AMC中的數(shù)論問題

1:Remembertheprimebetween1to100:

235711131719232931374143475359616771

7379838991

2:Perfectnumber:

LetPistheprimenumber.ifisalsotheprimenumber.thenistheperfectnumber.For

example:6.28,496.

33

3:Letn=abc9310isthreedigitalinteger.ifH=+Z?+C

ThenthenumberiscalledDaffodilsnumber.Thereareonlyfournumbers:

153370371407

Letn=abed,。10isfourdigitalinteger.ifn=CI4+b4+C4+d*

ThenthenumberiscalledRosesnumber.Thereareonlythreenumbers:

163482089474

4:TheFundamentalTheoremofArithmetic

Everynaturalnumberncanbewrittenasaproductofprimesuniquelyup(oorder.

i=l

5:Supposethataandbareintegerswithb=0.Thenthereexistsuniqueintegersqand

rsuchthat0<r<|b|anda=bq+r.

6:(1)GreatestCommonDivisor:Letged(a,b)=max{dGZ:d|aandd|b}.

Foranyintegersaandb,wehave

gcd(a,b)=gcd(b,a)=gcd(±a,±b)=gcd(a,b-a)=gcd(a,b+a).

Forexample:gcd(15(),60)=gcd(60,30)=gcd(30,0)=30

(2)Leastcommonmultiple:Letlcm(a,b)=min{d€Z:a|dandb|d}.

(3)Wehavethat:ab=gcd(a,b)lcm(a,b)

7:Congruencemodulon

Ifa-b=mq、m00,thcnwecallacongruencebmoduloinandwerewritea=bmodm.

(1)Assumea,b,c,d,m,k6Z(k>0,m0).

Ifa三bmodm,c三dmodmthenwehave

a±c=b+dmodm,ac=bdmodtn,ak=bkmodm

(2)Theequationax=b(modm)hasasolutionifandonlyifgcd(a,m)dividesb.

8:Howtofindtheunitdigitofsomespecialintegers

(l)Howmanyzeroattheendofn!

Forexample,whenn=1(X),LetNbethenumberzeroattheendof100!then

浦riool「100]「100]”4?

N=——+——+——=20+4=24

L5J[25」[125」

(2)Find(heunitdigit.Forexample,when

9:Palindrome,suchas83438,isanumberthatremainsthesamewhenitsdigitsarereversed.

Therearesomenumbernotonlypalindromebut

11

2

=121,

22

2

=484,

11

4

=14641

(1)Somespecialpalindromethatisalsopalindrome.Forexample:

12=1

1『=121

1112=12321

11ll2=1234321

???????

11111111I2=12345678987654321

(2)How(ocreateapalindrome?Almostintegerplusthenumberofitsreverseddigitsand

repeatitagainandagain.Thenwegetapalindrome.Forexample:

87+78=165165+561=726

726+627=13531353+3531=4884

ButwhetheranyintegerhasthisPropertyhasyettoprove

(3)Thepalindromeequationmeansthatequationfromlefttorightandrighttoleftitallsetup.

Forexample:

12x42=24x2112x231=132x21

112x124=1388888831=421x211

Letandaretwodigitalandthreedigitalintegers.Ifthedigitssatisfythe

3.Forthepositiveintegern,let<n>denotethesumofallthepositivedivisorsofnwiththe

exceptionofnitself.Forexample,<4>=1+2=3and<12>=1+2+3+4+6=16.Whatis?<6?>?

(A)6(B)12(C)24(D)32(E)36

8.Whatisthesumofallintegersolutionsto?

(A)10(B)12(C)15(D)19(E)5

c「M6

10Howmanyorderedpairsofpositiveintegers(M,N)satisfy(heequation—=——

6N

(A)6(B)7(C)8(D)9(E)10

l.Letandberelativelyprimeintegerswithand.Whatis?

