数学竞赛常用知识手册.pdf

1m56+chttp:/ 6=0),3q,va=bqaUb,Pb|a.ab,ba().eb 6=1,Kba.eaUb,KPb-a.XJat|b,at+1-b,t N,Patkb.2.u5(1)b|0,1|a,a|a(a 6=0).(2)eb|a,a 6=0,K1 6|b|6|a|.(3)ec|b,b|a,Kc|a.(4)eb|a,c 6=0,Kbc|ac.(5)ec|a,c|b,Kc|(ma+nb)(m!n Z).(6)ekPi=1ai=0,bUa1,a2,akk 1,KbU,.21.m,eabm,Kabm,Pa b(modm).2.5(1)a b(modm)m|(b a).(2)a b(modm)b=km+a(k Z).(3)a a(modm).(4)ea b(modm),Kb a(modm).2(5)ea b(modm),b c(modm),Ka c(modm).(6)ea b(modm),c d(modm),Ka c b d(modm),ac bd(modm),an bn(modm).(7)eac bc(modm),(c,m)=d,Ka b(modmd).(c,m)Lcm.AO/,?(c,m)=1,eac bc(modm),Ka b(modm).3.adum|8,z8uma(uma).du?mU0,1,m1m/,8UmXmf8:A0,A1,Am1.Ai=qm+i|m,q Z,i=0,1,m 1.kAi(i=0,1,m 1)vm1Si=0Ai=Z,m1Ti=0Ai=.4.XlmmaA0,A1,Am1,zaAiai,Ka0,a1,am1mX(mX).mX0,1,m 1,m?KX.w,mU?mX.31.u1,XJk1,();XJ1?k,.1Q.d,8Z+=1SS.2.u1k,?.3.a?ua.4.k.5.3Xf(n)=mPi=0aini,?g,n,f(n).6.%(Wilson)np7(p 1)!1(modp).4)1.)n()n)3zu1U)/,XJrUd?S(/),).2.n(n 1)IO)n=mQi=1pii.pi,i,i=1,2,m.3.nd(n)=Pd|n1Lu1nk,nIO)n=mQi=1pii,Kd(n)=mYi=1(1+i).4.n(n)=Pd|ndLu1nk,nIO)n=mQi=1pii,K(n)=mYi=1pi+1i 1pi 1.5.3n!IO),pPr=1?npr?.PxLLx.51.(1)ec|a1,c|a2,c|an,Kca1,a2,an.a1,a2,anka1,a2,an.P(a1,a2,an).(2)ea1,a2,anIO)a1=mQi=1pii,a2=mQi=1pii,an=mQi=1pii,pi,i,i,iK,i=1,2,m,K(a1,a2,an)=mQi=1ptii,ti=mini,i,i.(3)XJab,oab8b8.(4)XJab,K(a,b)=b.(5)abu1,d=ax0+by0/Xax+by(x!y)?,Kd=(a,b).(6)ab88.(7)m?,K(am,bm)=(a,b)m.(8)nab,K?an,bn?=(a,b)n.(9)ab(a b)va=bq+r,0 6 r b,q!r Z.K(a,b)=(b,r).4dda!b=.a=bq1+r1,0 6 r1 b.er1=0,K(a,b)=b.er16=0,Kqr1bb=r1q2+r2,0 6 r2 r1.er2=0,K(a,b)=(b,r1)=r1.er26=0,2r2r1r1=r2q3+r3,0 6 r3 r1 r2 r3 9ri(i=1,2,)K,K3?1,g,X1n+1grn+1=0.?durn6=0,Kk(a,b)=(b,r1)=(r1,r2)=(rn1,rn)=rn.d(5)/Xax+by?d=ax0+by0.2.?(1)ea1|b,a2|b,an|b,Kba1,a2,an.a1,a2,ank?a1,a2,an?.Pa1,a2,an.(2)ea1,a2,anIO)a1=mQi=1pii,a2=mQi=1pii,an=mQi=1pii,pi,i,i,iK,i=1,2,m,Ka1,a2,an=mQi=1prii,ri=maxi,i,i.(3)a1,a2,an?.(4)a,b=ab(a,b).6p!?nfn1.p(1)e(a1,a2,an)=1,a1,a2,anp(p).np(p).AO/,1?p;p;p;p,epUa,Kpap.(2)e(a,b)=1,K(a b,a)=1,(a b,ab)=1.(3)e(a,b)=1,a|bc,Ka|c.(4)ea|c,b|c,(a,b)=1,Kab|c.(5)e(a,b)=1,K(b,ac)=(b,c).(6)e(a,b)=1,c|a,K(c,b)=1.(7)e(a,b)=1,K(a,bk)=1.(8)ea1,a2,amzb1,b2,bnzp,K(a1a2am,b1b2bn)=1.52.:?ummp.(Euler),P(m).em=nQi=1pii,K(m)=mnQi=1?1 1pi?.pi,i(i=1,2,n).?m,(m)=m 1.5:(1)(m)5,=(a,b)=1,K(a)(b)=(ab).(2)ep,K(p)=p 1,(pk)=pk pk1.(3)m=p11p22pkk,K(m)=m?1 1p1?1 1p2?1 1pk?.(4)d1,d2,dT(m)mk,KT(m)Pi=1(di)=m.3.n?n(1).nm 2,(a,m)=1,(m).,Ka(m)1(modm).(2)(Fermat)?np,(a,p)=1,Kap1 1(modp).5:?n.n?mA.4.fnm1,m2,mkkp.K|x b1(modm1),x b2(modm2),x bk(modmk)k)x M01M1b1+M02M2b2+M0kMkbk(modM).M=m1m2mk,Mi=Mmi,i=1,2,k,M0iMi 1(modmi),i=1,2,k.5:fnqIn.71.eU2,K;e21,K.6882a.2.