DEExam
TexasA&MHighSchoolMathContestOctober24,2015
Answersshouldincludeunitswhenappropriate.
1.Twoferryboatsplybackandforthacrossariverwithconstantspeeds,turningatthebankswithoutlossoftime.Theyleaveoppositeshoresatthesameinstant,meetforthe?rsttime700feetfromoneshore,continueontheirwaytothebanks,returnandmeetforthesecondtime400feetfromtheoppositeshore.Determinethewidthoftheriver.
2.Threemen,A,B,andC,workingtogether,doajobin6hourslesstimethanAalone,in1hourlesstimethanBalone,andinone-halfthetimeneededbyCwhenworkingalone.WhatisthetimeneededbyAandB,workingtogethertodothejob.√n?1<0.01.3.Findthesmallestintegernsuchthat
4.Findallpossiblevaluesoftheexpression
|x+y|+|z+x
||y|+|z|
8.Leta,bbepositiverealnumbers.Findthevaluesofmforwhichtheequation
|x?a|+|x?b|+|x+a|+|x+b|=m(a+b)
hasatleastonerealsolution.
9.Ifcos2α=m,?ndsin6α+cos6α.
10.Findtheremainderobtainedbydividingx2015byx2?3x+2.11.Solvethesystem??logyx+logxy
xy
??√3=5/2=2712.Solvetheequation??x4=13.5.
13.Findallpairs(x,y)ofrealnumberssuchthatx2+2xsin(xy)+1=0.
1
14.Acircleistangenttothecoordinateaxesandtothehypotenuseofthe30??60??90?triangleABCasshown,whereAB=1.Findtheradiusofthecircle.
15.InaquadrilateralABCD,itisgiventhat∠A=120?,anglesBandDarerightangles,AB=13,and
AD16.Findallvaluesofcsuchthatthepolynomialscx3?x2?x?(c+1)andcx2?x?(c+1)haveacommonroot.
17.Findallxsuchthat0≤x<2πandsin3x=2sinx.
18.InatriangleABC,ADandAEtrisect∠BAC.ThelengthsofBD,DE,andECare2,3,and6,
19.InthequadrilateralABCD,segmentsABandCDareparallel,themeasureofangleDistwicethatofangle
segmentsADandCDareaandbrespectively.FindthemeasureofAB.
20.Supposethatthefunctionf(n)satis?esf(x)+f(y)=f(x+y)?xy?1foreverypairx,yofrealnumbers.Iff(1)=1,?ndallintegersnsuchthatf(n)=n.
21.IfP(x)denotesapolynomialofdegreensuchthatP(k)=k
DEExam,Solutions
TexasA&MHighSchoolMathContestOctober24,2015
Answersshouldincludeunitswhenappropriate.
1.Lett1andt2bethetimesfromstarttilltheboatsmeetforthe?rstandthesecondtime,respectively.Ifwisthewidthoftheriver,andvisthesumofspeedsoftheboats,thenvt1=wandvt2=3w,hencet2=3t1.Oneferrytraveleddistance700feetattimet1,andattimet2ittraveledw+400=3·700feet.Therefore,w=3·700?400=1700feet.
2.Leta,b,andcbeproductivitiesoftheworkers(jobperhour).Thenthestatementsoftheproblemare:
1
a
andweareaskedto?nd
a+b+c1?6=12c,=1
a?6(a+b)=
a?6c=
b1b1b????1c?
c?3?(a+b)=?c=11bandb·b2
=1224.Discardingthenegativeroot,wegetc=3/4,hence1c=4/3hours=80minutes.
√?(n?1)√√n?1=n
n,henceweareaskedto?ndthesmallestintegern3.Wecanrewriten?1√+√√n?1>100.Ifn?1>100,then2suchthat√n+n?1=100.Therefore,theanswerisn=2501.