(A)I(B)2(C)3(D)4(E)5

15.Thefiguresandshownarethefirstinasequenceoffigures.For,isconstructedfromby

surroundingitwithasquareandplacingonemorediamondoneachsideofthenewsquarethan

hadoneachsideofitsoutsidesquare.Forexample,figurehas13diainonds.Howmanydiamonds

arethereinfigure?

?區(qū)向

(A)40式(B)碗(C)585(D)626(E)761

18.Positiveintegersa,b.andcarerandomlyandindependentlyselectedwithreplacementfrom

(heset{1,2,3，…，2023}.Whatis(heprobabilitythatisdivisibleby3?

12911廠13

(A)-（。義(D)—(E)—■

J2727

24.Letandbepositiveintegerswithsuchthatand.Whatis?

(A)249(B)250(0)251(D)252(E)253

5.Inmultiplyingtwopositiveintegersaandb,Ronreversedthedigitsofthetwo-digitnumber

a.Hiserroneousproductwas161.Whatisthecorrectvalueo:theproductofaandb?

(A)116(B)161(C)204(D)214(E)224

23.Whatisthehundredsdigitof?

(A)1(B)4(C)5(D)6(E)9

9.Apalindrome,suchas83438,isanumberthatremainsthesamewhenitsdigitsarcrcvcrscd.Thc

numbersxandx+32arcthree-digitandfour-digitpalindromes,rcspcctivcly.Whatisthesumof

thedigitsofx?

(A)20(B)21(C)22(D)23(E)24

21.Thepolynomialhasthreepositiveintegerzeros.Whatisthesmallestpossiblevalueofa?

(A)78(B)88(C)98(D)108(E)118

24.Thenumberobtainedfremthelasttwononzerodigitsof90!Isequalton.Whatisn?

(A)12(B)32(C)48(D)52(E)68

25Jimstartswithapositiveintegernandcreatesasequenceofnumbcrs.Each

successivenumberisobtainedbysubtractingthelargestpossibleintegersquarelessthanor

equaltothecurrentnumberuntilzeroisreached.Forexample,ifJimstartswithn=55,then

hissequencecontain5numbers:

55

551=6

6-22=2

2-l2=1

1-12=0

LetNbethesmallestnumberforwhichJim'ssequencehas8numbers.Whatistheunitsdigitof

N?

(A)1(B)3(C)5(D)7(E)9

21.Whatistheremainderwhenisdividedby8?

(A)0(B)I(C)2(D)4(E)6

5.Whatisthesumofthedigitsofthesquareof?

(A)18(B)27(C)45(D)63(E)81

25.For.let,wheretherearezerosbetweenthelandthe6.Letbethenumberoffactorsof2inthe

primefactorizationof.Whatisthemaximumvalueof?

(A)6(B)7(C)8(D)9(E)10

24.Lct.Whatistheunitsdigitof?

(A)0(B)1(C)4(D)6(E)8

AMCaboutalgebraicproblems

一、Linearrelations

(I)Slopey-interceptform:y=kx+b(kis(heslope,bisthey-intercept)

(2)Standardform:Ax+By4-C=0

(3)SlopeandonepointP(x(),y()),k{slope)y-=k(x-x0)

(4)Twopoints戶(％,y),P(x,,月)~~~—=~~——

X-XiX-X2Xj-x2

(5)x.y-interceptform:P(a,0),Q(0,〃),(aW0,Z?W0)—+—=I

ab

二、therelationsofthetwclines/,:^x+B[y+C=0J2:A2x+B^y+C=0

(1)4II/?OA&—A?4二o，G&-Ggr0

(1)I]±4。4&一片與二。

三、Specialmultiplicationrules:

cr-b1=(a-b)(a+b\

(a±b)2=a2±lab+b2

/-b^=(a-b){aJ+ab+b')

a3+by=(a+b)(a2-ab+b2)

an-bn=(a-b)(an'1+an~2b+…+abn~2+bn~})(n>2)