(),().?().().35.3.?.e?k,K.81.ea,Ka2a.2.U0,1,4,5,6,9.3.4.5,2,z.5.XJ6,o.6.U4;41.7.804;81.8.eU3,KU3;eU3,K31.9.eU5,KU5;eU5,K5+11.10.r,XJ,2r,U0,1,4,7,9.11.mUk.12.k,L5.13.XJp,op2.95A1.U27.2.U47U4.3.U5705.74.U37iU3.5.U97iU9.6.U117iiU11.7.U10n 1(n)7r?,2n,U10n 1,=rAA=10 x+y,y 0,1,9,K(10n 1)|A(10n 1)|(x+ny).dd?AU9,19,29,39,.8.U10n+1(n)7r?,2n,U10n+1.=rAA=10 x+y,y 0,1,9,K(10n+1)|A(10n+1)|(x ny).dd?AU11,21,31,41,.10?P1.A?LA=nPi=1ai10i,ai 0,1,9,i=0,1,n1,an 1,2,9.2.AnguAng,=An an0(mod10).3.An4y.4.AiS(A)=nPi=0aiu9,=A nPi=0ai(mod9).5.AiS(A)=nPi=0aivS(A+B)6 S(A)+S(B),S(AB)6 S(A)S(B).6.eab?K,K12a 5b?mk.7.e1nkk?m,Kn=2a 5b,a!bK.8.31n?m,!un 1.9.e(n,10)=1,K1n!r,rv10r 1(modn)?.11k?P1.k 2?(),K?A/kL,=Xe/:A=d0+d1k+d2k2+dnkn=nPi=0diki.di 0,1,k1,i=0,1,n1,dn 1,2,k1.2.Ak?LPA=(dndn1d1d0)k.3.BX?,KB/kL,=Xe/:B=d1k1+d2k2+dnkn+di 0,1,k 1,i=1,2,n,85:eBk?,Kk;eB?,K.121.?gax+by=c(1)ax+by=c(a!b!c)k)7(a,b)|c.(2)e(a,b)=1,(x0,y0)ax+by=c|),Kx=x0+bt,y=y0 at(t).2.x2+y2=z2)(1)ex=a,y=b,z=c(a!b!c)x2+y2=z2|),(a,b)=1,|)|).(2)ex=a,y=b,z=cx2+y2=z2|),KabTk,c.(3)x=a,y=b,z=cx2+y2=z2|),ba,K3mn,m n,(m,n)=1,m 6 n(mod2),a=2mn,b=m2 n2,c=m2+n2.(4)ea=2mn,b=m2 n2,c=m2+n2,Ka!b!cx2+y2=z2|);XJkm n 0,(m,n)=1m 6 n(mod2),Ka!b!c|).3.(Pell)(1)x2 dy2=1(d),.(2)do,x=1,y=0),|).(3)d 0,Kx2 dy2=1k).(4)n 0,(x1,y1)x2 dy2=1),qxnynde(x1dy1)n=xn+dyn,K(xn,yn)x2 dy2=1).13:3IX,!pI:,:.aq/,mIX:.1.:/:3:/(=g?/),S,/S:N,/:L,KS=N+L2 1.2./S:(1)1uI/,XJS:,1.(2)S:/,2.9(3)S:/4.3.S:KA(r)Lx2+y26 r2:,r,KA(r)=1+4r+4X16s6rpr2 s2A(r)=1+4r+8P16s6r2r2 s2 4?r2?2.,xLLx.d?,?r,x2+y26 r2:A(r)Cur.4.3:n/.5.?n 5,3:n/.14x1.x R,KxLLx.2.x5(1)y=x8R,8Z.(2)x=x+r,0 6 r 1.(3)x 1 x 6 x nPi=1xi.(7)x1,x2,xnk?nQi=1xi?nQi=1xi.AO/,x9nkxn xn,x nxn.(8)x!ykhyxi6yx.(9)n,Khxni=?xn?.(10)x,kx=x;x,kx=x 1.(11)mn,umn?khmni.10(12)xxX?,=x=x x.y=xkXe5:(i)x 0,1).(ii)x1?.(iii)n+x=x(n).(13)p N,v2|(2p)!M=2p 1.d(11)M=?2p2?+?2p22?+?2p23?+=2p1+2p2+2+1=2p 1.15?1.?(a,m)=1,k?,a 1(modm),ak6 1(modm),0 k Oa!b!c,nSOA!B!C,S!?nOr!R!r1!r2!r3,p,npOha!hb!hc,nOma!mb!mc,nOta!tb!tc,A?t0a,BCph,pBCY,S.S%!?%!%!R%OI!O!G!H,n%OI1!I2!I3.1.1.1unasinA=bsinB=csinC=2R.1.1.2un11a2=b2+c2 2bccosA,b2=c2+a2 2cacosB,c2=a2+b2 2abcosC.1.1.3n/(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=r2?cotA2+cotB2+cotC2?;(7)S=pr=(p a)r1=(p b)r2=(p c)r3.1.1.4en/q,Kuq;en/k,KuApp;en/kp,KupA.1.1.5r=4RsinA2 sinB2 sinC2;r1=4RsinA2 cosB2 cosC2;r2=4RcosA2 sinB2 cosC2;r3=4RcosA2 c