2007中国数学奥林匹克

2007abc,|a+b|=m,|a-b|=n,mn0.:max{|ac+b|,|a+bc|}mnm2+n2.():(1)2n-1,na1,a2,,an,ij{1,2,,n},ai+aj(ai,aj)2n-1;(2)2n-1,na1,a2,,an,ij{1,2,,n},ai+aj(ai,aj)2n-1.,(x,y)xy.()a1,a2,,a1111,2007.1,2,,20072007.22:i,1i11,ai,12i22,ai-11.1,2,,2007,;1,2,,2007,.:,?:1,2,,nx1,x2,,xn,ij(xi-xj);.()OIABC,ABCBCCAABDEF,FDCAP,DEABQ,MNPEQF.:OIMN.(){an}n1an2n+2006k=nakk+1+12n+2007,n=1,2,.:an1n,n=1,2,.()9n,na1,a2,,an(),9ai1,ai2,,ai9(1i1i2i9n)bi{4,7}(i=1,2,,9),b1ai1+b2ai2++b9ai99.()1:max{|ac+b|,|a+bc|}|b||ac+b|+|a||a+bc||b|+|a||b(ac+b)-a(a+bc)||a|+|b|=|b2-a2||a|+|b||b+a||b-a|2(|a|2+|b|2),m2+n2=|a-b|2+|a+b|25220074=2(|a|2+|b|2),,max{|ac+b|,|a+bc|}mnm2+n2.2:ac+b=1+c2(a+b)-1-c2(a-b),a+bc=1+c2(a+b)+1-c2(a-b).=1+c2(a+b),=1-c2(a-b),|ac+b|2+|a+bc|2=|-|2+|+|2=2(||2+||2).,(max{|ac+b,a+bc|})2||2+||2=1+c22m2+1-c22n2.,1+c22m2+1-c22n2m2n2m2+n2,1+c22m4+1-c22n4+1+c22+1-c22m2n2m2n2.1+c22m4+1-c22n4+1+c22+1-c22m2n221+c21-c2m2n2+1+2c+c24+1-2c+c24m2n2=1-c22+1+2c+c24+1-2c+c24m2n21-c22+1+2c+c24+1-2c+c24m2n2=m2n2..3:m2=|a+b|2=(a+b)(a+b)=(a+b)(…a+€b)=|a|2+|b|2+a€b+…ab,n2=|a-b|2=(a-b)(a-b)=(a-b)(…a-€b)=|a|2+|b|2-a€b-…ab.|a|2+|b|2=m2+n22,a€b+…ab=m2-n22.c=x+yi(xyR).,|ac+b|2+|a+bc|2=(ac+b)(ac+b)+(a+bc)(a+bc)=|a|2|c|2+|b|2+a€bc+…ab€c+|a|2+|b|2|c|2+…abc+a€b€c=(|c|2+1)(|a|2+|b|2)+(c+€c)(a€b+…ab)=(x2+y2+1)m2+n22+2xm2-n22m2+n22x2+(m2-n2)x+m2+n22=m2+n22x+m2-n2m2+n22-m2+n22m2-n2m2+n22+m2+n22m2+n22-12(m2-n2)2m2+n2=2m2n2m2+n2.(max{|ac+b|,|a+bc|})2m2n2m2+n2,max{|ac+b|,|a+bc|}mnm2+n2.(1)2n-1p,(a1,a2,,an)=1.i(1in),p|ai,ji,p8j.p8(ai,aj),ai+aj(ai,aj)ai(ai,aj)p=2n-1.(ai,p)=1(i=1,2,,n),ijp8(ai,aj).1,2,,p-1n-1{1,p-1},{2,p-2},,{n-1,n}.,ij,aiaj(modp)ai+aj0(modp).aiaj(modp),ai+aj(ai,aj)ai-aj(ai,aj)p=2n-1;ai+aj0(modp),ai+aj(ai,aj)p=2n-1.(1).