1、1 数学奥赛辅导 第一讲 奇数、偶数、质数、合数 知识、方法、技能 g265.g6984g6980g11352g3867g1610g5627 g4570g1852g1319g6984g6980g2010g1038g1016g12879g712g1973g71712g11352g1505g6980g11352g6980g12228g1038g1610g6980g712g2554g2029g12228g1038g3867g6980.g3252g8504g712g1231g980g1610g6980g2499g15932g10382mg708mg281Zg709g712g1231g980g3867g
2、6980g2499g15932g10382m+1g61222mg7131g11352g5430g5347.g3867g451g1610g6980g1867g7389g3926g991g5627g17148g726 g7081g709g3867g6980g102g3867g6980=g1610g6980g727g1610g6980g102g1610g6980=g1610g6980g727 g3867g6980g102g1610g6980=g3867g6980g727g1610g6980g104g1610g6980=g1610g6980g727 g3867g6980g104g1610g6980=g
3、1610g6980g727g3867g6980g104g3867g6980=g3867g6980g727 g7082g709g3867g6980g11352g5191g7053g18129g2499g15932g10388m+1g5430g5347g712g1610g6980g11352g5191g7053g18129g2499g15932g10388mg61228m+4g11352g5430g5347g708mg281Zg709. g7083g709g1231g1321g980g1022g8503g6984g6980ng712g18129g2499g1209g1901g6116 lnm2=
4、g11352g5430g5347g712g1866g1025mg1038g19762g17139g6984g6980g712lg1038g3867g6980. g17837g1135g5627g17148g7094g12628g2345g2460g7138g7186g712g9994g13792g4439g2376g14033g16311g1927g6980g4410g12466g17199g1025g980g1135g19602g20076. g266.g17148g6980g994g2524g6980g451g12651g7427g3534g7424g4462g10714 g3835g11
5、221g11352g6984g6980g6365g4439g1867g7389g3252g6980g11352g5785g1929g2460g2499g2010g1038g17148g6980g994g2524g6980g1016g12879. g980g1022g3835g11221g11352g6984g6980g712g3926g7536g19512g11141g2656g4439g14270g17535g1209g3818g8821g7389g1866g1194g8503g3252g4388g712g2029g12228g8504g6980g1038g17148g6980g6122g1
6、3044g6980g712g2554g2029g712g12228g1038g2524g6980. g7186g9994g7121g7094g993g7171g17148g6980g1075g993g7171g2524g6980g7272g7171g7380g4579g11352g1000g7171g5811g980g11352g1610g17148g6980. g4462g10714g726g708g8503g6984g6980g11352g5811g980g2010g16311g4462g10714g712g2460g2495g12651g7427g3534g7424g4462g10714
7、g709g1231g1321g3835g11221g11352g6984g6980Ag18129g2499g1209g2010g16311g6116g17148g6980g11352g1068g12227g712g14521g993g16757g17837g1135g17148g6980g11352g8437g5219g712g2029g17837g12193g17148g3252g4388g2010g16311g15932g12046g5347g7171g5811g980g11352g712g17839g13792Ag2499g1209g1901g6116g7643g1946g2010g16
8、311g5347g726 nanaapppA L2121= g708*g709. g1866g1025inpppp ,21g709g712g11004iipg19512g12573g5347nnnanaappppppLL21122121= g1016g12483g5483g726 .11111111niiniiniinipppppppLLLL+= 2 g8504g5347g7186g9994g993g6116g12447g708g3252g5050g12483g7171ip g11352g1505g6980g712g13792g2503g12483g993g7171g709.g6937ii =
9、 g4557g980g2011i=1g7122g712g258g712ng3355g6116g12447.g5811g980g5627g5483g16789. g6524g16782g726g708g2524g6980g11352g3252g4388g1022g6980g16757g12651g1856g5347g709g14521nnpppAL2121= g1038g7643g1946g2010g16311g5347g712g2029Ag11352g6164g7389g3252g4388g708g2265g63361g2656Ag7424g17535g709g11352g1022g6980g
10、12573g1122 ).1()1)(1(21+n L g708g12628g16772g1038=+nii1)1( g709 g17837g7171g3252g1038g712g1068g122272222212111()1()1(21nnpppppppp + LLL)nnp+L g11352g8611g980g20045g18129g7171Ag11352g980g1022g3252g4388g712g6937g1861g7389=+nii1)1( g1022. g4462g10714g726g17148g6980g11352g1022g6980g7171g7092g12363g11352
