AMC10美国数学竞赛真题 xx年.docx
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AMC10美国数学竞赛真题 xx年.docx
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AMC10美国数学竞赛真题xx年
AMC10美国数学竞赛真题xx年
Problem1
Themedianofthelist
is
themean?
Solution
.Whatis
Problem2
Anumberismorethantheproductofitsreciprocalanditsadditiveinverse.Inwhichintervaldoesthenumberlie?
Solution
Problem3
Thesumoftwonumbersis.Supposeisaddedtoeachnumberandtheneachoftheresultingnumbersisdoubled.Whatisthesumofthefinaltwonumbers?
Solution
Problem4
Whatisthemaximumnumberforthepossiblepointsofintersectionofacircleandatriangle?
Solution
Problem5
Howmanyofthetwelvepentominoespicturedbelowhaveatleastonelineofsymmetry?
Solution
Problem6
Let
and
denotetheproductandthesum,respectively,ofthedigitsof
and
.Suppose
isa
?
theinteger.Forexample,
two-digitnumbersuchthat.Whatistheunitsdigitof
Solution
Problem7
Whenthedecimalpointofacertainpositivedecimalnumberismovedfourplacestotheright,thenewnumberisfourtimesthereciprocaloftheoriginalnumber.Whatistheoriginalnumber?
Solution
Problem8
Wanda,Darren,Beatrice,andChiaretutorsintheschoolmathlab.Their
scheduleisasfollows:
Darrenworkseverythirdschoolday,Wandaworkseveryfourthschoolday,Beatriceworkseverysixthschoolday,andChiworkseveryseventhschoolday.Todaytheyareallworkinginthemathlab.Inhowmanyschooldaysfromtodaywilltheynextbetogethertutoringinthelab?
Solution
Problem9
ThestateincometaxwhereKristinlivesisleviedattherateof
ofannualincomeplus
ofanyamountabove
ofthefirst
.Kristinofher
noticedthatthestateincometaxshepaidamountedtoannualincome.Whatwasherannualincome?
Solution
Problem10
If,
andarepositivewith
is
and
Solution
then
Problem11
Considerthedarksquareinanarrayofunitsquares,partofwhichisshown.The?
rstringofsquaresaroundthiscentersquarecontainsunitsquares.Thesecondringcontainsunitsquares.Ifwecontinuethisprocess,thenumberofunitsquaresintheringis
Solution
Problem12
Supposethatistheproductofthreeconsecutiveintegersandthatby.Whichofthefollowingisnotnecessarilyadivisorof?
isdivisible
Solution
Problem13
Atelephonenumberhastheform,whereeachletterrepresentsadifferentdigit.Thedigitsineachpartofthenumbersareindecreasingorder;thatis,,,and.Furthermore,,,andareconsecutiveevendigits;,,,andareconsecutiveodddigits;and.Find.
Solution
Problem14
Acharitysellsbenefitticketsforatotalof.Someticketssellforfullprice(awholedollaramount),andtherestsellsforhalfprice.Howmuchmoneyisraisedbythefull-pricetickets?
Solution
Problem15
Astreethasparallelcurbsfeetapart.Acrosswalkboundedbytwoparallelstripescrossesthestreetatanangle.Thelengthofthecurbbetweenthestripesisfeetandeachstripeisfeetlong.Findthedistance,infeet,betweenthestripes?
Solution
Problem16
Themeanofthreenumbersismorethantheleastofthenumbersandlessthanthegreatest.Themedianofthethreenumbersis.Whatistheirsum?
Solution
Problem17
Whichoftheconeslistedbelowcanbeformedfromaradiusbyaligningthetwostraightsides?
sectorofacircleof
Solution
Problem18
Theplaneistiledbycongruentsquaresandcongruentpentagonsasindicated.Thepercentoftheplanethatisenclosedbythepentagonsisclosestto
Solution
Problem19
Patwantstobuyfourdonutsfromanamplesupplyofthreetypesofdonuts:
glazed,chocolate,andpowdered.Howmanydifferentselectionsarepossible?
Solution
Problem20
Aregularoctagonisformedbycuttinganisoscelesrighttrianglefromeachofthecornersofasquarewithsidesoflength.Whatisthelengthofeachsideoftheoctagon?
Solution
Problem21
Arightcircularcylinderwithitsdiameterequaltoitsheightisinscribedinarightcircularcone.Theconehasdiameterandaltitude,andtheaxesofthecylinderandconecoincide.Findtheradiusofthecylinder.
Solution
Problem22
Inthemagicsquareshown,thesumsofthenumbersineachrow,column,anddiagonalarethesame.Fiveofthesenumbersarerepresentedby,,,,and.Find.
