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2009IMO1.DABCBC,CAD=CBA.OBD,ABADEF,BFDEG,MAG.:CMAO.()2.n(n2).(n):a0,a1,,an0=a0a1anai12(ai+1+ai-1)(i=1,2,,n-1),(ni=1iai)2(n)ni=1a2i.()3.:p,p|(n!+1)ncp23,,cp.()4.abb-a2.:[a,b)mn,[ab,(a+1)(b+1))()S,xSxmn.()5.m1,n3n2m.ai,j(ijN,1im,1jn)(1)j(1jn),a1,j,a2,j,,am,j1,2,,m;(2)ij(1im,1jn-1),|ai,j-ai,j+1|1.M=max1imnj=1ai,j.()6.:40,2k+3l(klN).()1.1,EFBCP,GPADK,ACL.12,APQ,PQF=AEF=ADB.2AEFQFDPQ.Or.AP2=AQAP+PQAP=AFAD+PFPE=(AO2-r2)+(PO2-r2).AG2=(AO2-r2)+(GO2-r2).AP2-AG2=PO2-GO2.,PGAO.33,PFDAEBDAAFFEEPPBBD=1.PFDGDKKFFEEPPBBD=1.DAAF=DKKF.AKFD,AFKD=ADFK.AKFD=AFKD+ADFK=2AFKD.BDFEDBA=EFA.CAD=CBA,CAF=EFA,ACEP.,CPPD=AFFD.ACD,LPKALLCCPPDDKKA=1.ALLC=2.,AGL,MCAGAL,MC,MCGL.GLAO,,MCAO.2.(n)=n(n+1)24.,a1=a2==an=1.(n)n(n+1)24.:a0,a1,,an,(ni=1iai)2n(n+1)24ni=1a2i.:a1a22ann.,2iaii(ai+1+ai-1)i(i=1,2,,n-1).l(1ln-1),i(i=1,2,,l)(l+1)allal+1,allal+1l+1l(l=1,2,,n-1).:ijk{1,2,,n},ij,2ik2i+k2jk2j+k.,2ik2(j+k)2jk2(i+k),(i-j)k30,..ij(1ijn),aiaj.aiiajj,jai-iaj0.ai-aj0,,(jai-iaj)(aj-ai)0,aiajii+ja2j+ji+ja2i.(ni=1iai)2=ni=1i2a2i+21ijnijaiajni=1i2a2i+21ijni2ji+ja2j+ij2i+ja2i=ni=1a2ink=12ik2i+k.bi=nk=12ik2i+k.b1b2bn.a21a22a2n,ni=1a2ibi1n(ni=1a2i)(ni=1bi).,(ni=1iai)21n(ni=1a2i)(ni=1bi).ni=1bi=ni=1nk=12ik2i+k=ni=1i2+21ijni2ji+j+ij2i+j=ni=1i2+21ijnij=(ni=1i)2=n2(n+1)24,,(ni=1iai)2n(n+1)24ni=1a2i..,(n)=n(n+1)24.3.,n1np-1.nn1n2nk.k12p23.k12,.k12.ni+1-ni(1ik-1)112k-1.k-1i=1i=k-1i=1(ni+1-ni)=nk-n1p.:s1,|{1ik-1:i=s}|s,sis.,ni+1-ni=s.ni!+1ni+1!+10(modp).(p,ni!)=1.(ni+s)(ni+s-1)(ni+1)1(modp).,nis(x+s)(x+s-1)(x+1)1(modp).p,,s.ni+1-ni=snis.,.:l,l(l+1)2+1k-1,l(l+1)2+1l+1.,l(l+1)2+1l,1,2,,l(l+1)2+11l.,1,2,,l(l+1)2+1,11,22,ll,1l1+2++l=l(l+1)2,l(l+1)2+11,2,,l(l+1)2+1l.mm(m+1)2+1k-1.m(m+1)2+1k-1(m+1)(m+2)2+1.k-1i=1im-1i=0i(i+1)2+1+i(i+1)2+2++(i+1)(i+2)2m-1i=0(i+1)i(i+1)2+1m-1i=0(i+1)2=m(m+1)(2m+1)6m33.k12,m4.,k2+(m+1)(m+2)24m24(3k-1i=1i)234(3p)23..4..uauu+1b.[ab,(a+1)(b+1))xy,xyu(u+1).:vabu,vabuvabu+1]abuvab+u(ab+a+b+1).ab(u+1)v=uv+vab+u+abu+1ab+a+b+1(aub).,,f(t)=t+abt(atb)t=ab.,uv(u+1)vI=[ab,(a+1)(b+1)).x=uv,y=(u+1)v,xyu(u+1)=v2..mn.amn-1b.,k=m,m+1,,n-1,[ab,(a+1)(b+1))xkyk,Ak,xkykk(k+1)=A2k.n-1k=mxkykmn(m+1)2(n-1)2=n-1k=mA2k.Sxiyi(min-1).S,,xSxmn.S,mn.a+b2ab,ab+a+b+1ab+2ab+1,(a+1)(b+1)ab+1,[ab,(a+1)(b+1)).