a,l+bn=(a+b)(an-l-an-2b+-+(-\)n-2abn-2+(nisoddn>1)

a1+b2+C1=ab+bc^-ac<^>(a-b)2+(Z?-c)2+(c-?)2a=b=c

四、quadraticequationsandPolynomial

Thequadraticequationsy=UA~十bx十c(u工0)hastworoots勺,A2thenwehas

bc

X)+x2=——x1x2=—

a'a

Moregenerally,ifthepolynomialxn++axH~2'+-----Fa-\x+an~。has

2nroots

M,x2,,xn,thenwehave:

玉十工2十工3十?一十X〃=~a\

X.x2+x2xi+-xn_Kxn=a2

玉電工3?-一%〃=(一1)〃〃“

開方時開方、估計開方數(shù)的大小

絕對值方程

ArithmeticSequence

an=tZ1+(n-V)d=a2-\-(n-2)d=ci3+(n-3)J=???=am+(n-m)d

〃(4+%)_n{a2+a?_,)_〃(%+an_2)_〃(%+an_/n+})

3”———

“2222

n(n-V)d

s〃=〃4+---

Ifn=2k,thenwehavesn=k(ak+6fA+1)

Ifn=2k+1,thenwehavesn=nak^

Geometricsequence

%=%q"'=%q"2=

i-q

Somespecialsequence

1,1,2,3,5,8,…

9,99,999,9999,...

i,ii,in,mi,...

Exercise

A.WhenRingoplaceshismarblesintobagswith6marblesperbag,hehas4marblesleft

over.WhenPauldoesthesamewithhismarbles,hehas3marblesleftover.RingoandPaulpool

theirmarblesandplacethemintoasmanybagsaspossible,with6marblesperbag.Howmany

marbleswillbeleftover?

(A)l(B)2(C)3(D)4(E)5

7Forascienceproject,Sammyobservedachipmunkandasquirrelstashingacornsinholes.The

chipmunkhid3acornsineachoftheholesitdug.Thesquirrelhid4acornsineachoftheholesit

dug.Thcyeachhidthesamenumberofacorns,althoughthesquirrelneeded4fewerholes.Hew

manyacornsdidthechipmunkhide?

(A)30(B)36(C)42(D)48(E)54

21.Fourdistinctpointsarearrangedonaplanesothatthesegmentsconnectingthemhave

lengths/yZ/Zand/.Whatistheratioof/to/?

(A)x/3(B)2(C)>/5(D)3(E)z

6.Theproductoftwopositivenumbersis9.Thereciprocalofoneofthesenumbersis4timesthe

reciprocaloftheothernumber.Whatisthesumofthetwonumbers?

(A)Y(B用(C)7(D)y(E)€

8.Inabagofmarbles,ofthemarblesareblueandtherestarered.Ifthenumberofredmarblesis

doubledandthenumberofbluemarblesstaysthesame,whatfractionofthemarbleswillbered?

(A)I(B)|(C)J(D)|(E)|

13.Aniterativeaverageofthenumbers1,2,3,4,and5iscomputed(hefollowingway.Arrange

thefivenumbersinsomeorder.Findthemeanofthefirsttwonumbers,andthenfindthemeanof

thatwiththethirdnumber,thenthemeanofthatwiththefourthnumber,andfinallythemeanof

thatwiththefifthnumber.Whatisthedifferencebetweenthelargestandsmallestpossiblevalues

ihaicanbeobtainedusingthisprocedure?

(A)|(B)2(C)^(D)3(E)|

16.Thrcerunnersstartrunningsimultaneouslyfromthesamepointona500-metercircular

(rack.Theyeachrunclockwisearoundthecoursemaintainingconstantspeedsof4.4,4.8,and5.0

meterspersecond.Therunnersstoponcetheyarealltogetheragainsomewhereonthecircular

course.Howmanysecondsdotherunnersrun?

