2006年IMO中国国家集训队选拔考试

2006IMOHABC,DEFABC,ADBECF,STUDEFBCCAAB.:STUH.n.c,:1()c,n,1.M,abcd,MabcdM+49,ad=bc,M,M..k(k3).:kf(x),:(1)f(0)=0,f(1)=1;(2)n,n=f(x1)+f(x2)++f(xs)x1,x2,,xs,s2k-1.(x,f(x)Z,f(x).)mab,(a,b)=1.A,nanAbnA.A,|A{1,2,,m}|.ABCM.:ABC,M,M..11,OHABC,PABC,PBCQ.QHAPOH.:AABCAA,AABCHBC.,QHAP.APAP,QHAP.DNAPQH,AP=2OD,QH=2NH.APOD,,ODNH.QHAPOH..2,DBCABCP.ADBECF,PECA,PFAB.2PDBC,SDBC,,PBCS.,ABCO,OHM.,APMHS.,BPCPMHTHU.APBPCPP,,HSHTHU.STUH.c=n+12.a1,a2,,as,a1=a2==as=2,cn+12.n:si=11ain+12,a1,a2,,asn,1.(1)n=1,.(2)n-1,.n.a12,1a112.t,t-1i=11ai12.t=s+1,.ts,t-1i=11ai12ti=11ai.ti=11ai12+1at1,t=s,.ts,si=t+11ain+12-ti=11ain2.at+1,,asn-1,1.a1,a2,,at1,a1,a2,,asn,1.,n1.576,443.abcdabcd,ad=bc,uv,a(u-1)(v-1)uvd.(uv),a(u-1)(v-1)(u-1)vu(v-1)uvd,[(u-1)(v-1)][uv]=[(u-1)v][u(v-1)].:ad=bc,a(a,c)d(b,d)=b(b,d)c(a,c).a(a,c),c(a,c)=1,d(b,d),b(b,d)=1,a(a,c)=b(b,d)=s,d(b,d)=c(a,c)=t.,a=(a,c)s,b=(b,d)s,c=(a,c)t,d=(b,d)t.ab,(a,c)(b,d).ac,st.u=(b,d),v=t,a=(a,c)s(u-1)(v-1),d=uv...(1)576.576=24242425=25242525=625,576.M577.M,uv(uv),M(u-1)(v-1)uvM+49.uv-(u-1)(v-1)49,u+v50.,577M(u-1)(v-1)u+v-222,(u+v-2)22308482.,u+v-249,u+v51,.,576.(2)1M288,n,13nM+4913(n+1).13nM+49337.,n25.,12(n-1)=13(n+1)-n-25M+50-n-25M,M12(n-1)13nM+49.,1M288,M.{(ui,vi)}23i=1={(13,26),(14,25),(19,19),(14,26),(15,25),(19,20),(15,26),(20,20),(17,24),(19,22),(20,21),(13,33),(18,24),(20,22),(21,21),(15,30),(19,24),(16,29),(18,26),(19,25),(20,24),(21,23),(14,35)}.,i=2,3,,23,uivi(ui-1-1)(vi-1-1)+50,(ui-1-1)(vi-1-1)(ui-1)(vi-1),(u1-1)(v1-1)=300,u1v1=338,(u23-1)(v23-1)=442.288M300,M(u1-1)(v1-1)u1v1M+49.(ui-1-1)(vi-1-1)M(ui-1)(vi-1),M(ui-1)(vi-1)uivi(ui-1-1)(vi-1-1)+50M+49,i=2,3,,23.,288M442,M.443.443,uv(uv),443(u-1)(v-1)uv492.,uv-(u-1)(v-1)49,u+v50.443(u-1)(v-1)u+v-222,u+v45.443(u-1)(v-1)=uv-u-v+1uv-2uv+1=(uv-1)2,uv22,uv484.uv=484,485,486,487,488,489,490,491,49245u+v50(u,v)=(14,35),(18,27).1334=442,1726=442(u-1)(v-1)443.,443..kf(x),,f(0)=0,f(1)=1,f(x)0(mod2k),x;1(mod2k),x.:f(0)=0,f(1)=1kf(x)f(x)=akFk(x)+ak-1Fk-1(x)++a1F1(x),,Fi(x)=x(x-1)(x-i+1)i!,ak,ak-1,,a1,ak0,a1=1.Fi(x+2)=Fi(x)+2Fi-1(x)+Fi-2(x).f(x+2)-f(x)=2akFk-1(x)+k-1i=1(2ai+ai+1)Fi-1(x).ak,ak-1,,a22ak=2k,2ai+ai+1=0,1ik-1.