[ab,(a+1)(b+1)).r2[ab,(a+1)(b+1))(rZ),S={r2}.xSxmn.5.n=2l+1.3n2m,1lm-1.M.(1)i0(1i0m),ai0,l+1=m.ai0,lai0,l+2.1:ai0,lai0,l+2m.ai0,l=m.(2)ai0,l-1m-1,ai0,l-2m-2,ai0,1m-l+1,ai0,l+2m-1,ai0,l+3m-2,ai0,2l+1m-l.Mnj=1ai0,j(m-l)+2[(m-l+1)+(m-l+2)++m]=(2l+1)m-l2.2:ai0,lai0,l+2m.(1)i1(1i1m,i1i0),ai1,l=m.(1)(2)ai1,l+1=m-1,ai1,l+2=m.(2)ai1,l-1m-1,ai1,l-2m-2,ai1,1m-l+1,ai1,l+3m-1,ai1,l+4m-2,ai1,2l+1m-l+1.Mnj=1ai1,j2[(m-l+1)+(m-l+2)++m]+(m-1)=(2l+1)m-(l2-l+1).12M(2l+1)m-l2.,ai,j=f(2i+j)=2i+j(2i+jm),ai,j=f(2i+j)=(2m+1)-(2i+j)(m+12i+j2m),ai,j=f(2i+j)=(2i+j)-2m(2m+12i+j3m),ai,j=f(2i+j)=(4m+1)-(2i+j)(3m+12i+j4m).,ij(1im,1jn-1),m8(2i+j),|ai,j-ai,j+1|=|f(2i+j)-f(2i+j+1)|=1;m|(2i+j),|ai,j-ai,j+1|=|f(2i+j)-f(2i+j+1)|=0.(2).(1).,j(1jn)k(1km),i(1im),ai,j=k.jk(mod2),1jn,1kmn2m,-2mk-jm.-mk-j2m,k-j2.,k-j2k-j2+m{1,2,,m},i.j¢k(mod2),1jn,1kmn2m,-2m(2m+1)-(j+k)2m.,-m(2m+1)-(j+k)2m,(2m+1)-(j+k)2.(2m+1)-(j+k)2(2m+1)-(j+k)2+m{1,2,,m},i.,M.(1),1i1i2m,1jn,f(2i1+j)f(2i2+j),3xy2m+ny-x2mxy,f(x)f(y).,i,ai,1,ai,3,,ai,2l+1,ai,2,ai,4,,ai,2l.,nj=1ai,j(m-l)+2[(m-l+1)+(m-l+2)++m]=(2l+1)m-l2.,M=max1imnj=1ai,j(2l+1)m-l2.,M(2l+1)m-l2=mn-n-122.6.2k+3l40,a,a+d,a+2d,,a+39d,,adN+.m=[log2(a+39d)],n=[log3(a+39d)],,[x]x.:a+26d,a+27d,,a+39d2m+3l2k+3n(klN).a+26d,a+27d,,a+39da+hd2m+3l2k+3n,,bc,a+hd=2b+3c.mnbm,cn.a+hd2m+3l2k+3n,,bm-1,cn-1.bm-2,a+hd2m-2+3n-1=142m+133n712(a+39d)a+26d,;cn-2,a+hd2m-1+3n-2=122m+193n1118(a+39d)a+26d,.,b=m-1,c=n-1,a+26d,a+27d,,a+39d2m+3l2k+3n.,14132m+3l2k+3n.,7..(1)72m+3l.2m+3l1,2m+3l2,,2m+3l7(l1l2l7).3l1,3l2,,3l7d147.13d3l7-3l135-133l213(3l2-3l1)13d,.(2)72k+3n.2k1+3n,2k2+3n,,2k7+3n(k1k2k7).2k1,2k2,,2k7d147.13d2k7-2k125-122k213(2k2-2k1)13d,.,..()20081.32+x2i2008=f(x)+ig(x)(f(x)g(x)).f(x)().(A)-32(B)32(C)-12(D)122.10sinx+6=x().(A)8(B)7(C)6(D)53.123456.,,.,().(A)568(B)1568(C)4568(D)52044.f(x)=ax3+x2+x+dad,|x|1,|f(x)|1.ad().(A)[-2,0](B)[0,2](C)[-1,0](D)[0,1]5.A={x|1x9,xZ},B={(a,b)|abA},BZf:(a,b)vab-a-b.(a,b)f11().(A)4(B)6(C)8(D)126.xy(x-3)2+4(y-1)2=4,x+y-3x-y+1().(A)1,0(B)0,-1(C)1,-1(D)12,-12(9,54)7.Sn{an}n,S17=170,a2000=2001.S2008=.8.z,|z|=1.u=|z2-z+1|.9.f(x)=cos2x+cosxsinx2.f(x).