(2).2n-1,1pq,2n-1=pq.n:a1=1,a2=2,,ap=p,ap+1=p+1,ap+2=p+3,,an=pq-p,,p,p+1pq-pn-p.1ijp,ai+aj(ai,aj)ai+aj2p2n-1.62p+1ijn,2|(ai,aj),,ai+aj(ai,aj)ai+aj2pq-p2n-1.1ip,p+1jn,:(i)ipjn,ai+aj(ai,aj)pq-12n-1;(ii)i=pj=n,(p,pq-p)=p,ap+an(ap,an)=pqp=q2n-1.,.3.,11i=1ai.:1,2,,nn,l:i,bi,biZ(1il).1,2,,n(),().f(b1,b2,,bl;n).f.,1ijl,bibjf.,ij;,ij,jbj,ji.,,(),f.,():,,P.:Pf(b1,b2,,bl;n).,,P,.i,,pq(1pqn),,l-i:pq,qp,,pq,,.,.,P,.m:a1,a2,,ammn,f(a1,a2,,am,-a1,-a2,,-am;n)=mj=1aj.m=1,P,a1a1,,-a1(1,2,,n),,a1,0..m-1.m.a1a2am.,f(a1,a2,,am,-a1,-a2,,-am;n)=f(a1,-a2,-a3,,-am,a2,a3,,am,-a1;n),,P,1a1a1,2a2,3a2+a3,,ma2++amn-a1,m+12m-1n-a1(,n-a11+2++(n-a1),n1,2,,n),2m1.,22m-2n-a1,1,2,,n-a1(),2m1,2,,n().,2m-2n-a1P()a1P().,f(a1,-a2,-a3,,-am,a2,a3,,am,-a1;n)=a1f(-a2,-a3,,-am,a2,a3,,am;n-a1).f(-a2,-a3,,-am,a2,a3,,am;n-a1)=f(a2,a3,,am,-a2,-a3,,-am;n-a1)=mj=2aj.m.,,.n=2007,m=11,722007411j=1aj.ac.ABCPFD,CPPAAFFBBDDC=1.,PAPC=AFFBBDDC=AFDC=p-ap-c.,PACA=p-aa-c.,PA=b(p-a)a-c.PE=PA+AE=b(p-a)a-c+p-a=2(p-c)(p-a)a-c,ME=12PE=(p-c)(p-a)a-c,MA=ME-AE=(p-c)(p-a)a-c-(p-a)=(p-a)2a-c,MC=ME+EC=(p-c)(p-a)a-c+(p-c)=(p-c)2a-c.,MAMC=ME2.MEMABC,,ME2M.MAMCMABC,MAMC=ME2,MABC.,MABC.,NABC.OIMN.bn=an-1n,bn2n+2006k=nbkk+1(n1).bn0.an,M,bnM.n100000,bn2n+2006k=nbkk+1M2n+2006k=n1k+1=M32nk=n1k+1+M2n+2006k=32n+11k+1M12+Mn2+200632n+167M.,mbn67mM.,bn0(n100000).bn0(n100000).,nN+1,bn0,bN0.bn0(n=1,2,),an1n(n=1,2,).a1=a2=1,a3=a4=3,a5==a12=0,9.,n13.n=13.,ma1,a2,,am(),3ai1ai2ai3b1b2b3{4,7},b1ai1+b2ai2+b3ai39,m67m8a1,a2,,am6ai1,ai2,,ai6b1,b2,,b6{4,7}9|(b1ai1+b2ai2++b6ai6).A1={i|1im,9|ai},A2={i|1im,ai3(mod9)},A3={i|1im,ai6(mod9)},A4={i|1im,ai1(mod3)},A5={i|1im,ai2(mod3)}.|A1|+|A2|+|A3|+|A4|+|A5|=m,(1)iA2,jA3,9|(4ai+4aj);(2)iA4,jA5,94ai+4aj4ai+7aj7ai+4aj(39);(3)ijkA2ijkA3,9|(4ai+4aj+4ak);(4)ijkA4ijkA5,94ai+4aj+4ak4ai+4aj+7ak4ai+7aj+7ak(39).,|Ai|2(1i5).|A1|1,|A2|+|A3|2,|A4|+|A5|2.,m=|A1|+|A2|+|A3|+|A4|+|A5|6.|A1|=0,m7,7m=|A1|+|A2|+|A3|+|A4|+|A5|8.,min{|A2|,|A3|}+min{|A4|,|A5|}3.(i)(ii),i1,i2,,i6A2A3A4A5,i1i2i6b1,b2,,b6{4,7}9|(b1ai1+b2ai2++b6ai6).,n=13.()82