(A)1,000(B)1,250(C)2.500(D)5,000(E)10.000

24.Letandbepositiveintegerswithsuchthatand.Whatis?

(A)249(B)250(C)251(D)252(E)253

I.Whatis?

14743

(A)-I(C)H①慌(E)T

10.Considerthesetofnumbers{i,10.102,103......1010}.Theratioofthelargestelementofthe

settothesumoftheothertenelementsofthesetisclosesttowhichinteger?

(A)1(B)9(C)10(D)11(E)101

19.Whatistheproductofalltherootsoftheequation?

(A)-64(B)-24(C)-9(D)24(E)576

4.LetXandYbethefollowingsumsofarithmeticsequences:

X=10+12+14+…+100.

Y=12+14+16+...+102.

WhatisthevalueofX—X?

(A)92(B)98(C)100(D)102(E)112

7.WhichofthefollowingequationsdoesNOThaveasolution?

(A)*+7)2=0(Bi|-3x|+5=0(C)\/^x-2=0

(D)4-8=0(Ei|-3x|-4=0

16.Whichofthefollowinginequalto?

(A)3VI(B)25(C)--(D)3G(E)6

2

13.Whaiisthesumofallthesolutionsof?

(A)32(B)60(C)92(D)120(E)124

14.Theaverageofthenumbers1,2.3...98,99,andxislOOx.Whatisx?

495015150

(A)——(B)——(D)——(E)—

10110110199

11.Thelengthoftheintenalofsolutionsoftheinequalityis10.Whatisb-a?

(A)6(B)10(C)15(D)20(E)30

13.Angelinadroveatanaveragerateof80kphandthenstopped20minutesforgas.Aftcrthestop,

shedroveatanaveragerateof100kph.Altogethershedrove250kminatotaltriptimeof3hours

includingthestop.Whichequationcouldbeusedtosolveforthetimetinhoursthatshedrove

beforeherstop?

Q

(A)80r4-100(--r)=250(B)80/=250(C)100r=250

3

Q

(D)90/=250(E)80(27)+100,=250

3

21.Thepolynomialhasthreepositiveintegerzeros.Whatisthesmallestpossiblevalueofa?

(A)78(B)88(C)98(D)108(E)118

15.Whenabucketistwo-thirdsfullofwater,thebucketandwaterweigh/kilograms.Whenthe

bucketisone-halffullofwaler(hetotalweightis/kilograms.Intermsof/and/,whatisthetotal

weightinkilogramswhenthebucketisfullofwater?

oi-iI%4

(A)江+紂(B)-id(C)%+b(D)^a+2b(E)3a-26

13.Supposethat/and/.Whichofthefollowingisequalto/foreverypairofintegers/?

(A)P2Q(B)Ppm(C)PnQam(D)P2mQ”(E)P2nQ^

16.Let///,and/berealnumberswith/,/,and/.Whatisthesumofallpossiblevaluesof/?

(A)9(B)12(C)IB(D)18(E)24

5.Whichofthefollowingisequaltotheproduct?

812164/?+42(X)8、

4812……4〃……2004'

(A)251(B)502(C)10()4(D)2023(E)4016

7,Thefractionsimplifiestowhichofthefollowing?

(A)1(B)9/4(C)3(D)9/2(E)9

13.Dougcanpaintaroomin5hours.Davecanpaintthesameroomin7hours.DougandDave

painttheroomtogetherandtakeaone-hourbreakfbrlunch.LetIbethetotallime,inhours,

requiredforthemtocompletethejobworkingtogether,includinglunch.Whichofthefollowing

equationsissatisfiedbyt?

(A)(l+l)(r+i)=i(B)(l+l)r+i=i(C)=l

575757

(D)((+；)?-1)=1(E)(5+7)r=l

15.YesterdayHandrove1hourlongerthanIanatanaveragespeed5milesperhourfasterthan

lan.Jandrove2hourslongerthanIanatanaveragespeed10milesperhourfasterthanIan.Han

drove70milesmorethankn.HowmanymoremilesdidJandrivethanIan?