(a1=1)ak=2k-1,ak-1=-2k-2,,a2=-2.,f(x+2)-f(x)=2kFk-1(x).,xf(x+2)-f(x)0(mod2k).f(0)=0,f(1)=1,f(x)=2k-1Fk(x)-2k-2Fk-1(x)+-2F2(x)+F1(x)(xk2k-1k!,k3)..n-1(mod2k),x1,x2,,xs,f(x1)+f(x2)++f(xs)=n,f(x1)+f(x2)++f(xs)-1(mod2k).,2k01.2k-1,s2k-1.(1)a=b=1,A{1,2,,m}={1,2,,m},|A{1,2,,m}|=m.(2)ab.A1={k|a|k,a+18k,}.A1.n,n=an1,a8n1.2|,an=a+1n1A1;28,bn=abn1A1.|A1{1,2,,m}|=i=1(-1)i+1mai.(3)n,cn=ab,ncnA.B={c1,2c2,3c3,}.,|A{1,2,,m}||B{1,2,,m}|.n,n=an1,a8n1,dn=a,2|;b,28.Bn={d1,2d2,,ndn,(n+1)cn+1,(n+2)cn+2,},B0=B.:|Bi{1,2,,m}||Bi+1{1,2,,m}|,i=0,1,.i.(i)i=0,c1=b,c1ici(i2),|B0{1,2,,m}|=1+|{2c2,3c3,}{1,2,,m}|,|B1{1,2,,m}|1+|{2c2,3c3,}{1,2,,m}|=|B0{1,2,,m}|.c1=a,B0=B1.,i=0,.(ii)i1,ci+1di+1,i+1=ani,a8ni.i)2|,di+1=a.,ci+1=b.,(i+1)ci+1=abni,a8bni.d1,2d2,,idia,2|,(i+1)ci+1d1,2d2,,idi.ji+1,(i+1)ci+1=(i+1)bjbjcj.,(i+1)ci+1m,|Bi{1,2,,m}|=1+|{d1,2d2,,idi,(i+2)ci+2,}{1,2,,m}||{d1,2d2,,idi,(i+1)di+1,(i+2)ci+2,}{1,2,,m}|=|Bi+1{1,2,,m}|.(i+1)ci+1m,(i+1)di+1=(i+1)a(i+1)b=(i+1)ci+1m.,.ii)28,di+1=b.,ci+1=a.,(i+1)di+1=abni=idi,i=a-1bnii+1.|Bi{1,2,,m}|=|{d1,2d2,,idi,(i+1)ci+1,}{1,2,,m}||{d1,2d2,,idi,(i+2)ci+2,}{1,2,,m}|=|{d1,2d2,,idi,(i+1)di+1,(i+2)ci+2,}{1,2,,m}|=|Bi+1{1,2,,m}|.,.i.n,ndnm.|A{1,2,,m}||B{1,2,,m}|=|B0{1,2,,m}||Bn{1,2,,m}|=|A1{1,2,,m}|=i=1(-1)i+1mai.,MABC(3)MABC(B),MB(4).34AC1B1=0,MC1.1,MABC;2,MABC.[1,2],1=0-1,2=0+2,MC1,,,()ACBC.C1B1=mf(),A1B1=nf(),f(0)=1,,mn.=B+C1B1A1(),AC=AB1+B1C=mf()sinsinA+nf()sinCsin(-).f()=ACsinAsinCmsinsinC+nsin(-)sinA=ACsinAsinCasin(+1),,a1.sin(+1),,.max{f(1),f(2)}f(0)=1.M1M2M,1,ABC..BB2=mf(),f(0)=1,AB2=BB2sinAsin,CB2=BB2sinCsin(B-).AC=mf()sinsinA+mf()sinCsin(B-)=f()asin(+1)sinAsinC.,f()=ACsinAsinCasin(+1)..()