数学奥林匹克问题

11931,ABCDE,EADCAB,EBDCBA.:ADE=BDC.(,246142)21942,PABC,PABCD,PA=PB+PC.:1PB+1PC=1PD.(,713100)193abcR+,a2+b2+c2+abc=4.m=a3+b3+c3,n=1+2abc,s=a+b+c.mns,mns.(,362600)194RtABC,ACB=90,BC=a,CA=b,CDAB,D.GABC,GACCDBCPTQ.PG=TQ,CT(ab),ba.(,510006)31913,ABCO,ACBC,D1D2AC,AD1=BCD2,AD1AD2CD1CD2.:AD1AD2=ACBC+CD1CD2.:2,BD2,D1D2BCE.AD1=BCD2,,AD1=BD2.ACD2D1CE,CAD2=CD1D2=CD1E.ABCD1,,D1CE=D1AB.D1AB=D1AD2+D2AB=D2CD1+D1CA=D2CA,D2CA=D1CE.,ACD2D1CE.ACCD2=CD1CE,CD1CD2=ACCE.ACD2BD2E,CAD2=D2BC=D2BE.ABD2D1,,BD2E=D1AB.D1AB=ACD2.,ACD2=BD2E.,ACD2BD2E.,ACAD2=BD2BE,BD2AD2=ACBE.AD1AD2=AC(BC+CE)84=ACBC+ACCE.AD1AD2=ACBC+CD1CD2.(,100028)192ABCD,EFABCD,AE=1mAB,DF=1nDC,DEAFG,GHAB,H.mnAHAB.:4,DEBCL,AFBCK,DAKBCP.4AE=1mAB,DF=1nDC,EB=m-1mAB,FC=n-1nDC.LBAD=BEAE=m-1mAB1mAB=m-1.,CKAD=n-1.LKAD=LBAD+BCAD+CKAD=m+n-1,ABAH=LDGD=LPKP=LPAD=LKAD+1=m+n.,AHAB=1m+n.(,152054)191ABC,:cos4A1+cos2A+cos4B1+cos2B+cos4C1+cos2C320.:cos4A1+cos2A125(9cos2A-1)Z25cos4A(9cos2A-1)(1+cos2A)Z16cos4A-8cos2A+10Z(4cos2A-1)20.,cos4A1+cos2A125(9cos2A-1).cosA=12,A=3,.,cos4B1+cos2B125(9cos2B-1),cos4C1+cos2C125(9cos2C-1).cos2A+cos2B+cos2C34,cos4A1+cos2A+cos4B1+cos2B+cos4C1+cos2C925(cos2A+cos2B+cos2C)-32592534-325=320.A=B=C=3,.(,225700)1925,PABC,PAB=PBC=PCA.APPBCA.BC.:SABC+SBCA+SCAB3SABC.5:ABC=APC=PCA+PAC=PAB+PAC=CAB,ACB=APB=PBA+PAB=PBA+PBC=CBA,,ABCCAB.,SABCSCAB=BCAB2.94200712007,20072007,,,;,;1.,2.,,,,1.2007430,:300071,:022-23504709,E2mail:[e-mail]2.,,,2006128,BCAABC,SBCASABC=CABC2;CABBCA,SCABSBCA=ABCA2.SABCSCAB+SBCASABC+SCABSBCA=BCAB2+CABC2+ABCA2,SABC+SBCA+SCABSABC=BCAB2+CABC2+ABCA2.BCAB2CABC2ABCA2,,BCAB2+CABC2+ABCA233BCAB2CABC2ABCA2=1.BCAB2+CABC2+ABCA23.SABC+SBCA+SCAB3SABC.(,246142)