(A)120(B)130(C)140(D)150(E)160

AMC中的幾何問題

一、三角形有關(guān)知識點

1.三角形B勺簡樸性質(zhì)與幾種面積公式

①三角形任何兩邊之和不小于第三邊；

②三角形任何兩邊之差不不小于第三邊；

③三角形三個內(nèi)角附和等于180。；

④三角形三個外角的和等于360。；

⑤三角形一種外角等于和它不相鄰的兩個內(nèi)角H勺和；

⑥三角形一種外角不小于任何一種和它不相鄰的內(nèi)角。

設(shè)三角形ABC的三個角A.B.C對應(yīng)時邊是a,b,c,以A為頂點的高為h。

則日勺三角形的面積公式有：

—cah

①SABC-"Q；

absinCacsinBbesinA

②SABC=

222

③SABC=""=rp(〃=；(〃+〃+”).其中r是內(nèi)切圓半徑；

④SABC)(p-b)(p-c)(p=g(a+b+c))

2.直角三角形的有關(guān)定理(勾股、射影)

直角三角形的識別:

①有一種角等于90。的三角形是直角三角形；

②有兩個角互余的三角形是直角三角形；

③勾股定理定理：兩個直角邊的平方和等于斜邊的平方

④勾股定理逆定理：三角形兩邊的平方和等于第三邊的平方.那么這個三角形是直角三

角形。

直角三角形的性質(zhì)：

①直角三角形的兩個銳角互余；

②直角三角形斜邊上的中線等于斜邊的二分之一；

③射影定理：如圖直角三角形中過直角點向斜邊做亙線AD

A

BD

則有AB2=BDBC,AC2=DC-BC,AD2=BD-DC

3.正三角形的）數(shù)據(jù)：

等邊三角形ABC如上圖，分別作ABC的內(nèi)切圓和夕橫圓.設(shè)等邊三角形的邊長為政！

AD=—aS=—a2R=OA=—ar=OD=-a

2ABCC436

4.其他特殊三角形：

等腰三角形性質(zhì)：

①等邊對等角；

②等腰三角形的頂角平分線、底邊上的中線、底邊上的高互相重疊；

③等腰三角形是軸對稱圖形，底邊的中垂線是它的對稱軸；

5.三角形的四心：

①三角形的三條中線交于一點、這個點叫做三角形的重心，重心將每一條中線提成1：2；

②三角形三邊的垂直平分線交于一點，這個點叫做三角形B勺外心，三角形的外心到各頂

點的距離相等；

③三角形的三條角平分線交于一點.這個點叫做三角形的內(nèi)心，三角形的內(nèi)心到三邊的

距離相等；

④三角形三條垂線交于一點，這個點叫做三角形的垂心。

6.三角形全等與相似:

定義：經(jīng)過旋轉(zhuǎn)平移可以重合的兩個三角形叫全等三角形

三邊對應(yīng)相等（SSS）

全等一角形J方法）兩邊一夾角對應(yīng)相等（SAS）

判定

力達兩角一邊對應(yīng)相等（AAS）

兩邊或者一角一邊對應(yīng)相等的兩個直角二角形（HL）

f定義

兩對應(yīng)邊的比相等夾角相等

相似三角朝

判定方法兩個對應(yīng)角相等

三條對應(yīng)邊的比相等

二、正六邊開鄉(xiāng)ABCDEF設(shè)AB=a貝IJ

正六邊形ABCDEF被三條對角線提成了六個全等的等邊三角形.

