数学竞赛之窗

:O1244:A:0488-7395(2001)19-0045-02,(200062,E2mial:[e-mail]),(200231,E2mail:[e-mail])..2000(61)(),(215633,),.20001(Putnam,61thA22):n,n,n+1,n+2().:0=02+02,1=12+02,2=12+12.2(Putnam,61thA23)P1P2P88,P1P3P5P75,P2P4P6P84.8P1P2P8.3(Putnam,61thA25),r3,:3,3r.4(Putnam,61thB21)aj,bj,cj,1jN.j,aj,bj,cj.:r,s,t{raj+sbj+tcj|1jN},4N7.5(Putnam,61thB22)nm1,n,m,:(m,n)nnm.6(Putnam,61thA26)f(x),{an}:a0=0,an+1=f(an),n0.:m,am=0,a1a2.7(Putnam,61thB25)S0,S0,S1,S2,,:a-1Sn,aSn,aSn+1.:N,SN=S0{N+a|aS0}.8(Putnam,61thB26)Bn(n3),(1,1,,1),B2n+1n.:B,.1[1]m,n=4m4+542001194m2,:n=(2m2)2+(2m)2,n+1=02+(2m2+1)2,n+2=12+(2m2+1)2,.[2]x2-2y2=1,(x,y),n=2y2,:n=y2+y2,n+1=02+x2,n+2=12+x2,.:.235.:P1,P3P2P8,P2P4,P1P2P8.3a,b,cABC,SABC12(Pick,).,SABC=12absinC=abc4r12,,abc2r,a,b,c32r3r47(x,y,z),x,y,z{0,1}.:aj,bj,cj,Aj={xaj+ybj+zcj|x,y,z{0,1}}4,4,1jN.,x=y=z=0,xaj+ybj+zcj.,{xaj+ybj+zcj|x,y,z{0,1},x,y,z,1jN},4N.,,(x,y,z),x,y,z{0,1},x,y,z,{xaj+ybj+zcj|1jN}4N7.5,r,tZ,mr+nt=(m,n).,(m,n)nnm=rm+ntnnm=rmnnm+tnm=rn-1m-1+tnmZ.,.6,m,nZ,mn,m-n|f(m)-f(n).bn=an+1-an,,nN,bn|bn+1(bn=0,bn+1=0).a0=am=0,a1=f(a0)=f(am)=am+1,b0=bm.,b0=0,{an},a0=a1==am=0,.b00,b0|b1,b1|b2,,bm-1|bmb0=bm,n{1,2,,m-1},bn=b0,,b0+b1++bm-1=am-a0=0,b0,,bm-1,k{1,2,,m-1},bk-1=-bk,ak-1=ak+1,{an},nk-1,an=an+2.a0=am=am+2=f(f(am))=f(f(a0))=a2,a2=0,.7Sn:6aSnxa,,6aSn+1xa(1+x)6aSnxa(mod2),6aSnxa(1+x)n6aS0xa(mod2).,kN3,2kS0,N=2k,6aSNxa(1+x)2k6aS0xa=(1+2x+x2)2k-16aS0xa(1+x2)2k-16aS0xa(1+x2k)6aS0xa(mod2).SN=S0{N+a|aS0},.8BX,TX={Y|d(X,Y)=2,YRn}.d(X,Y)nRn.,TxRn,X,2.C={Y|Y=(y1,,yn),yi{1,-1}}.C,XB,CnYTX.(,YX,YTX),C2n,B2nn,,CY,YBP,Q,R2,P,Q,RY,,P,Q,R,P,Q,R.64200119