22,a2,a2+a32,a3,a3+a42,.():n(n4),Gn={1,2,,n}2P1,P2,,P2n-n-1,|PiPi+1|=2(i=1,2,,2n-n-2).()mn1,aij(i=1,2,,n;j=1,2,,m)0mn.f=nni=1mj=1aij2+mmj=1ni=1aij2ni=1mj=1aij2+mnni=1mj=1a2ij.()M(M0),n,a1,a2,,anb1,b2,,bn(1)nk=1bk=1,2bkbk-1+bk+1(k=2,3,,n-1);(2)a2k1+ki=1aibi(k=1,2,,n,an=M).()11,DMNK,ABACBCMNK.KDE=AEK=EFC,MKCG.BNCMDE.BEECCNNAAMMB=1.BKKCCNNAAMMB=1.BEKC=ECBK.1=BEECCFFGGDDB=BEECCFFGKCBK=CFFG.,CF=FG.xn=122[(3+22)n-(3-22)n](n1).anbnN+.(3+22)n=an+bn2.(3-22)n=an-bn2.xn=bn,a2n-2b2n=1(n1).q2,3.:q2p-1.q|xp,q|bp,,{bn}q.d,q|bd..n,d|n,q|bn.:abcd,a+b2c+d2(modq)ac(modq),bd(modq).d|n,n=du.an+bn2=(3+22)duaud(modq).bn0(modq).,q|bn,n=du+r(0rd).r1,an(3+22)n=(3+22)du(3+22)raud(ar+br2)(modq),audbr0(modq).a2d-2b2d=1,q|bd,q8a2d.q,,q8ad.,(q,aud)=1.q|br,d.,r=0,d|n...q,,Ciq(1iq-1)q.q2,3,3q3(modq),2q2(modq).,2q-121(modq).(3+22)q=qi=0Ciq3q-i(22)i3q+(22)q=3q+2q2q-122322(modq).,(3+22)q2(322)q3+22(modq).(aq2-1+bq2-12)(3+22)3+22(modq).,3aq2-1+4bq2-13(modq),2aq2-1+3bq2-12(modq).,q|bq2-1.q|bp,d|p.q,,d=1p.d=1,q|b1=2,.d=p.q|bq2-1,d|(q2-1),p|(q2-1).,p|(q-1)p|(q+1).q-1q+1,,q2p-1.1N+,.,c1c2cnN+.ai=c2i-1(iN+),a1a1+a22a2a2+a32a3.2iN+,jN+(ji),ii+j2j,.,a1=1,a1.kN+,a1a1+k2k,,a2=k;lN+,a2a2+l2l,,a3=l;,a1a1+a22a2a2+a32a3.3N+,i0N+,jN+(ji0),i0i0+j2j(1,2),.,i0=1.,{ii0|i=1,2,}.1.jN+(j2),j2j-1.,N+.a1.:a1a1a2an,a1a1+a22a2a2+a32a3.n.n=1,a1.a1a2an.an+1.(1)iN+,an+ian+2i.an+1,an+1an,anan+an+12an+1.,N+.i,an+i,,an+2i.an=2k+1.2k+1,2k+i+1,2k+2i+1.2(2k+i+1)-1=4k+2i+1.(2k+1)+(4k+2i+1)2=3k+i+1.2(3k+i+1)-1=6k+2i+1.,{2nk+2i+1}n=1,,,.an+1.(2)iN+,an+ian+2i.an=2k+1.(i)an+ian+2i,an+1=an+2i.(ii)an+ian+2i,2(2k+i+1)-1=4k+2i+12(2k+2i+1)-1=4k+4i+1.,(2k+1)+(4k+4i+1)2=3k+2i+1,an+1=4k+4i+1;(2k+1)+(4k+4i+1)2=3k+2i+1,2(3k+2i+1)-1=6k+4i+1.,(2k+1)+(6k+4i+1)2=4k+2i+1,an+1=6k+4i+1.3..,n3,n:n(n3),Gn={1,2,,n}P1,P2,,P2n-1,i{1,2,,2n-2},|PiPi+1|=1,P1={1},P2n-1=Gn.n=3,{1},{1,2},{2},{2,3},{1,3},{3},{1,2,3}.n=k(k3),GkP1,P2,,P2k-1,,P1={1},P2k-1=Gk.n=k+1,:P1,P2k-1,P2k-2{k+1},P2k-3,P2k-4{k+1},P2k-5,,P