11、. g463g16789g7138g464g1563g4462g17148g6980g11352g1022g6980g2494g7389g7389g19492g3822g1022 ,21 nppp L g13783g4531g6984g6980 .121+=npppa L g11013g11221a g1000g2460g993g14033g15999 ),2,1( nipiL= g19512g4625g712g1122g7171g11013g12651g7427g3534g7424g4462g10714g11705g712a g5529g14033g1901g6116g980g1135g17
12、148g6980g11352g1068g12227g712g13792g17837g1135g17148g6980g5529g5334g1122 ),2,1( nipiL= g712g17837g994g1563g4462g11695g11474.g6937g17148g6980g7389g7092g12363g3822g1022. 赛题精讲 g13751g714g16786g8503g6984g6980dg993g12573g11222g7125g71213.g16789g7138g3324g19610g2524g7912g7125g71213g712dg793g1025g2499g1209
13、g6226g2052g1016g1022g1815g13044a,bg712g1363g5483abg7131g993g7171g4448g1852g5191g7053g6980. g271g130727g2077IMOg3275g3477g272 g463g16311g464g11013g11222g1045g7131=32g7122g10413g7131=52g7125g10413g7131=82g712g3252g8504g712g2494g19668g16789g71382dg7131g7125dg7131g71213dg7131g1025g14279g4581g7389g980g10
14、22g993g7171g4448g1852g5191g7053g6980. g11004g2465g16789g8873g712g1563g16786g4439g1216g18129g7171g4448g1852g5191g7053g6980g712g1208 2dg7131=x2g311 5dg7131=y2g312 13dg7131=z2g313 x,y,zg281N*g11013g311g11705g712xg7171g3867g6980g712g16786x=2kg7131g712g1122g71712dg7131=g7082kg7131g7092g712g2375d=2k2g7132
15、k+1g712g17837g16840 g7138dg1075g7171g3867g6980.g3252g8504g712g1889g11013g312g712g313g11705g712y,zg3355g7171g1610g6980. g16786y=2m,z=2n,g1207g1849g313g451g314g712g11468g1955g712g19512g12094g5483g7122d=n2g713m2=(n+m)(ng713m)g712g1186g13792n2g713m2g1038g1610g6980g712ng712mg5529g2528g7171g1610g6980g712g
16、1122g7171m+ng994mg713ng18129g7171g1610g6980g712g17837g76912dg4613g71714g11352g1505g6980g712g2375dg1038g1610g6980g712g17837g994g990g17860dg1038g3867g6980g11695g11474.g6937g2641g20076g5483g16789. g13752g714g16786 ag451bg451cg451dg1038g3867g6980g712 bcaddcba = g1186g13792addadacbadmk 4)()(,(22+=+ g1122
17、g7171g3252g103822)(4)( cbbccb +=+ .g1889g11013 ,222,2,22ababbcadbcadkmmk= g2499g5483 g3252g13792 )()2(2 abababmkm+=g311 g7186g9994g712 abab + , g1038g1610g6980g712 abmk 2 g1038g3867g6980g712g5194g1000 abab + g2656 g2494g14033g980g1022g10384ng3423 g1610g6980g712g980g1022g10384n+2g3423g1610g6980g708g2
18、554g2029g4439g1216g11352g5058g5224g10384g11352g1505g6980g712g9994g13792g4439g1216g11352g5058g12573g11222ag993g71714 g11352g1505g6980g709g712 g3252g8504,g3926g7536g16786 feabmk=2 g712g1866g1025e,fg1038g3867g6980g712g18039g1052g11013g311g5347g2462 abab + , g11352g10317g5627g4613g7389 g708g265g709=+.2,
19、21fabeabmg6122g708g266g709=+.2,21eabfabmg11013 fabababefmk222 +nmmmlfnffffnffnftg g6164g1209g2029g6937=+.,2);)(2)(,0,2,3;,11g1866g1194g5785g5430g3926g990g1000g1038g3867g6980g5415g1038g3867g6980g7114g5415tgtgtmtnnlmmng13755g714g16786ng7171g8503g6984g6980g712kg7171g993g4579g11222g11352g6984g6980.g1680
