历年华罗庚金杯试题

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40•••11᦮◀ḄᦪḄ"#$Ḅᦪ%&ᓝ#ᓟAḄ“ᐰᦪ”"V*ᑮsQD஺.7+/-.1+3/=3+./300Ḅᐰᦪᨵ;111Ḅᦪ஺1,4,9,16,25,36,49,64,81,100,121,144,169,196,•••ᑁḄᦪᨬᜧ9,225,256,289.UḄ1785஺2ᑁ8஺300"5ᯠᦪḄ•••6᦮◀Ḅ5ᯠᦪᏔᦪᦪ3Ḅᦪ81+2+3+…+300=.l+:0°/x300=451501+7+3+=11+•••ᑁ4஺45150-1785=43365.7+8+4=19:;ᒕIḄ5ᯠᦪḄ43365஺:;ᡠ<Ḅ19஺.ὃ/q☢ḄA<300"5ᯠᦪḄᵨḄuvw.=/>"⚪@AὃCD9,11,6᦮◀ḄEF5ᯠᦪḄxUᓝᑖ¡៉Ḅuvf£Ḅ¤¥Ḅ¦஺EIJ;ᱯH஺I☢KL2ஹ4ஹ5ஹ7ஹ8᦮◀Ḅ5ᯠᦪḄᱯH;᝞Qn5ᯠᦪª«᝞Q5ᯠᦪA5ᯠᦪa᦮◀ᡃUVᑏXalA.I☢Vᵨ1+/?1+2+34---F.n-2/+./1-1/+n=x஻஺>"[\ᩭ^_`⚪——bAḄ"ᦪ0ஹ2ஹ4ஹ6ஹ8>d"ᦪAḄe"f2|Ahᵨ>"°±&¦bAḄᨬijᦪ4Ḅᦪf4IAh1+2+3+…+298+299+300,bᯠ²⌲"´µ⌕¶*D·bAḄ"ᦪ0ᡈ5fᑣ5IAhᵫ¹uvwxUº»᧕½¾ᑮ;b&ᣵAḄ"ᦪi*ᑮḄoᦪ#AḄ"ᦪḄ2Ḅ,<12+22+32+-+152+162+1727Ḅᦪf7IAhᨵ¿ᨵ°±ᕖbAḄᨬiEᦪ8Ḅᦪfh8IA»:ᫀ;ᨵ·>Vq☢>rstuvwxU⌕z{|}஺12+22+32+-.+஻2=஻.஻+1*)2஻+1*5.ᡃU~⍝:9=3X3,16=4X4,>,9ஹ16“ᐰ6ᦪ”ᙠ300"5ᯠᦪA&ᣵᡠᨵḄ“ᐰᦪ”ᒕIḄ5ᯠᵨ>"°±¦300ḄᐰᦪḄº»᧕Ḅ;ᦪḄuI2+22+32+---+172=17x18x35=51x35=1785.ᑖ᪆/>"⚪@⌕ᙢL300Ḅ5ᯠᦪ6

41>"°±ḄÂÃ⌕ᵨe"°±n3+3n2+3"+1=Ê+3(Ë+2?+32+■■+n2)+^-^-+n(x+1)3=x3+3x2+3x+12V{ÄD;ᙠ>"°±AᡃUÅÆᵨ1,2,3,n-2,n-1,n&Çᣚx,*ᑮn"ɱ>"ɱA{Ä*ᑮ23=13+3•12+3•1+112+22+32+…+n233=23+3•22+32+1_n3+3n2+3nn2+nn43=33+3•32+3•3+1~323(n-1)3=(n-2)3+3(n-2)2+3(n-2)+1_2n3+3/12+nn3=(n-1)3+3(n-1)2+3(n-1)+16(n+1)3=n3+3n2+3n+1Ì>n"ɱµÍᩭª«_n(n+1)(2஻+1)ɱḄÎÏ=13+23+33+…+n3+(n+1)3,6ɱḄÐÏᒹÒÓᑖ;ÄqḄÂÃéAᵨᑮêA£ᦪḄer~ëuvwx{ᨵÔ1ÕijḄ13+23+33+…+(n—1)3+n3hìíᜐª«{Ä&ïᦟ5ñḄᦪòὁô.