Solution
Problem23
Aboxcontainsexactlyfivechips,threeredandtwowhite.Chipsarerandomlyremovedoneatatimewithoutreplacementuntilalltheredchipsaredrawnorallthewhitechipsaredrawn.Whatistheprobabilitythatthelastchipdrawniswhite?
Solution
Problem24
Intrapezoid
,
and
areperpendicularto,and.Whatis
with
?
Solution
Problem25
Howmanypositiveintegersnotexceeding?
aremultiplesoforbutnot
Solution
Problem1
Themedianofthelist
is
themean?
Solution
.Whatis
Problem2
Anumberismorethantheproductofitsreciprocalanditsadditiveinverse.Inwhichintervaldoesthenumberlie?
Solution
Problem3
Thesumoftwonumbersis.Supposeisaddedtoeachnumberandtheneachoftheresultingnumbersisdoubled.Whatisthesumofthefinaltwonumbers?
Solution
Problem4
Whatisthemaximumnumberforthepossiblepointsofintersectionofacircleandatriangle?
Solution
Problem5
Howmanyofthetwelvepentominoespicturedbelowhaveatleastonelineofsymmetry?
Solution
Problem6
Let
and
denotetheproductandthesum,respectively,ofthedigitsof
and
.Suppose
isa
?
theinteger.Forexample,
two-digitnumbersuchthat.Whatistheunitsdigitof
Solution
Problem7
Whenthedecimalpointofacertainpositivedecimalnumberismovedfourplacestotheright,thenewnumberisfourtimesthereciprocaloftheoriginalnumber.Whatistheoriginalnumber?
Solution
Problem8
Wanda,Darren,Beatrice,andChiaretutorsintheschoolmathlab.Their
scheduleisasfollows:
Darrenworkseverythirdschoolday,Wandaworkseveryfourthschoolday,Beatriceworkseverysixthschoolday,andChiworkseveryseventhschoolday.Todaytheyareallworkinginthemathlab.Inhowmanyschooldaysfromtodaywilltheynextbetogethertutoringinthelab?
Solution
Problem9
ThestateincometaxwhereKristinlivesisleviedattherateof
ofannualincomeplus
ofanyamountabove
ofthefirst
.Kristinofher
noticedthatthestateincometaxshepaidamountedtoannualincome.Whatwasherannualincome?
Solution
Problem10
If,
andarepositivewith
is
and
Solution
then
Problem11
Considerthedarksquareinanarrayofunitsquares,partofwhichisshown.The?
rstringofsquaresaroundthiscentersquarecontainsunitsquares.Thesecondringcontainsunitsquares.Ifwecontinuethisprocess,thenumberofunitsquaresintheringis
Solution
Problem12
Supposethatistheproductofthreeconsecutiveintegersandthatby.Whichofthefollowingisnotnecessarilyadivisorof?
isdivisible
Solution
Problem13
Atelephonenumberhastheform,whereeachletterrepresentsadifferentdigit.Thedigitsineachpartofthenumbersareindecreasingorder;thatis,,,and.Furthermore,,,andareconsecutiveevendigits;,,,andareconsecutiveodddigits;and.Find.
Solution
Problem14
Acharitysellsbenefitticketsforatotalof.Someticketssellforfullprice(awholedollaramount),andtherestsellsforhalfprice.Howmuchmoneyisraisedbythefull-pricetickets?
Solution
Problem15
Astreethasparallelcurbsfeetapart.Acrosswalkboundedbytwoparallelstripescrossesthestreetatanangle.Thelengthofthecurbbetweenthestripesisfeetandeachstripeisfeetlong.Findthedistance,infeet,betweenthestripes?
Solution
Problem16
Themeanofthreenumbersismorethantheleastofthenumbersandlessthanthegreatest.Themedianofthethreenumbersis.Whatistheirsum?
Solution
Problem17
Whichoftheconeslistedbelowcanbeformedfromaradiusbyaligningthetwostraightsides?
sectorofacircleof
Solution
Problem18
Theplaneistiledbycongruentsquaresandcongruentpentagonsasindicated.Thepercentoftheplanethatisenclosedbythepentagonsisclosestto
Solution
Problem19
Patwantstobuyfourdonutsfromanamplesupplyofthreetypesofdonuts:
glazed,chocolate,andpowdered.Howmanydifferentselectionsarepossible?
Solution
Problem20
Aregularoctagonisformedbycuttinganisoscelesrighttrianglefromeachofthecornersofasquarewithsidesoflength.Whatisthelengthofeachsideoftheoctagon?
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