AD=BE=CF=2a

AC=BD=CE=DF=EA=FB=y/3a

,_3732

dABCDEF&

三、正四面體數(shù)據(jù)

A

如上圖，設(shè)正四面體ABCD的棱長為a,則有：

1.正四面體是由四個全等正三角形圍成的空間封閉圖形。它有4個面，6條棱,4個

頂點。

正四面體是最簡樸的正多面體。

正四面體的重心、四條高的交點、外接球、內(nèi)切球球心共點，此點稱為中心。

正四面體有一種在其內(nèi)部的內(nèi)切球和一種外切球

正四面體有四條三重旋轉(zhuǎn)對稱軸.六個對稱面C

正四面體可與正八面體填滿空間，在任意頂點周圍有八個正四面體和六個正八面體。

2.有關(guān)數(shù)據(jù)

當(dāng)正四面體的犢長為a時，某些數(shù)據(jù)如下:

高：。中心0把高分為1:3兩部分。

表面積：

體積：

對棱中點的連線段時長:

外接球半徑：，

內(nèi)切球半徑：，

兩鄰面夾角：

正四面體的對棱相等。具有該性質(zhì)的四面體符合如下條件：

I.四面體為對棱相等的四面體當(dāng)且僅當(dāng)四面體每對對棱的中點的連線垂直于這

兩條棱。

2.四面體為對棱相等的四面體當(dāng)且僅當(dāng)四面體每對對棱中點的三條連線互相垂

直。

3.四面體為對校相等的四面體當(dāng)且僅當(dāng)四條中線相等。

四、正方體

有關(guān)數(shù)據(jù)：

I.如圖.設(shè)正方體的棱長為a,則面對角線為.體對角線為，體對角線不僅與截面、垂直,并且

被截面與截面提成三等分。

2.正方體的八個頂點中的每四個面對角線的頂點構(gòu)成了一種棱長為V2a的正四面體。即

ACBQ1與ACiBD是一種棱長為V2a的正四面體。這兩個正四面體的相交部分是一種正

八面體（恰好是正方體六個面的中心的連線工

3.正方體六個面的中心的連線構(gòu)成一種棱長為的正八面體，體積是正方體的

4.正方體在各個方向的投影的面積最大為6a?

5截面的性質(zhì)：正方體的截面中會出現(xiàn)（見下圖）：三角形、正方形、梯形、菱形、矩形、

平行四邊形、五邊形、六邊形、正六邊形。其中三角形還分為銳角三角型、等邊、等腰三角

形。梯形分位非等腰梯形和等腰梯形。不也許出現(xiàn)：鈍角三角形、百角三角形、百角梯形、

正五邊形、

七邊形或更多邊形。

6.最大截面：最大截面四邊形，如圖所示的矩形:

五、正八邊形與正八面體:

IEA邊形：

設(shè)正八邊形的棱長為a,面積是為,，四邊形、是正方形。正八邊形有20條對角線（更一般的凸

邊形有條對角線.內(nèi)部有49個交點（這個推廣還沒有統(tǒng)一的結(jié)論，是一種較為困難的問題）。

IEA面依

和都是正方形，內(nèi)切球的半徑，夕HS球半徑，體積為

六、圓和球：

切割線、切割線定理

（1）相交弦定理：圓內(nèi)面弦相交.交點分得H勺兩條線段的乘積相等。

即：在c中，，,弦、相交于點.

:.PAPB=PCPD

(2難論：假如弦與直徑垂直相交，那么弦的二分之一是它分直徑所成的兩條線段的比例中

項。

即：在O中，?.直徑，

；.CE?=AEBE

(3切割線定理：從圓外一點引圓的切線和割線，切線長是這點到割線與圓交點的)兩條線段

長的比例中項。

即：在。中，?.是切線，是割線

PA2=PCPB

球的有關(guān)公式：

球的體積、表面積公式：、

Exercise

2Acircleofradius5isinscribedinarectangleasshown.Theratioofthelengthoftherectangle

(oitswidthis2:l.Whatistheareaoftherectangle?