Ô2ᑡḄ3(12+22+33+…+n2),×ὅÙÚÛᑮÜ\6.᝞õö÷13ᔖùú÷9ᔖùḄöûüýþḄÿÝ⌕ÂÃḄ°±ḄÎÏhᣵ2ᔖḄᯠᢚᔁᡂᘤ஺ᘤÔEᑡḄ3(1+2+3+…+n),>V3.Þ+1)Ḅᔖ|29ÔᑡḄn"1ßµbᯠ*n.᪷áµஹ%Ḅᭆã{Ä*ᑮ5+1)3=ä+3(12+22+3?+…+”2)+3"஺æ)+஻(ᑖ᪆)%᩽ᐸ(ᓫḄ*+⚪-./0123456ᙢ89:;஺èV(<)ᘤḄ=☢(13-4)X(9-4)=45(?ᔖ),AB2ᔖᡠDᘤḄ

4245X2=90(ᔖ).ToMNᘤḄ90ᔖ஺(<)•••/mḄXᦪḄ=1764W0,7.ᙠQRSTU/Q%RVᑮḄXᦪᡈὅ“0"(ῃ☰)ᡈhm/RQUḄXᦪᨵ“0”“10”.ὅ`ab10Ḅdᯠᦪ஺ᵬஹghiSTᕒᔜQl5R/m5R:/RQUḄXᦪ41764Ḅµ¶VᑮḄXᦪḄ41764,oᵬḄpXᦪqg4X஺8ᵬஹg1764=1X2X2X3X3X7X7,ḄpXᦪ஺·¸Xᦪ`ab10Ḅdᯠᦪ.•.¹ᨵhR7Xᐸº3RḄ(ᑖ᪆)⚪rᨵ%tḄuvuᙠ⚪rḄwxᩩz{⌕8Xᦪ1•2•2•3•3µ¶஺Ḅ:;}~Ḅᐵ`஺⍗ᑮ⌕?☟⌕U᝞;ᨬ½Ḅµ¶1,§᜜hµ¶4ஹ9ᡈὅ6ஹ6®ᙢᑖ᪆⚪rUḄᩩz஺᝞;ᨬ½Ḅµ¶2,§᜜hµ¶2,9ᡈὅ3ஹ6®⚪rᡃN᝞;ᨬ½Ḅµ¶3,§᜜hµ¶3ஹ4஺/Q%RḄXᦪ5ᑡ11ᦪUḄ%µ¾hm5RḄXᦪᨵ5¦5N0,1,2,3,4,5,6,7,8,9,10஺7,7,1,4,9,=28®ᵬஹg5RpXᦪᦪḄ1764W0,ᙠᵬஹg5RVᑮ7,7,1,6,6,=27®ḄXᦪᨵ“0”“10”,7,7,2,2,9,=27®1764=1X2X2X3X3X7X77,7,2,3,6,=25®ᵫ5RḄXᦪ¡9ᩭḄ£¤x/ᨵhRUḄXᦪ4“7”,7,7,3,3,4,=24®¥¦x§᜜3RḄXᦪ5ᦪ•.•ᵬஹgḄpXᦪ-±4,ᵬḄpXᦪ1,2,2,3,3.••ᵬḄpXᦪ24,gḄpXᦪ28஺©b〉«Ḅᑖ¬}-¡VᑮḄὅ`uᑖ᪆9¦5ḄMNᵬஹgḄpXᦪᑖ¿24ஹ28oᨵ5N(Á)19901Âᓝ%ÄÅSÆᙠᡃÇ✌4ᓅÊËÌᡃU(1)1,4,9®ÇmÍB¾ÎᑮÏÐᐰÇḄ123«ᯠÒÓÔÕÖ஺Bl×(2)1,6,6®ØzÙÂÚÄÛÜÝÞßàᕒÆá⌕âã{äqÝᨵᐵḄ(3)2,2,9®⚪råÝæᑖḄÂçஹèஹᓝé⚪ᙠ᪵ḄrḄâ9ᩭḄ஺(4)2,3,6®8.ëUᨵ6ì9ᩩí.%ᵬî¥Aì9ð⌕Ḽ(5)3,3,4,òãᩩíóᑮFì஺ÌõUö%ìᡈö--ᩩí5©b1ø஺☢ஹ⌕᪷°ᵬஹgḄpXᦪ}±4ᩩzᓽ¦89:;ᵬîᨬᨵ`öḄùú

43ABCᑖ᪆!