4.Wehave|x+y|≤|x|+|y|,hencethesumislessthan3.Ontheotherhand,twoofthenumbers,sayxandy,areofthesamesign.Then|x+y|=|x|+|y|,andoneofthesummandsisequalto1.Hence,thesumSsatis?es1≤S≤3.Ifwesetx=1,y=1,z=?a,where0<a≤1,thentheexpressionisequalto1+21?a1+atakesallvaluesintheinterval[0,1)when0<a≤1.Therefore,ouroriginalexpressiontakesallvaluesintheinterval[1,3)forx=y=1,z=?a.Thevalue3isattainedforx=y=z=1.Itfollowsthatthesetofpossiblevaluesoftheexpressionistheclosedinterval[1,3].
5.Fromthesecondequationy?2xy=0wegety(1?2x)=0,hencey=0orx=1/2.Ify=0,thenthe?rstequationbecomesx=x2,whichhassolutionsx=0orx=1.Ifx=1/2,thenthe?rstequationis1/2=1/4+y2,hencey2=1/4,whichhassolutionsy=1/2ory=?1/2.Thereforetheansweris(x,y)=(0,0),(1,0),(1/2,1/2),or(1/2,?1/2).
6.Since0≤x<π,sinxispositive,andwecanwritethe?rstequationas
??,5
or11?cos2x.
12±√27121/4
Takingsquareofbothsides,weget
1
5
or
2cos2x?
cosx+cos2x=1?cos2x,
2
25=0.
Solvingitasaquadraticequationforcosx,weget
2
4
cosx=
25
5
±
2
1+48
10
,
??
hencecosx=4/5orcosx=?3/5.Thensinx=
√
1?16/25=3/5or
ab
.
8.Letusassumethata≤b.Thecaseb≤awillbesimilar.If?a≤x≤a,then?b≤x≤b,and|x?a|+|x+a|=2a,|x?b|+|x+b|=2b,henceform=2solutionsareallnumbers?a≤x≤a.
Ifa≤x≤b,then|x?b|+|x+b|=2b,and|x?a|+|x+a|=x+a+x?a=2x,sothattheequationbecomes2x+2b=m(a+b),whichhassolutionx=m2b.Itmustsatisfya≤x≤b,whichisequivalentto2a+2b≤2x+2b≤4b,whichisequivalentto2(a+b)≤m(a+b)≤4b,i.e.,2≤m≤4bsatisfym(a+b)a+b.
Thecases?b≤x≤?aandx≤?barereducedtothepreviouscasesbyreplacingxby?xintheequation.
Itfollowsthatthesetofvaluesforwhichtheequationhasasolutionis[2,+∞).9.Wehavecos2α=2cos2α?1,hencecos2α=
1+m
2
4
,whichmust
==
1?m
4
4
.
?1?m
2
4
1+2m+m2?1+m2+1?2m+m2
10.Theremainderisapolynomialoftheformax+bsatisfying
x2015=P(x)(x2?3x+2)+(ax+b)
forsomepolynomialP(x).Thepolynomialx2?3x+2hasroots1and2,therefore
1=a+b,
22015=2a+b
Subtractingthe?rstequationfromthesecond,wegeta=22015?1.Then,fromthe?rstequationwegetb=2?22015.Therefore,theansweris(22015?1)x+(2?22015).
1
11.Wehavelogyx=
t
+t=5/2,
2
hence
t2?
2
5
±
??
4
?4
2
±3
,2
hencet=2ort=1/2.Ift=2,thenlogxy=2,i.e.,y=x2.Ift=1/2,thenx=y2.Inthe?rstcasethesecondequationgivesx3=27,inthesecondcasewegety3=27.Hence,theansweris(x,y)=(3,9)or(9,3).
√√
√√4=12+312.Wehave333
2x/3
=27
4
13.Letusreplacesin(xy)byanothervariablea.The
discriminantD=4a2?4.Sincexhastobereal,D≥0.cases:??
sin(xy)2
x+2x+1and
??
sin(xy)x?2x+1
2
quadraticequationx2+2ax+1=0inxhasBut?1≤a≤1,soa=±1.Hence,wehavetwo==
10
??x
isincreasing.
