高中数学竞赛常用知识汇集

1(b6=0),q,a=bqab,bja.ab,ba().b6=§1,ba.ab,b-a.atjb,at+1-b,t2N,atkb.2.(1)bj0,§1ja,aja(a6=0).(2)bja,a6=0,16jbj6jaj.(3)cjb,bja,cja.(4)bja,c6=0,bcjac.(5)cja,cjb,cj(ma+nb)(mn2Z).(6)kPi=1ai=0,ba1,a2,¢¢¢,akk¡1,b.21.m,abm,abm,a´b(modm).2.(1)a´b(modm),mj(b¡a).(2)a´b(modm),b=km+a(k2Z).(3)a´a(modm).(4)a´b(modm),b´a(modm).2(5)a´b(modm),b´c(modm),a´c(modm).(6)a´b(modm),c´d(modm),a§c´b§d(modm),ac´bd(modm),an´bn(modm).(7)ac´bc(modm),(c;m)=d,a´b(modmd).(c;m)cm.,(c;m)=1,ac´bc(modm),a´b(modm).3.m,m(m).m0,1,¢¢¢,m¡1m,,mm:A0,A1,¢¢¢,Am¡1.Ai=fqm+ijm,q2Zg,i=0;1;¢¢¢;m¡1.Ai(i=0;1;¢¢¢;m¡1)m¡1Si=0Ai=Z,m¡1Ti=0Ai=?.4.mmA0,A1,¢¢¢,Am¡1,Aiai,a0,a1,¢¢¢,am¡1m(m).m0,1,¢¢¢,m¡1,m.mm.31.1,1,();1,.1.,Z+=f1gSfgSfg.2.1,.3.apa.4..5.f(n)=mPi=0aini,n,f(n).6.(Wilson)p(p¡1)!´¡1(modp).41.()31,(),.2.n(n1)n=mQi=1p®ii.pi,®i,i=1;2;¢¢¢;m.3.d(n)=Pdjn11n,nn=mQi=1p®ii,d(n)=mYi=1(1+®i).4.¾(n)=Pdjnd1n,nn=mQi=1p®ii,¾(n)=mYi=1p®i+1i¡1pi¡1.5.n!,p1Pr=1·npr¸.[x]x.51.(1)cja1,cja2,¢¢¢,cjan,ca1,a2,¢¢¢,an.a1,a2,¢¢¢,ana1,a2,¢¢¢,an.(a1;a2;¢¢¢;an).(2)a1,a2,¢¢¢,ana1=mQi=1p®ii,a2=mQi=1p¯ii,¢¢¢,an=mQi=1p±ii,pi,®i,¯i,¢¢¢,±i,i=1;2;¢¢¢;m,(a1;a2;¢¢¢;an)=mQi=1ptii,ti=minf®i;¯i;¢¢¢;±ig.(3)ab,abb.(4)ab,(a;b)=b.(5)ab1,d=ax0+by0ax+by(xy),d=(a;b).(6)ab.(7)m,(am;bm)=(a;b)m.(8)nab,µan;bn¶=(a;b)n.(9)ab(ab)a=bq+r,06rb,qr2Z.(a;b)=(b;r).4ab.a=bq1+r1,06r1b.r1=0,(a;b)=b.r16=0,r1bb=r1q2+r2,06r2r1.r2=0,(a;b)=(b;r1)=r1.r26=0,r2r1r1=r2q3+r3,06r3r2.,br1r2r3¢¢¢ri(i=1;2;¢¢¢),,n+1rn+1=0.rn6=0,(a;b)=(b;r1)=(r1;r2)=¢¢¢=(rn¡1;rn)=rn.(5)ax+byd=ax0+by0.2.(1)a1jb,a2jb,¢¢¢,anjb,ba1,a2,¢¢¢,an.a1,a2,¢¢¢,ana1,a2,¢¢¢,an.[a1;a2;¢¢¢;an].(2)a1,a2,¢¢¢,ana1=mQi=1p®ii,a2=mQi=1p¯ii,¢¢¢,an=mQi=1p±ii,pi,®i,¯i,¢¢¢,±i,i=1;2;¢¢¢;m,[a1;a2;¢¢¢;an]=mQi=1prii,ri=maxf®i;¯i;¢¢¢;±ig.