(A)50(B)100(C)125(D)150(E)200

3Thepointinthexy-planewithcoordinates(1000,2023)isreflectedacrosstheliney=2O23.What

arethecoordinatesofthereflectedpoint?

(A)(998,2012)(B)(1000,1988)(C)(1000,2024)(D)(1000,4012)(E)(1012,2012]

12PointBisdueeastofpointA.PointCisduenorthofpointB.ThedistancebetweenpointsA

andCis/,and/=45degrees.PointDis20metersdueNorthofpointC.ThedistanceADis

betweenwhichtwointegers?

(A)30and31(B)31and32(C)32and33(D)33and34(E)34and35

14Twoequilateraltrianglesarecontainedinsquarewhosesidelengthis/.Thebasesofthese

trianglesaretheoppositesideof(hesquare,andtheirintersectionisarhombus.Whatistheareaof

therhombus?

(A)|(B)瓜(C)2V2-1(D)8v/3-12(E)竽

16Threecircleswithradius2aremutuallytangent.Whatisthetotalareaofthecirclesandthe

regionboundedbythem,asshowninthefigure?

(A)l(hr+4\/3(B)l加一百(C)12r+v^3(D)l(hr+9(E)13^r

17Jessecutsacircularpaperdiskofradius12alongtworadiitoformtwosectors,thesmaller

havingacentralangleof120degrees.Hemakestwocircukircones,usingeachsectortoformthe

lateralsurfaceofacone.Whatistheratioofthevolumeofthesmallerconetothatofthelarger?

<A>5(B)；9)零(D)浮(玲挈

23Asolidtetrahedronisslicedoffawoodenunitcubebyaplanepassingthroughtwo

nonadjacemverticesononefaceandonevertexontheoppositefacenotadjacenttoeitherofthe

firsttwovertices.Thetetrahedronisdiscardedandtheremainingportionofthecubeisplacedona

tablewiththecutsurfacefacedown.Whatistheheightofthisobject?

(A)警(B)竽(C)l(D)竽(E)V2

24.Thekeystonearchisanancientarchitecturalfeature.ltiscomposedofcongruentisosceles

trapezoidsfittedtogetheralongthenon-parallelsides,asshown.Thebottomsidesofthetwoend

trapezoidsarchorizontal.Inanarchmadewith/trapezoids,let/betheanglemeasureindegreesof

(helargerinteriorangleofthetrapezoid.Whatis/?

(A)100(B)102(C)104(D)106(E)108

lO.Maiydividesacircleinto12sectors.Thecentralanglesofthesesectors,measuredindegrees,

arcallintegersandtheyformanarithmeticscqucncc.Whatisthedegreemeasureofthesmallest

possiblesectorangle?

(A)5(B)6(C)8(D)10(E)12

11.ExternallytangentcircleswithcentersatpointsAandBhaveradiioflengths5and3,

respectively.AlineexternallytangenttobothcirclesintersectsrayABatpointC.WhatisBC?

(A)4(B)4.8(C)10.2(D)12(E)14.4

15.Threeunitsquaresandtwolinesegmentsconnectingtwopairsofverticesarcshown.Whatis

(heareaof/?

(A)|(B)|(C)|(D)；⑻?

UOO*4

21.Letpoints/=/,/=/,/=/,and/=/.Points/,/,/,and/aremidpointsoflinesegments/and

/respectively.Whatistheareaof??

(A)V2(B)竽(C)苧(D)瓜(E)竽

9.TheareaofAEBDisonethirdoftheareaof3-4-5AABC.SegmentDEisperpendicularto

segmentAB.WhatisBD?

4r94A/35

(A)-(B)J3(C)-(D)沫(E)-

16.Adartboardisaregularoctagondividedintoregionsasshown.Supposethatadartthrownat

theboardisequallylikelytolandanywhereontheboard.Whatisprobabilitythatthedartlands

withinthecentersquare?

