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2022cìN¥ém [\Á（最后一卷）ë)1.n´u2½ê, 1a1a2an´ê,÷vPi2f0;1giai´ØÓê,¦ynXi=11ai2{{{1:e¡y²r:nXi=11ainXi=112i 1(1)Ak=fa1;a2;;akgfa1;a2;;ang,duAk´f8k2k 1.dK:ùf8ÚpØÓ.a1+a2++ak2k 1;1kn@okSk=kXi=1(ai 2i 1)0(1)ª dnXi=112i 1 1ai0;()nXi=1ai 2i 12i 1ai0()Si Si 12i 1ai0()n 1Xi=1Si12i 1ai 12iai+1+Sn2n 1an0(2)duaiai+1=)12i 1ai12iai+1;i=1;2;;n 1(ÜSi0;i=1;2;;n,(2)ª¤á!{{{2::Ó{1:a1+a2++ak2k 1;k=1;2;;nÚÚÚnnn:0kab,Kk1a+1b1a k+1b+ky.()(a k)(b+k)ab()k2+k(b a)0w,¤á.£¯K:Pf(a1;a2;;an)=1a1+1a2++1an,dÚnf(a1;a2;;an)f(1;a2+a1 1;a3;;an)f(1;2;a1+a2+a3 (1+2);;an)18Bf(a1;a2;;an)f(1;2;22;;2n 2;a1+a2++an (1+2+22++2n 2))f(1;2;22;;2n 1)=2 12n 1{{{3:Ó{1:kXi=1ai2k 1;k=1;2;;n@okE¼êf(x)=1x;(x0),@of0(x)= 1x20;f00(x)=2x30|^k.êAØ ªíØnXi=11ainXi=112i 1=2 12n 12{{{4:E1¼êP=(1+xa1)(1+xa2)(1+xak)=1+xa1+xa2++xa1+a2++ak1+x+x2+@oP=kYi=1(1+xai)11 x=)kXi=1ln(1+xai) ln(1 x)(0x1)=kkXi=1xai x2ai2+x3ai3 x+x22+x33+üرxkXi=1xai 1 x2ai 12+x3ai 13 1+x2+x23+ü3(0;1)þÈ©nXi=11ai1 122+132 1+122+132++2=k212nXi=11ai!26k1a1+1a2++1an2:::µµµ: K´²;¯K:@´5gErdos,¦31975crd¯Ku3\CMB,þproblem220,1994cþ°½¿m,1999cRomanianMathematicalOlympiad)þ d¯K; Ky35w ´k.êAØ ª,ém·@Ð^{1CCÃ{Ú{2ÛÜ8B{'T.{4´|^²;¼ê=(2)=P+1n=11n2=26,@´Ík¿g'{ük:|ÜêØ ª¯K,´ÁgdKÏ!2.¦¤kêk1,¦3áõê±L«üêkg ,´Ã{L«üêkgÚ):¤¦êk´u1¤kê.ndXe:Äê2nk(2k 1)=(2n+1)k 2nke¡y²:§(2k 1)2nk=uk+vkÃê)(u;v).(1):n=0.Kk2k 1=uk+vk,@ouk2k 12k=)u2@ou=1.Ónv=1,dw,Ã).(2):n2N+n´÷v§uk+vk=(2n+1)k (2n)kên,Kk2juk+vk=)uv(mod2)(I):euv0(mod2),u=2u1;v=2v1,Kk2(n 1)k(2k 1)=(u1)k+(v1)kn5ØÎ.(II):euv1(mod2)a):e2jk,Kuk+vk2(mod4),2nk(2k 1)0(mod4)gñ.3b):e2-k,duuk+vku+v=uk 1 uk 2v++vk 1k1(mod2),v2(uk+vk)=v2(u+v),(Üuk+vk=2nk(2k 1),@ouk+vku+vj2k 1,@ou+v2nk2k2k 1uk+vku+v=(u+v)2uk+vkØuv,eu3,Kkuk+vk3uk 1+vk=uk 1+2uk 1+vku2+2uv+v2=(u+v)2u=1,@ov=1gñ.nþ:¤¦k´u1 Nê{{{2:2jk.Äêp3(mod4);(p;k)=1.p=4t+3,a=2t+2;b=2t+1,Kp=4t+3=(2t+2)2 (2t+1)2=a2 b2dLTEÚnvp(ak bk)=vp(a2 b2)+vp(k2)=1y3bak bk=ck+dk,Kpjck+dk,5¿k´óê, p3(mod4),Kvp(ak bk)=vp(ck+dk)óê,gñ!(2):ek1(mod2).d)|X½n,3áõêp2(modk), 2pk(p+1)k,a=p+1;b=1.Kpjak bkbak bk=ck+dk,Kck( d)k(modp).epjd,Kpjc,Kkpkjck;pkjdk,@oak bk=ck+dk2pk(p+1)kgñ¤±p-d,dp2(modk),p 2=kq,dpjak bk=ck+dk=)(c( d) 1)k1(modp)=)(c(d 1))kq1(modp)=)(c(d 1))p 21(modp)Kpjc+d4(I):ec+d=p,Kkck+dk=ck+(p c)kpk(p+1)k 1gñ(II):ec+d2p,dck+dk2c+d2k2pkgñ.nþ:k2êþ÷v^.:::µµµ:dK5gu(250Problemsinelementarynumbertheory)(1971cÑ),ù SK8KÑ~k¿g,éõÑ´5g[¯KÚ),Ö.