(3)a1,a2,¢¢¢,an.(4)[a;b]=ab(a;b).61.(1)(a1;a2;¢¢¢;an)=1,a1,a2,¢¢¢,an().n().,1;;;p,pa,pa.(2)(a;b)=1,(a§b;a)=1,(a§b;ab)=1.(3)(a;b)=1,ajbc,ajc.(4)ajc,bjc,(a;b)=1,abjc.(5)(a;b)=1,(b;ac)=(b;c).(6)(a;b)=1,cja,(c;b)=1.(7)(a;b)=1,(a;bk)=1.(8)a1,a2,¢¢¢,amb1,b2,¢¢¢,bn,(a1a2¢¢¢am;b1b2¢¢¢bn)=1.52.:mm(Euler),'(m).m=nQi=1p®ii,'(m)=mnQi=1µ1¡1pi¶.pi,®i(i=1;2;¢¢¢;n).m,'(m)=m¡1.:(1)'(m),(a;b)=1,'(a)'(b)='(ab).(2)p,'(p)=p¡1,'(pk)=pk¡pk¡1.(3)m=p®11p®22¢¢¢p®kk,'(m)=mµ1¡1p1¶µ1¡1p2¶¢¢¢µ1¡1pk¶.(4)d1;d2;¢¢¢;dT(m)m,T(m)Pi=1'(di)=m.3.(1)m2,(a;m)=1,'(m),a'(m)´1(modm).(2)(Fermat)p,(a;p)=1,ap¡1´1(modp).:m.4.m1,m2,¢¢¢,mkk.x´b1(modm1),x´b2(modm2),¢¢¢¢¢¢x´bk(modmk)x´M01M1b1+M02M2b2+¢¢¢+M0kMkbk(modM).M=m1m2¢¢¢mk,Mi=Mmi,i=1;2;¢¢¢;k,M0iMi´1(modmi),i=1;2;¢¢¢;k.:.71.2,;21,.62.2.(),().().()..3..,.81.a,a2a.2.0,1,4,5,6,9.3..4.5,2,.5.6,.6.4;41.7.804;81.8.3,3;3,31.9.5,5;5,5+1¡1.10.,,,,0,1,4,7,9.11..12.,.13.p,p2.91.2.2.44.3.505.74.33.5.99.6.1111.7.10n¡1(n),n,10n¡1,AA=10x+y,y2f0;1;¢¢¢;9g,(10n¡1)jA,(10n¡1)j(x+ny).A9,19,29,39,¢¢¢.8.10n+1(n),n,10n+1.AA=10x+y,y2f0;1;¢¢¢;9g,(10n+1)jA,(10n+1)j(x¡ny).A11,21,31,41,¢¢¢.101.AA=nPi=1ai10i,ai2f0;1;¢¢¢;9g,i=0;1;¢¢¢;n¡1,an2f1;2;¢¢¢;9g.2.AnAn,An´an0(mod10).3.An4.4.AS(A)=nPi=0ai9,A´nPi=0ai(mod9).5.AS(A)=nPi=0aiS(A+B)6S(A)+S(B),S(AB)6S(A)S(B).6.ab,12a£5b.7.1n,n=2a£5b,ab.8.1n,n¡1.9.(n;10)=1,1nr,r10r´1(modn).11k1.k2(),Ak,:A=d0+d1k+d2k2+¢¢¢+dnkn=nPi=0diki.di2f0;1;¢¢¢;k¡1g,i=0;1;¢¢¢;n¡1,dn2f1;2;¢¢¢;k¡1g.2.AkA=(dndn¡1¢¢¢d1d0)k.3.B,Bk,:B=d¡1k¡1+d¡2k¡2+¢¢¢+d¡nk¡n+¢¢¢d¡i2f0;1;¢¢¢;k¡1g,i=1;2;¢¢¢;n;¢¢¢8:B,;B,.121.ax+by=c(1)ax+by=c(abc)(a;b)jc.(2)(a;b)=1,(x0;y0)ax+by=c,x=x0+bt,y=y0¡at(t).2.x2+y2=z2(1)x=a,y=b,z=c(abc)x2+y2=z2,(a;b)=1,.