2016滑铁卢竞赛试题答案

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1、www.linstitute.netTheCENTREforEDUCATIONinMATHEMATICSandCOMPUTINGcemc.uwaterloo.ca2016EuclidContestTuesday,April12,2016(inNorthAmericaandSouthAmerica)Wednesday,April13,2016(outsideofNorthAmericaandSouthAmerica)Solutions©2016UniversityofWaterloowww.linstitute.net1.(a)Theaverageis5+15+2

2、5+35+45+556=(5+55)+(15+45)+(25+35)6=60+60+606=30(b)Sincex2=2016,then(x+2)(x−2)=x2−4=2016−4=2012.(c)SincepointsP,QandRlieonastraightline,thentheslopeofPQequalstheslopeofPR.2a−5TheslopeofPQequalsandtheslopeofPRequalsa−72a−5Therefore,=5andso2a−5=5(a−7).a−7Thisgives2a−5=5a−35or3a=30,ands

3、oa=10.30−512−7=255=5.2.(a)Ifn9=25,thenn2=25(9)=225.Therefore,n=15orn=−15.n(Wecancheckbysubstitutionthateachofthesevaluessatisfiestheoriginalequation.)(b)Whenweexpandtheleftsideof(x−3)(x−2)=6,weobtainx2−5x+6=6.Thus,x2−5x=0orx(x−5)=0,whichgivesx=0orx=5.(Wecancheckbysubstitutionthateacho

4、fthesevaluessatisfiestheoriginalequation.)(c)Letabethecost,indollars,of1appleandletbbethecost,indollars,of1banana. Fromthegiveninformation,2a=3band6a+12b=6.30.Since3b=2a,then12b=4(3b)=4(2a)=8a.Therefore,6a+8a=6.3or14a=6.3,whichgivesa=0.45. Inotherwords,thecostof1appleis$0.45.3.(a)Solu

5、tion1Sincethesumoftheanglesinanytriangleis180◦,thenthecombinedsumoftheanglesin4ABD,4FBGand4CBEis3·180◦or540◦.Thenineanglesinthesetrianglesincludethosewithmeasures,indegrees,ofp,q,r,s,t,u aswellasthethreeangles∠ABD,∠FBGand∠CBE.Theselastthreeanglesformastraightangle,andsotheirsumis180◦

6、.Therefore,thesumoftheremainingsixanglesmustbe540◦−180◦=360◦. Inotherwords,p+q+r+s+t+u=360.Solution2Werepeatedlyusethefactthatthesumoftheanglesinanytriangleis180◦.From4ABD,∠ABD=180◦−p◦−q◦. From4FBG,∠FBG=180◦−r◦−s◦.From4CBE,∠CBE=180◦−t◦−u◦.SinceABCformsastraightlinesegment,then∠ABD+∠F

7、BG+∠CBE=180◦whichgives(180◦−p◦−q◦)+(180◦−r◦−s◦)+(180◦−t◦−u◦)=180◦ Rearranging,weobtain360=p+q+r+s+t+uandsothevalueofp+q+r+s+t+u is360.www.linstitute.net(b)Solution1Theintegerequalto1020consistsofthedigit1followedby200s.Theintegerequalto1020−1thusconsistsof209s.Now,n=1020−20is19lessth

8、an1020−1whic