3,P2{k+1},{k+1},P1{k+1},P2,P3{k+1},P4,,P2k-2,P2k-1{k+1}.,P1=P1={1},P2k+1-1=P2k-1{k+1}=Gk+1.P1P2k-1=P1,Pr+1(Pr{k+1})=Pr+1Pr,(P2{k+1}){k+1}={k+1}={k+1}(P1{k+1}),Pr(Pr+1{k+1})=PrPr+1,r2k-2.,,.,..:n(n4),Gn={1,2,,n}2P1,P2,,P2n-n-1,|PiPi+1|=2(i=1,2,,2n-n-2),P2n-n-1={1,n}.n=4,{1,3},{1,2,3},{2,3},{1,2,3,4},{1,2},{1,2,4},{2,4},{2,3,4},{3,4},{1,3,4},{1,4}.n=k(k4),P1,P2,,P2k-k-1,P2k-k-1={1,k}.n=k+1,Gk+1=Gk{k+1}:k+1,k+1.,Gkq1,q2,,q2k-1,i{1,2,,2k-2},|qiqi+1|=1,q1=Gk,q2k-1={1}.,P1,P2,,P2k-k-1,q1{k+1},q2{k+1},,q2k-1{k+1}.,,P2k+1-(k+1)-1=q2k-1{k+1}={1,k+1}.,.f1.f1.nni=1mj=1aij2+mmj=1ni=1aij2ni=1mj=1aij2+mnni=1mj=1a2ij.G=ni=1mj=1aij2+mnni=1mj=1a2ij-nni=1mj=1aij2-mmj=1ni=1aij2.G0.aijnm,aijij.4apqaprasqasr,,[psqr],,1psn,1qrm.G3=[psqr](apq+asr-apr-asq)2,,[psqr].:G=G3.,a2ij.ij,Ga2ijmn+1-m-n.G3,aij(m-1)(n-1),,G3a2ij(m-1)(n-1)=mn+1-m-n.GG3a2ij.,aijaik(jk).G-2(n-1),G3[psqr]sn-1,,G3-2(n-1).GG3aijaik(jk).,,GG3aikajk(ij)-2(m-1).,GG3apqast(ps,qt)2.,.,G0,.aij(i=1,2,,n;j=1,2,,m)1,f=1,f1.fm+nmn+min{m,n}.:fm+nmn+min{m,n}.nm.,fm+nmn+n.S=n2(m+1)m+nni=1r2i+mn(m+1)m+nmj=1c2j-ni=1mj=1aij2-mnni=1mj=1a2ij,,ri=mj=1aij(i=1,2,,n),cj=ni=1aij(j=1,2,,m).,S0.ni=1aibi2=ni=1a2ini=1b2i-1kln(akbl-albk)2,ai=ri,bi=1(i=1,2,,n)-ni=1mj=1aij2=-nni=1r2i+1kln(rk-rl)2.SS=mn(n-1)m+nni=1r2i+mn(m+1)m+nmj=1c2j-mnni=1mj=1a2ij+1kln(rk-rl)2.mn=mn(n-1)m+n+mn(m+1)m+n,,SS=mn(n-1)m+nmj=1ni=1aij(ri-aij)+mn(m+1)m+nmj=1ni=1aij(cj-aij)+1kln(rk-rl)2.aij0,ri-aij0,cj-aij0(i=1,2,,n;j=1,2,,m),S0,.a11=a22==ann=1,0,f=m+nmn+n,,fm+nmn+m.,nm,fm+nmn+m.,fm+nmn+min{m,n}.max1kn{ak}2,max1kn{bk}2n-1.:L=max1kn{ak}.(2)L21+L,,L2.bm=max1kn{bk}.(1)bk(k-1)bm+(m-k)b1m-1k-1m-1bm,1km;(k-m)bn+(n-k)bmn-mn-kn-mbm,mkn.1=nk=1bk=mk=1bk+nk=m+1bkm2bm+n-m-12bm=n-12bm.bm2n-1...f0=1,fk=1+ki=1aibi(k=1,2,,n).fk-fk-1=akbkbkfk.fk-fk-1bkfkfk+fk-1=bk12+fk-fk-12(fk+fk-1)2bk12+2bk2(fk+fk-1)2bk12+bk4bk12+12(n-1).k1nM=anfnf0+nk=1bk12+12(n-1)=32+12(n-1).nMmax32.M=32.:ak=1+k2n,bk=1n(k=1,2,,n),a2k=1+k2n21+ki=11n1+i2n.,M32.()