20、9g16789g726kn g2499g15932g12046g6116ng1022g11468g13499g3867g6980g11352g2656. g463g16789g7138g464g4557kg11004g6980g4410g5414g13447g8873. g5415k=2g7114g712g3252 ),12(312+= nn L g2641g20076g3324g12447. g1563g16786k=mg7114g6116g12447g712g2375 ,)12()3()1(2nnanaaanm+=+= L g708ag1038g7588g19762g17139g6980g
21、709 g2029 ,)()(2221nnnnannnnannnmm+=+=+g14521g16772 nnnab +=2g708g7186g9994bg1038g19762g17139g1610g6980g709g712g1122g7171 1),12()3()1(21+=+=+=+mknbbbnnbnm即L g7114g712g2641g20076g6116g12447g712g6937g2641g20076g5483g16789. g13756g714g3324g5191g19766g990g1231g11023g980g7477g6164g7389g20042g9869g18129g7
22、171g7696g9869g11352g19393g6252g13459g712g5194g1000g2520g14422g11352g19283g11468g12573.g14033g1363g17837g19393g6252g13459g11352g14422g6980g1038g3867g6980g731g16789g7138g1332g11352g13479g16782. g271g2669g3342g2227g3316g3888g2121g3127g3275g3477g272 g463g16311g464g1208g12538g2524g20076g16786g7477g1226g1
23、1352g19393g6252g13459g1038A1A2g258AnA1g712g2029g6164g7389g20042g9869iA g11352g3364g7643g708iiyx , g709g12538 g2524 ).,2,1(, niZyxiiL= g5194g1000 CniCYXii,2,1(22L=+ g1038g980g3278g4462g11352g8503g6984g6980g709g712g1866g1025 ),2,1(,111111yyxxniyyYxxXnniiiiii=+L g2029g11013g5062g11705g7389 =niiX1,0 g31
24、1 =niiY1,0 g312 5 2222222121 nnYXYXYX +=+=+ L g313 g993g3964g16786iiYX 和 g1025g14279g4581g7389g980g1022g1038g3867g6980g708g3252g1038g16786 mtXimi,2= g7171g6363g6980g7380g4579g11352g712tig1038g3867g6980g712g110042mg19512g6164g7389g11352g6980g2530g712g1866g2842g1185g9397g17287g311g451g312g451g313g5347
25、g709g712g1122g7171g4439g1216g11352g5191g7053g2656Cg2494g14033g10384k+1g61224k+2. g5415 C=4k+2 g7114g712g11013g313g11705g712g6164g7389g6980g4557iiYX 与 g18129g5529g20047g7171g3867g6980g712g3252g8504g712g7693g6466g311g451g312g5347g11705g712ng5529g1038g1610g6980. g5415 C=4k+1 g7114g712g11013g313g11705g7
26、12g6164g7389g6980g4557iiYX 与 g18129g5529g980g3867g980g1610g712g13792g11013g311g11705g712Xig1025g1038g3867g6980g11352g7389g1610g6980g1022g708g16786g1038 2ug709g712g1325g991g11352 ng7132u g1022g1038g1610g6980g708g994g1055g4557g5224g11352 Yig5529g1038g3867g6980g709g712g1889g11013g312g11705g712g17837g12
27、193g3867g6980g11352Yig1075g5224g7389g1610g6980g1022g708g16786g1038 un 22 = g709g712g6937 )(2 += un =g1610g6980. g13520g990g6164g17860g712g993g14033g1328g1998g9397g17287g20076g16786g7477g1226g13792g7389g3867g6980g1022g14422g11352g19393g6252g13459. g13757g714g8726g1998g7380g4579g8503g6984g6980ng712g13
28、63g1866g5700g7389144g1022g993g2528g11352g8503g3252g6980g712g1000g1866g1025g738910g1022g17842g13505g6984g6980. g271g130726g2077IMOg4167g3882g3477g709 g463g16311g464g7693g6466g20076g11458g16213g8726g712ng717110g1022g17842g13505g6984g6980g12227g11352g1505g6980g712g3252g13792g5529g9994g14033g159992g7123
29、g712g258g71210g6984g6980.g11013g11228=23g7129=32g71210=2g1045g712g6937g1866g7643g1946g2010g16311g5347g1025g712g14279g4581g2559g738923g10332g1035g1037g11352g3252g5347g712g3252g8504g712g14521g16786 ,11753254321L=n g2029 .1,1,2,34321 g11013 ,144)1)(1)(1)(1(4321=+ L g13792 ,482234)1)(1)(1)(1(4321=+ g693