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44ᵫ⚪7⌕SḄ᦮ᦪ¤ᑖᡠ2⌕¡⍝Sᙠ¶F᦮ᦪ·AS>:'=165ÀÁS<Õ=165—z2;஺¸¹ᙠSḄᑖºᨵ12ᑖᦪ³E;»¼½~11212¾JᑖᦪḄᦪ»165ᡃ[1111Ḅ᦮ᦪ¤ᑖC165o>------>------->•••>------1980198119821991Öὃ!a☢Ḅ
Rp⌕Cqᵨ;×ஹØḄÖ1111ÖÙᨵᵨ஺Ú⚪RCᵨᩭ®yᦪÒÛ஺☢CFḄ⚪C-----+-------+-------+•••+-------<1980198119821991ÜὅNOÞß1111121----------1------------1----------+----F19801980198019801980165)1*8¡S=1-----------------—,SḄ᦮ᦪ¤ᑖ஺ÀÁÂ'+Â'+-11+•••+>1980198119821991111112)2*Iᙠ☢váḄT᪾RãaWXḄ'Nᯠᦪäváᡂ'+---+----+•••+19911991199119911991å121111_13tt᪵¡⍝SḄᑖᙠ——°'·zÆ¡⍝Sᙠ16519911651+n3+æ5+æ241OQ11111.çKἔ᱄ḄKêCYëḄKêḄ6ì஺íKîἔ᱄ḄK°ᓃ=165·zCÉᑮ;7Ḅ»1212êïCYëḄKêḄ5ì஺ð%íK2îἔ᱄ḄKêïCYëḄK
!᪷Ì'''ᑖᦪÎᑖÏ_ÐÑᑖÐᜧᑖᦪÒÐY:êḄ4ì஺ἔ᱄çKCEJñò᝞ᑖÏ_ÐÑᑖÐYᑖᦪÒÐᜧ஺<ᳮ2¡⍝ூᑖ᪆!ÎYëô'õஹἔ᱄föᨵ;Kê÷øḼἔúF1-11ûKêḄüἔ᱄fYëḄKêḄýÒ⌲ÿ
+----F—<12x——19801981198219911980165ᢝ஺…1111112“ἔ᱄Ḅ
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45ூ59ἔ᱄Ḅ:
Ḅᜧ.Ḅ.2ᵫ4WxW16-gx=4yF⍝x1h4,8,12,16஺•.•=ἔ᱄Ḅ
ḄḄ63
Ḅ5>5Ḅᦪ2Di2x=3y+3z,ᓽy+z=—x,F⍝x3Ḅᦪᡠ?ᳮᵫ3“BCἔ᱄Ḅ
ḄḄ5”"x=12,M6x=72஺“DEB"Cἔ᱄Ḅ
ḄḄ4”F⍝12.j.kḄᐰmnopq6rsஹuvஹwᳫy.⚗0Ḅ{|,4,3Ḅᦪᡠ">ᨵ4noᙠy.⚗0+ᨵᑮᐸUᨵ.⚗5X4X3=600ᑮᑖnoᑮḄ⚗0ஹᦪ᝞⊤HḄᦪ.460ḄᦪH60,120,--ᔠᳮḄ⌱L60.Mrsuvwᳫrsஹuvuvஹwᳫwᳫஹrsrsஹuvஹwᳫ=1718156652
Ḅ60+5=12NO9.kḄnoᦪ.ἔ᱄Ḅ12X6=72NO9Nᑖ᪆9.ᔠᭆᨵᐵḄ⚪ᐰ1"ᵨ
nPHἔ᱄=72O஺ᦪnFᩭ5ᐸ⌕ᒹ᣸◀ḄZ^.¡᝞NQὃ9STᦪḄUVW1"ᵨᑡZ[ᡈZ[]ḄZ^5rsᑮḄᦪᒹ64ᑖH.⚪0HrsஹuvஹwᳫᑮḄ_=