-| 5kÆ)JÑ2019c LdK.ù´·vk5¿!3.bn/ABC¥÷vjABjjACj,Ù ,I´S%,:D;E©O´ þüØÓ:, E3`lBACþ,÷vAD?DI;jEBjtan\B=jECjtan\C:F´÷vDFu , jBFj=jCFj,¦y:A;E;Fn:.{{{1:Xã:AH?BCuH,O (ABC) %,KBHCH=ccos\Bbcos\C=tan\Ctan\B=EBECKEH²©\BECM´lBC¥:,N´lBAC¥:,KE;H;M;A;I;M;M;O;Nn:þ©O.5A0Aé»:,KDIA0.F0=EA\MN,dF0MEF0AN,F0MAF0EN,KkF0NF0E=ANEM;F0EF0M=ENAM@okF0NF0M=ANENAMEM, ¡:d\ADI=\MDN=90;\DAI=\DNM=)DMNDIAKkFNFM=DN2DM2=DA2DI2=MA02IM2=AN2IM2=ANEMANMHqANAM=2SANM=2SHNM=HMEN@oFNFM=ANEMENAM=F0NF0MF=F0¤±A;E;F{{{2:Xã::::000 :H;N;L´A;I;J3BCþÝK,G´D'uFMé¡:,S;T´I3AB;ACþ6ÝK,J´ABC %.Ó{1:E;H;Mn:,d\ADI=\ASI=\ATI=2A;D;S;I;T §q\DSA=\DTA;\DBA=\DCA=)DSBDTC@oDBDC=BSCT=BNCN=)DN²©\BDC,KD;N;M.duBN=AB+BC AC2=CLM;L;Gn:.dFD´,D;G'uFMé¡,KFG .y²:E;A;Fn:,@oIy²EDAG´NÚo/,=I÷vMK;MH;MN;ML´NÚå,duH;N;K;L´Rv,@oI`²A;K;I;J´NÚ:,ù´·Ù!{{{3:Xã:ëDMBCu:U,LIIV?BCuV:,A0´A:é»:,EA7L:DuF0.ëF0M.Ó{1:M;H;En:,dî.½n2Rr=R2 OI2=AIIM=)IVAA0=AIIM=)IVIM=AIAA0q\VIM=\AMO=\A0AI,(Üþ¡:IVMAIA0=)\IMV=\AA0I=\AMDD;V;M;Uo:.du\UAI=\OAI=\UDI,D;A;U;I .\AUI=\IDA=90,¤±UHVI´Ý/.ÚÚÚnnn:F0AF0E=MHMEy²:ÏF0DAF0ED=)F0AF0E=F0AF0DF0DF0E=AD2ED2qd/9 .kMHME=MB2=MI2=)MHME=MH2MI2@oIy²MHMI=ADED=sin\VMIsin\VMH()MHsin\UMH=MIsin\IMVduo/UHVI´Ý/:þªw,!£¯K:dÚn:F0M==AH,KF0M?BC,F0=F555:dKÄ g´´1^=zE;H;M,){1Ì|^Ó{(Üq'~,{2Ì|^A²; ./ÏNÚ:y²,{3|©EÝ/(ÜÓ{=,, K±|^üy²(Ø´' , Úc¡{aq),dK5´IMO2021MalaysianTrainingCamp:http-s://artofproblemsolving.com/community/c6h2395519p196544914.p;q´u1½ê, (p;q)=1.êP;Q÷vPq 1;Qp 1,·¡8ÜA=fa0;a1;;aP+Qg(a0a1aP+Q)´²ï,´a0;a1;;aP´|¤ú p ê,aP;aP+1;;aP+Q´|¤ú q ê,¦ØÓ²ï8ê,¦Ù?¿ü8þ.8YYY:ØØØÓÓÓ²²²ïïï888êêêPPPn,KKKnmax=8:P+Q+bPqc(p q)+1pqP+Q+(q p)bQpc+1pq½Â:·P²ï8Ab´±Äb8Ü.ØpqÚÚÚnnn:8ÜAbAc(bc)´k8¿^´3êx;y¦c b=xp+yq;0xP;0yQ777555:ec b=xp+yq(0xP;0yQ),Kkc+(P x)p=b+Pp+yq3Ab¥,3Ac¥,=AbTAc6=?¿¿¿©©©555:eAbTAc6=?dcb,K½3oê:k1;k2;l1;l2,÷vPk1k20;Ql1l20,¦b+k1p+l1q=c+k2p+l2q=)c b=(k1 k2)p+(l1 l2)q=xp+yqÙ¥x=k1 k22[0;P];y=l1 l22[0;Q]y.!£££¯¯¯KKK:y3Ab1;Ab2;;Abn´n÷v^²ï8(=?¿ü8), Øb1b2bn,dÚn:3Kê i; i(i=1;2;;n 1),¦bi+1 bi= ip+ iq; i+ i1 =n 1Xi=1 i; =n 1Xi=1 i;i=1;2;;n 1@o + =n 1Xi=1( i+ i)n 1, 5¿bn b1=xp+yq;0xP;0yQ Pp+Qq