30、7g7380g3822g17836g7389g980g1022,2),5(0 jjj g1000g1038g1363ng7380g4579g712g14270g9994g4464g2474 .025 g11013 )0(144)1)(1)(1)(1()0(144)1)(1)(1)(1)(1(54321554321g7114g6122g7114 =+=+ ,g13783g15397144g11352g2499g14033g2010g16311g712g5194g8616g17751g11468g5224ng11352g3835g4579g712g2499g11705g2524g1058g1621
31、3g8726g11352g708g7380g4579g709,2,521= ,1543= g6937g6164g8726g11352 .11088011753225=n g991g19766g16774g980g1022g3324g6363g4462g19610g2524g1881g11352g256g2524g6980g257g11352g19394g20076.g17837g12193g2524g6980g994g17902g5132g11352g2524g6980g7389g2318g2047g712g20076g1025g11352g256g13044g1815g13044g257g7
32、171g6363g3324g16825g19610g2524g1881g11352g13044g6980g712g1075g994g17902g5132g11352g13044g6980g7389g2318g2047. g1375 8g714g16786n2g1038g13485g4462g11352g8503g6984g6980g712 .,1*NkknVn+= g16809g16789g726g4396g3324g980g6980 ,nVr g17837g1022g6980g2499g11004g993g2494g980g12193g7053g5347g15932g12046g6116g6
33、980g19610Vng1025g13044g1815g13044g11352g1068g12227. g271g130719g2077IMO g3275g3477g272 g463g16789g7138g464g11013g1122Vng1025g11352g6980g18129g993g4579g1122),2(1 + nng3252g13792nVnnnn )12()1(,)12(,)1(22. g7186g9994 )12()1(,)1(2 nnn g7171Vng1025g11352g13044g1815g13044.g2460g14521g7082ng7131g7092g993g7
34、171Vng1025g13044g1815g13044g712g2029g7389 ,)12()1()1(,12=+ nbnanba g1363 g11013g8504g7389 ,44 baabnn += g1122g7171 ,31 ab g1186g137926 b=1,a=1;b=1,a=2,b=1,a=3,g4557g8504g4613g7389 ,8,28,2=n g6937 n=8.g17837g16840g7138 g712g54152)12(,8 nn 时 g4613g7171 Vng1025g13044g1815g13044. g5415 )12)(1()12()1(,.)
35、12()1(,82222= nnnnrVrnnrnng1000g7186g9994g1208g7114 ).12)(1( nn g5415n=8g7114g712g73891089=136g1048+1=9g104121=33g10433g712g137929g712121g71233g281V8. g13520g990g11705g712g2641g20076g5483g16789. g13759g714g5062g11705ng3052g712g8726g16789g726g3926g7536 nkk +2g4557g1122g6984g6980kg70830nk g709g7171g17
36、148g6980g712g2029 nkk +2g4557g1122g6164g7389g6984g6980 )20( nkk g18129g7171g17148g6980. g271g130728g2077g2711987g272g1715g1949g3316g3888g813g2431g2852g2232g3275g3477 6g272 g463g16789g464g16786 m g7171g1363 nkk +2g1038g2524g6980g11352g7380g4579g8503g6984g6980.g14521nmmpnmn+ g3252g1038 nmpmp + )1()1(2
37、g1038g2524g6980g712g6164g1209 .12,1 + mpmmp g11013 ,122nmmpm + g2375 ,01332+ nmm g11013g8504g5483363123 nnm 且, g712 g11013g5114g1325g19512g8873g7389 =+=nnrrrb L g712g13792 b g1038g7389g19492g6980g712g3252g8504g712g5529g7389g980g1022g7380g3822g993g17241g17819bg11352g8503g6984g6980ng4396g3324g712g1363
38、g5483 0nr g712g13792 01=+nr g712g6937g11013g4462g10714g1120g5483g726 ).,(),(),(),(11211babrrrrrrrrnnnnn=+（）L g1375g3926g712g7083960g712756g709=g708756g712180g709=g708180g71236g709=36. g1867g1319g12651g5347g3926g991g726 5g708q1g709 3960g708ag709 756g708bg709 4g708q2g709 3780 720 180g708r1g709 36g708r
39、2g709 5g708q3g709 180 0g708r3g709 g11013g4462g1053g2656g990g17860g8726g8873g993g19602g5483g1998g7380g3835g1856g13434g6980g2656g7380g4579g1856g1505g6980g11352g3926g991g5627g17148g726 g7081g709 ),(),(, bambmamNm = 则 . g7082g709g16786 bac ,为 g11352g1856g13434g6980g712g2029 .),(),(cbacbca= g10317g2047g3