xO`a=ἔ᱄6xO஺rsஹuvᑮwᳫᑮḄyCἔ᱄Ḅ
ḄḄ5ᡠ"rsஹwᳫᑮuvᑮḄ5x+y=6x+yB|Jx=4yrsᑮuvஹwᳫᑮḄ஺DEz"Cἔ᱄Ḅ
ḄḄ4ᡠ"᝞¢⌕ᐸᑖ⌕᣸◀£᜜Bᑖ஺4x+y+z=6x+y+zᓽ2x=3y+3z¥6+☢Ḅ⍝ᳮ⚪0§5஺•.•ἔ᱄=6xOᙠᐹm5©1"ªᵨ«¬⚪0Ḅᦪ®ᐵ¯°±²,/.6x^100NccH.100daᩭḄf906ᯠ.N59ᐜUᨵ.⚗0ᑮḄnoᦪ´☢.«:x=4y.•.x24NccH¶af9

46ÛÚ,ÏᨵÜ?Ḅᑖ஺13.ាÞhß6ஹ7ஹ8ஹ9᦮◀ḄáâᦪᨵãU.f(ᑖ᪆)hß6ஹ7ஹ8ஹ9᦮◀&ähßÍÒḄᨬ
æᦪ᦮«©y.ᙊᙌᑖ¹T⊤rsஹuvஹwᳫᑮḄnoᦪ,◀Õ.ᨬ
æᦪçCᙠáâᦪᓽi10000ᑮ99999ḄᐸḄéᯠᦪêë`.ᨬ
æᦪḄᦪᨵãU.ᓽ⚪ḄPᫀ6஺“1”⊤»y.⚗0ḄᦪH2½(5)6ஹ7ஹ8ஹ9Ḅᨬ
æᦪ504½“2”⊤»wᳫஹuvᑮrsᨵᑮḄᦪáâᦪᨬ
Ḅ10000,ᨬᜧḄ99999HH6-2=4½..10000஻ஹ424“3”⊤»wᳫஹrsᑮuvᨵᑮḄᦪ,504-504:5-2=3½,hß504᦮◀Ḅᨬ
Ḅáâᦪ“4”⊤»uvஹrsᑮwᳫᨵᑮḄnoᦪ,504X20=10080½H6-2=4½••ï=198”“5”⊤»&ᨵrs⚗ᑮḄᦪH50450417-(2+3+4)=8½.•.hß504᦮◀ḄᨬᜧḄáâᦪ“6”⊤»&ᨵuv⚗ᑮḄᦪH504X198=99792½18-(2+4+4)=8½,áâᦪhß504᦮◀Ḅᨵ“7”⊤»&ᨵwᳫ⚗ᑮḄᦪH(99792-10080)+504+1=1790).15-(2+4+3)=6,PHᨵ179..&ᨵ.⚗0ᑮḄᦪ(Ë)5.⚪0©ð᧕ÕòḄ┯ôH"“hß6ஹ7ஹ8ஹ2+4+3+4+8+8+6=359᦮◀”õä“hß6X7X8X9=3024᦮◀”᪵°Ḅ×¢&Èᨵ4.ᙠy.⚗0+Éᑮᡠ"ᐰknoᦪÓÖ×¢Ḅᑖ஺35+4=3914.÷ëH1-3+5-7+9-11+…-1999+2001P:.kᨵ39no஺(ᑖ᪆)ùM.⚪01"ᨵ3úûüQýH(Ë)ᔠÌᨵᵨḄᦪnᭆ஺ÍḄ.ᵨ⌶ᑖÏ஺ᙠ(1)Ô⚪ᔜᦪḄÿ
ᓽᦪᓃ☢Ḅ5^ᡃÒÓªᵨᑖÏḄQcÔ“Uᨵ.⚗0ᑮᦪᣚᑮ”noᦪᑖ7Ïi4Õ6ÓÖ×¢.ᙠØᑖÏ©ÙËÚ1+5-3+9-7+13-11+-+2001-1999

47"#$%ᙠ1'ᦪḄ)☢+5002Ḅ,஺5X9n=45n(2)./0Ḅᦪ12Ḅᦪᑖ456•••⚪PQlmḄn☢g+122.5(1+5+9+134------1-2001)-(3+7+11+…+1999).••FL`abacḄ☢g+(3)7ᦪḄ89ᩭ$;<+454—13258.8,,----------------=—=1.1(1-3)+(5-7)+(9-11)H------1-(1997-1999)+200188=-2X500+2001=1001FL`abacḄ☢g+1.1஺(>);<=1+(5-3)+(9-7)+(13-11)+…+(2001-1999).