6-1(B)7(C)￥(D)T(E)2-6

~T

17.In(hegivencircle,(hediameterEBisparalleltoDC,andABisparalleltoED.TheanglesAEB

andABEareintheratio4:5.WhatisthedegreemeasureofangleBCD?

(A)120(B)125(C)130(D)135(E)140

20.RhombusABCDhassidelength2andZB=120°.RegionRconsistsofallpointsinsidethe

rhombusthatareclosertovertexBthananyoftheotherthreevertices.WhatistheareaofR?

V36c26

(A)—(B)—(C)--(D)1+—(E)2

333

22.Apyramidhasasquarebasewithsidesoflengthlandhaslateralfacesthatarcequilateral

triangles.Acubeisplacedwithinthepyramidsothatonefaceisonthebaseofthepyramidandits

oppositefacehasallitsedgesonthelateralfacesofthepyramid.Whatisthevolumeofthiscube?

(A)5\/2-7(B)7-4百(C)&

279

11.SquareEFGHhasonevertexoneachsideofsquareABCD.PointEisonABwith

AE=7EB.WhatistheratiooftheareaofEFGHtotheareaofABCD?

①)手

(c)V14

i(E)-

o

24.Twodistinctregulartetrahedrahavealltheirverticesamongtheverticesofthesameunit

cube.Whatisthevolumeoftheregionformedbytheintersectionofthetetrahedra?

1V2y/3172

（A）正

⑻五?五(D)-(B)T

16.AsquareofsidelengthIandacircleofradiussharethesamecenter.Whatistheareainside

thecircle,butoutsidethesquare?

(B)

(A)f-1普-冬(C)1(D)7(E)V

19.AcirclewithcenterOhasarea156lT.TrianglcABCisequilateral,BCisachordonthecircle,,

andpointOisoutside△ABC.Whatisthesidelengthof△ABC?

(A)2y[3(B)64(C)4>/3(D)12(E)18

20.TwocircleslieoutsideregularhexagonABCDEF.ThefirstistangenttoAB,andthesecondis

tangenttoDE,BotharetangenttolinesBCandFA.Whatistheratiooftheareaofthesecond

circletothatofthefirstcircle?

(A)18(B)27(C)36(D)81(E)108

14.TriangleABChasAB=2AC.Le(DandEbeonABandBCrespectively,suchthatZBAE=Z

ACD.LetFbetheintersectionofsegmentsAEandCD,andsupposethat△CFEis

equilateral.WhatisZACB?

(A)60°(B)75°(C)90°(D)105°(E)120°

17.Asolidcubehassidelength3inches.A2-inchby2-inchsquareholeiscutintothecenterof

eachface.Theedgesofeachcutareparalleltotheedgesofthecube,andeachholegoesallthe

waythroughthecube.Whatisthevolume,incubicinches,oftheremainingsolid?

(A)7(B)8(C)10(D)12(E)15

20.Aflytrappedinsideacubicalboxwithsidelength1meterdecidestorelieveitsboredomby

visitingeachcornerofthebox.ltwillbeginandendinthesamecornerandvisiteachoftheother

cornersexactlyoncc.lbgetfromacornertoanyothercorner,itwilleitherflyorcrawlina

straightline.Whatisthemaxinnimpossiblelength,inmeters,ofitspath?

(A)4+4>/2(B)2+4>/2+2>/3(C)2+3^2+35/3

(D)4V2+4V3(E)35/24-55/3

9.Segment/and/intersectat/,asshown,/,and/.Whatisthedegreemeasureof/?

(A)52.5(B)55(C)57.7(D)60(E)62.5

13.Asshownbelow,convexpentagon/hassides/,/,/,/,and/.Thepentagonisoriginally

positionedintheplanewithvertex/attheoriginandvertex/onthepositive/-axis.The

pentagonisthenrolledclockwisetotherightalongthe/-axis.Whichsidewilltouchthepoint

/onthe/-axis?

(A)AB(B)BC(C)CD(D)DE(E)

1