40、332g712g14521 1),(),( =cbcabac 则 . g7083g709g16786naaa ,21L g7171g1231g5859ng1022g8503g6984g6980g712g3926g7536nnncaccaccaa =),(,),(,),(1332221L g712 g2029nncaaa =),(21L . g325221121111|,|,|,|,|,| nnnnnnnnnnnnccacccacccac 故而 g712g3926g8504g12879g6524g5483g1998nc g14033g69845 g19512nnncaaa 于是,11Lg7171
41、g4439g1216g11352g980g1022g1856g13434g6980.g2460g16786naaac ,21L为 g11352g1231g980g1856g13434g6980g712g202921|,| acac g712g3252g137922| cc g712g2528g10714g2499g6524g19983| cc g712g3926g8504g12879g6524g7380g2530g2499g5483ncc | . g1122g7171nccc | g712g6937nc g7171g7380g3835g1856g13434g6980. g7084g709g14
42、521 cba =),( g712g2029g980g4462g7389g6984g6980 yx和 g712g1363g5483 cbyax =+ . g10317g2047g3332g712 = 1),( ba g4396g3324 1, =+ byaxyx 使得 . g17837g2499g11013g17771g17728g11468g19512g8873g11352g313g5347g17882g6524g13792g5483 byaxrcn+= . g7085g709g14521 ),(),(,1),( bcbacba = 则 . g7086g709 Nba, g311 )(,=
43、Nkbakbkak g727 g312 bam ,为 g11352g1231g980g1856g1505g6980g712g2029 mba |, g727 g313 abbaba =,),( g712g10317g2047g3332g712g14521 abbaba = ,1),( 则 . g311g2499g11013g313g11464g6521g5483g2052g712g312g2499g11013g7380g4579g1856g1505g6980g4462g1053g5483g712g313g7693g6466g311g451g312g5347g11705g712 =,),( ba
44、ba =+=niniiiiiabppii11),min( . g7087g709g16786naaa ,21L g7171g1231g5859ng1022g8503g6984g6980.g14521 =,1332221 nnammammaa L mng712g2029nnmaaa =,21L . g17837g7171g980g1022g8726g3822g1022g6984g6980g11352g7380g4579g1856g1505g6980g11352g7053g8873.g4439g2499g11004g16789g7138g313g12879g1296g11352g7053g8873
45、g7481g16789g7138. g267.g7053g5142g19394g20076 g980g1022g8503g6984g6980ng14033g2554g15932g6116mg1022g6984g6980g11352kg8437g7053g2656g11352g19394g20076g12228g1038g7053g5142g2656g19394g20076.g10317g2047g3332g712g5415 1=m g7114g12228g1038kg8437g7053g19394g20076g712g5415 2=k g7114g712g12228g1038g5191g705
46、3g2656g19394g20076. g14033g15932g1038g7588g6984g6980g11352g5191g7053g11352g6980g12228g1038g4448g1852g5191g7053g6980.g12628g12228g5191g7053g6980g712g1863g1122g5191g7053g6980g712g7138g7186g7389g3926g991g980g1135g12628g2345g11352g5627g17148g2656g13479g16782g726 g7081g709g5191g7053g6980g11352g1022g1313g
47、6980g4395g2494g2499g14033g71710g7121g7124g7125g7126g7129. g7082g709g1610g6980g11352g5191g7053g6980g71714g11352g1505g6980g712g3867g6980g11352g5191g7053g6980g159998g19512g13251g712g2375g1231g1321g5191g7053g6980g159994g19512g11352g1325g69806 g2494g14033g71710g61221. g7083g709g3867g6980g5191g7053g11352g
48、2325g1313g6980g4395g7171g1610g6980. g7084g709g2325g1313g6980g4395g7171g3867g6980g11352g5191g7053g6980g11352g1022g1313g6980g980g4462g71716. g7085g709g993g14033g159993g6984g19512g11352g6980g11352g5191g7053g159993g19512g13251g712g14033g159993g6984g19512g11352g6980g11352g5191g7053g14033g159993g6984g19512.g3252g13792g712g5191g7053g6980g159999g19512g11352g1325g6980g10380g7121g7124g7127g712g1000g8504g5