Inthe?rstcasewegetx=?1andy=?π/2+2kπ,k∈Z.Inthesecondcasewegetx=1andy=?π/2+2kπ,k∈Z.
14.LetA1,B1,andC1bethefeetoftheperpendicularsfromthecenterOofthecircletothelinesAC,CB,andAB√,respectively.ThenOA1CB1isasquare.Denotetheradiusbyr.ThenBB1=r?1/2,
3/2)=1.ItfollowsthatAA1=√
r?
3+
r==?1=0
15.SinceBandDarerightangles,wecancircumscribeacirclearoundABCD,andACwillbeitsdiameter.LetObethecenterofthecircle,andritsradius.Then∠DOB=120?,since∠DCB=60?.Usingthelawofcosines,weget
DB2=462+132+2·46·13·
√
1
3,∠DOB=120?.ItfollowsthatDB=2·r·
√2
=
31
16.Inotherwords,weareaskedtosolvethesystem
??cx3?x2?x?(c+1)=cx2?x?(c+1)=00
Wehavecx3?x2=x+c+1=cx2,hencex2(cx?1?c)=0.Itfollowsthateitherx=0,orcx?1?c=0.Inthe?rstcasewegetc+1=0,hencec=?1.Inthesecondcasewehavec=0,andx=1+c
cintotheequations,weseethatbothofthemaresatis?ed.Itfollowsthatthepolynomialshaveacommonrootforallc=0.Notethatforc=0thepolynomialsare?x2?x?1and?x?1,sotheyhavenocommonroots.Answer:allc=0.
17.Wehavesin3x=sinxcos2x+sinxcos2x=sinxcos2x+2sinxcos22x.Therefore,theequationisequivalentto
sinx(cos2x+2cos2x?2)=0.
Ifsinx=0and0≤x<2π,thenx=0orπ.Ifsinx=0,then
cos2x+2cos2x?2=0
whichcanbewritten
2cos2x?1+2cos2x?2=0,√or4cos2x=3,hencecos2x=3/4,orcosx=±6,5π6,11π
924x+y2?3xya=9
4x2?xya=5,xya=5
weget24x?5,hencey2=524x?5intothe?rstequation,
x2+5
2
4x2+10=4,
1
40
.
19.LetDD1bethebisectoroftheangleD,whereD1isapointonthesegmentAB.Then∠AD1D=∠D1DC=∠ADD1,hence△DAD1isisosceles,sothatAD1=a.ThelineDD1isparalleltoBC,henceDD1BCisaparallelogram,sothatD1B=DC=b.WeconcludethatAB=a+b
.
20.Wehavef(n)+f(1)=f(n+1)?n?1,hencef(n+1)=f(n)+n+2.Itfollowsf(2)=1+3,
.f(3)=1+3+4,f(4)=1+3+4+5,e.t.c.,f(n)=1+3+4+5+···+n+1=(n+1)(n+2)
2Notethat(replacingnbyn?1)wegetf(n?1)=f(n)?n?1,whichgivesaproofbyinductionfor2theformulaf(n)=n+3n?2
2=n,
n2+3n?2=2n
n2+n?2=0.
Rootsaren=1,?2.
21.Theconditionimpliesthatthepolynomial(x+1)P(x)?xhasroots0,1,2,...,n.Since(x+1)P(x)?xhasdegreen+1,itfollowsthat(x+1)P(x)?x=cx(x?1)(x?2)···(x?n)forsomenon-zeronumberc.Consequently,P(x)=cx(x?1)(x?2)···(x?n)+x
(n+1)!x(x?1)(x?2)···(x?n)+x
n(n+1)!(n+1)n(n?1)···1+(n+1)n+2.In
otherwords,P(n+1)=
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