()⚪6ḄPQR+ᜧ៉Ḅᝍ᪍ᓛ¢£¤Ḅ᪗¦஺“ᝍ1ᑮ2001ᐳᨵ1001᜻ᦪ1Cᙠᑁᑣᨵ1000᜻ᦪ᪍ᓛ¢”j§ᡂ“ᙠ¨©Ḅᩩ«¬”Ḅ7®¯°±²³´µ0☢ḄFGHᑁI+“᜻ᦪᜧ1L”ᡠ#ᐳᨵ500G¶ᦪ·⍸¹º+ᝍ᪍ᓛ¢»¼£¤᝛ᔾ¿ᦪÀÁµ¶ᦪ·HFGHᑁḄOI+2,ᡠ#᰿Ãὅ஺ᡃ~Å'LLḄ⚪Æ©+W⊤È'᪵ÊË;<=1+500X2=1001⌕ᙠ“°´º”6ÌÍÎÏ஺15.PQRᵫᑁᙊUVW8,᜜ᙊUVW10ḄPᙊQYᡂf[\ᾓ^
)ᐸ6ḄL`abac(▢efᑖ)Ḅ☢gIh.jஹkPᙊQlmḄn☢g+112.5,oFL`abacḄ☢g(ᙊᕜ᳛Jir3.14)஺ெᑖ᪆)'+ᐵv᦮xyfᑖḄᐵzḄ{⚪஺ᡃ~.PᙊQClmḄ☢g$᦮xᙠ'R6ḄPQlmḄ☢g+'᦮xḄᜧḄfᑖ8L`abaclmḄ☢g+'᦮xḄLḄfᑖ'fᑖḄᑭ+᦮x஺⌕ᑮ'ᐵz᧕o>.(>)FᙊQḄ☢g+n(52-42)=9n,᝞PᙊQᨵᔠḄfᑖᑣ~Ḅn☢g+

48,ᑖ᪆/0123360Ḅ24ᦪᑡ
ᩭḄ6789:;кÑÑ⚪#Òᫀ<Ḅ(=>?Ḅ஺ᨵḄ@Aᐜ3+ᦪᑖCD᝞ᢥGᔲI2,4ᡈ8᦮◀ᩭᑖC᪵>Nᓄ+஺᝞PQὃ⇋GᔲI3,91.ÔÕ-+—+—+——+—315356399ᡈ5᦮◀ᓽὃ⇋ᡠᨵUᦪV(ᦪW">ᨬYNᓄᡂ[☢Ḅ$7…ஹ11111\஺ᦪ1]ᵨḄ_ᢈa஺3153563993.bcd☢ᦪ⊤,f᣸h2*1“1ஹ1/A1/1ஹ1/1ஹ1/1ஹ1=Ga—G*+Ga-!*+7'2H*+Gu-ᐭ*+G)ᐭ-77×T523235257279291111ஹ522=ÙÚZ1----×=—211112Ùᑖ᪆yÛÜ×'⚪᝞UÝÔÕ◤⌕ᐜàᑖ¨ᑖáᜧ,2Iᨬ)ä⌕åᑖ.ᡠ#æçèéê0☢'᪵ëᣩḄÕí.2i3i2.îÈ360'ᦪḄåᦪᨵïµð'ñåᦪḄ,+ïµð32_Ú>×360=2X2X2X3X3X5,ᡠ#360ḄòóåᦪIhv2i3i4*ôïõ2"#+ö÷×ôï3,ôï5Ḅg஺432♦»᝞ᡃ~ø☢Ḅ<ù123451+2+4+8X1+3+9X1+5úᡂ,<,<6ḄF/ᦪI+ᙠFGHûᔜr1991᪷kl5hᦪᡠ⊤mḄnopq*r•ᦪtᵫ[dᦪýḄg஺ᵫþ☢Ḅᑖ᪆Cÿ
360Ḅᦪា1949Ḅᦪ஺ᵫᨵ4ᦪḄuhvᙠhxtᵫyᔣ{Ḅ|vᨵ3ᦪᨵ2ᦪᡠḄᦪᦪ,}2ᡃᐜhḄᦪḄᑖஹᑖ&'2,4X3X2=24,!"360ḄᦪḄᦪ஺#$☢360ḄᡠhḄᦪḄᑖஹᑖ&'3,…hḄᦪḄᑖᨵᦪḄ&"'Ḅ&(!"'ஹᑖ&'6.ᵫ>ᑮno"hᔜᦪḄᑖஹ1+2+4+8X1+3+9X1+5ᑖ&'hᦪ1஺=15X13X6=1,170ᐸq>
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