(2)x=a,y=b,z=cx2+y2=z2,ab,c.(3)x=a,y=b,z=cx2+y2=z2,a,mn,mn,(m;n)=1,m6´n(mod2),a=2mn,b=m2¡n2,c=m2+n2.(4)a=2mn,b=m2¡n2,c=m2+n2,abcx2+y2=z2;mn0,(m;n)=1m6´n(mod2),abc.3.(Pell)(1)x2¡dy2=1(d),.(2)d,x=§1,y=0,.(3)d0,x2¡dy2=1.(4)n0,(x1;y1)x2¡dy2=1,xnyn(x1¡pdy1)n=xn+pdyn,(xn;yn)x2¡dy2=1.13,,.,.1.(),S,N,L,S=N+L2¡1.2.(1),,1.(2),2.9(3)4.3.A(r)x2+y26r2,r,A(r)=1+4[r]+4X16s6r[pr2¡s2]A(r)=1+4[r]+8P16s6rp2[pr2¡s2]¡4·rp2¸2.,[x]x.,r,x2+y26r2A(r)¼r.4..5.n5,n.14[x]1.x2R,[x]x.2.[x](1)y=[x]R,Z.(2)x=[x]+r,06r1.(3)x¡1[x]6x[x]+1.(4)y=[x],x16x2,[x1]6[x2].(5)n2Z,[n+x]=n+[x].(6)·nPi=1xi¸nPi=1[xi].(7)x1;x2;¢¢¢;xn·nQi=1xi¸nQi=1[xi].,xn[xn][x]n,[x][npx]n.(8)xyhyxi6[y][x].(9)n,hxni=·[x]n¸.(10)x,[¡x]=¡[x];x,[¡x]=¡[x]¡1.(11)mn,mnhmni.10(12)fxgx,fxg=x¡[x].y=fxg:(i)fxg2[0;1).(ii)fxg1.(iii)fn+xg=fxg(n).(13)p2N,2¸j(2p)!¸M=2p¡1.(11)M=·2p2¸+·2p22¸+·2p23¸+¢¢¢=2p¡1+2p¡2+¢¢¢+2+1=2p¡1.151.(a;m)=1,¸,a¸´1(modm),ak6´1(modm),0km,¸am.¸6'(m),¸j'(m).2.¸='(m),am'(m),,am.3.¸(1)am¸,a0;a1;¢¢¢;a¸¡1,m.(2)¸m,at´1(modm)t,¸jt.11.14ABCabc,ABC,rRr1r2r3,p,hahbhc,mambmc,tatbtc,\At0a,BCh,BC®,S.IOGH,I1I2I3.1.1.1asinA=bsinB=csinC=2R.1.1.211a2=b2+c2¡2bccosA,b2=c2+a2¡2cacosB,c2=a2+b2¡2abcosC.1.1.3(1)S=12aha=12bhb=12chc;(2)S=12absinC=12bcsinA=12casinB=12ahsin®;(3)S=abc4R=2R2sinA¢sinB¢sinC=R22(sin2A+sin2B+sin2C);(4)S=a2sinB¢sinC2sin(B+C)=b2sinC¢sinA2sin(C+A)=c2sinA¢sinB2sin(A+B);(5)(Heron)S=pp(p¡a)(p¡b)(p¡c);(6)S=r2µcotA2+cotB2+cotC2¶;(7)S=pr=(p¡a)r1=(p¡b)r2=(p¡c)r3.1.1.4,;,;,.1.1.5r=4RsinA2¢sinB2¢sinC2;r1=4RsinA2¢cosB2¢cosC2;r2=4RcosA2¢sinB2¢cosC2;r3=4RcosA2¢cosB2¢sinC2;r1+r2+r3=r+4R.1.1.6\BIC=90±+\A2,\CIA=90±+\B2,\AIB=90±+\C2,\BI1C=90±¡\A2,\CI2A=90±¡\B2,\AI3B=90±¡\C2.1.1.7