2024年全国高中数学联赛加试A卷评分细则.pdf

2024c?I)c?mmAK352024c?I)c?mm2024c?IpmAY9IO6(e5Y6),?K?).gr?,IO,?5K6.c?,?1.zK8Uk),u?(?)A.u?(?)XkO)a?),V(10).?YzK?K?)=U8c?.,?eZK|?),.2024c?I)c?mmK|12024A1K:?KUen?.(A)3r,yCmax=12.(10)(B)3r,?E,C=r2(r+1).(10)(C)3r,yC r2(r+1).(20)3r,yC r2(r+1)r?.”?a1(0,1),r=2m,?a1?r?a1=0.b1b2b3,bi 0,1,2m 1(i 1).Kkaik=k0.bibi+1k.e3bi m+1,Kkaik 1 m+12m=m12mm2m+1.e3bi m 2,Kkaik m12mm2m+1.eb?zbi m 1,m.ekYbi=bi+1=m,Kkaik 1 m2mm4m2m2m+1.ekYbi=bi+1=m 1,Kkaik m12m+m4m2 0),?B0(2kb,2b),D0(2kd,2d).BC?”L:A,B0,C0?,P?%O0B,KxO0B=c,yO0B=b+k(c (kb)=(1+k2)b+kc,=O0B(c,(1+k2)b+kc).nkO0D(c,(1+k2)d kc).BC?”L:A,B0,D0?,?%I(x0,y0)vd k(x0 kd)=b+k(x0+kb)=y0,)?x0=1+k22k(d b),y0=1+k22(b+d).EF?”L:A,E0,F0?,dEF1uBD,?%Ik/O0EF(x0,y0).3?:Q0,A,E0?;AE0L:A,B0,C0?L:A,E0,F0?:,l?AE0 O0BO0EF.ukO0BO0EF=(1+k2)b+kc y0c x0,?xQ0=yQ0kAE0=yQ0kO0BO0EF=2d(1+k2)b+kc y0c x0=2(1+k2)bd+kcd 1+k22(b+d)dc x0,nxP0=2(1+k2)bd kbc 1+k22(b+d)bc x0.duCrL4?CL4?,dB,D,P,Q?B0,D0,P0,Q0?B0D0Q0P0?F/,=IyxP0 xQ0=xD0 xB0=2k(b+d).?d?O,xP0 xQ0=2(1+k2)bd+kbc+1+k22(b+d)bc x0+2(1+k2)bd+kcd 1+k22(b+d)dc x0=2k(b+d)c+1+k22(b+d)(b d)c 1+k22k(d b)=2k(b+d).y.4?(Menelaus+n?O)?BP,DQOCA?u:L,L0.AE,AC,AFOBD?u:X,Y,Z.4AY Z?BPLdMenelaus?,ALLYY BBZZPPA=1.4AY X?DQL0dMenelaus?,AL0L0YY DDXXQQA=1.uL=L0ALLY=AL0L0YY BBZZPPA=Y DDXXQQA.duZPPA=BZsinPBZBAsinPBA,?Y BBZZPPA=Y BBZBZ sinPBZBA sinPBA=Y BBAsinPBZsinPBA.aq?Y DDXXQQA=Y DDAsinQDXsinQDA.dS?5kY BBA=Y DDA,?IysinPBZsinPBA=sinQDXsinQDA.(10)?ABD=x,ADB=y,ABP=PAL=ZAY=u,ADQ=QAL0=XAY=v.y3Iysin(x+u)sinu=sin(y+v)sinv cosx+cotusinx=cosy+cotv siny.?EFAC?uT.dEFkBD,XYY Z=ETTF=BYY D=ABAD.duABAD=sinysinx,XYY Z=AXsinvAZsinu=sinv sinx+y2u2sinusiny+x2v2,ksinysinx=sinv cosxy+2u2sinucosyx+2v2,siny sinucosy x+2v2=sinxsinv cosx y+2u2,siny sinu(cosy x2cosv siny x2sinv)=sinxsinv(cosx y2cosu sinx y2sinu),(siny sinucosv sinxsinv cosu)cosx y2=sinusinv siny x2(sinx+siny)=sinusinv siny x2 2sinx+y2cosx y2.w,|xy2|?cosxy2?siny sinucosv sinxsinv cosu=sinusinv(cosx cosy).?sinusinv?=?cosx+cotusinx=cosy+cotv siny.(50)52023A1nK:?KUe?.(A)?E/,y?8S,S?x?Ln.(20)(A1)=Y9/,vky/?K n,?.(A2)=U?yn?5e(,?10.(B)?/,yo38S,?Sx?un.(30)(B1)?vkm,yL?0.)(A)/?,?xm/?,k1?/UK n.?y?”5Y6k?/?0,2?O,?,?O.xm/.?S8,1 i n2,?S(i)S12i1,2i?8.K|S(i)b S(i)w|2.S(i)b 3,S(i)w 3,e|S(i)b S(i)w|3,KS(i)b=3,S(i)w=0,S(i)b=0,S(i)w=3,?S?.|S(i)b S(i)w|2.en=2m,K|Sb Sw|mXi=1|S(i)b S(i)w|2m=n.en=2m+1,n=1w,|Sb Sw|1.n 1,e|Sb Sw|n,b?1,1,Kz1 i m,kS(i)b S(i)w=2,S3?T,1,n3,nuS,?2,nuS.?1,n 13,n 1uS,S(m)b S(m)w=2g.62023A1oK:?),ky?e(?u.(A)y?en S,K?k 1,kAkn+B S.(10)5:10u(4)?r.evk+B,Xen S,KAkn S.vk?Akn+B SU?.(B)Akn+B?(J,k?,?,Ap?n1.(10)5:?n1Ap(5YD,e)n1),?Ak 1(mod n1),?Akn+B?n1n+B.k10.(A)(B),k(B)7k(A).)e)5Y6?,n?(K3?.PA1=A(A,B),B1=B(A,B),K(A1,B1)=1,d(1)(2)?:u?K?k,Ak1 S;d(4)(2)?:u?n S,A1n+B1 S.k,n?n=n1n2,(n1,A1)=1,?n2?zfA1?f.d,3?pq,?Ap1 1(mod n1),n2|Aq1.eg?(an),(bn):a1=A1 Ap11+B1 S,?(a1,A1)=(B1,A1)=1;b1=a1 Ap1i S;a2=A1 b1+B1 S,?(a2,A1)=(B1,A1)=1;b2=a2 Ap1i S;.;.ak=A1 bk1+B1 S,?(ak,A1)=(B1,A1)=1;bk=ak Ap1i S.5?ak=Akp1+A(k1)p1B1+Ap1B1+B1,ak 1+kB1(mod n1),d(B1,n1)=(B1,n)=1,3k,?n1|ak.?d(3),ak Aq1 S,|n|ak Aq1,d(2)=?(.7