2017年滑铁卢大学欧几里得数学竞赛答案

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1、TheCENTREforEDUCATIONinMATHEMATICSandCOMPUTINGcemc.uwaterloo.ca2017EuclidContestThursday,April6,2017(inNorthAmericaandSouthAmerica)Friday,April7,2017(outsideofNorthAmericaandSouthAmerica)Solutions©2017UniversityofWaterloo2017EuclidContestSolutionsPage21.(a)Since5(2)+3

2、(3)=19,thenthepairofpositiveintegersthatsatises5a+3b=19is(a;b)=(2;3).(b)Welisttherstseveralpowersof2inincreasingorder:n12345678910112n24816326412825651210242048Eachpowerof2canbefoundbymultiplyingthepreviouspowerby2.Fromthetable,thesmallestpowerof2greaterthan5is23=8a

3、ndthelargestpowerof2lessthan2017is210=1024.Since2nincreasesasnincreases,therecanbenofurtherpowersinthisrange.Therefore,thevaluesofnforwhich5<2n<2017aren=3;4;5;6;7;8;9;10.Thereare8suchvaluesofn.(c)Eachofthe600EurosthatJimmyboughtcost$1.50.Thus,buying600Euroscost600$1:

4、50=$900.WhenJimmyconverted600Eurosbackintodollars,theratewas$1:00=0:75Euro.$1:00$600Therefore,Jimmyreceived600Euros==$800.0:75Euros0:75Thus,Jimmyhad$900$800=$100lessthanhehadbeforethesetwotransactions.2.(a)Sincex6=0andx6=1,wecanmultiplybothsidesofthegivenequationbyx

5、(x1)to5x(x1)x(x1)x(x1)obtain=+or5=(x1)+x.x(x1)xx1Thus,5=2x1andso2x=6orx=3.Thismeansthatx=3istheonlysolution.(Wecansubstitutex=3intotheoriginalequationtoverifythatthisisindeedasolution.)(b)Thesumoftheentriesinthesecondcolumnis20+4+(12)=12.Thismeansthatthesumof

6、theentriesineachrow,ineachcolumn,andoneachdiagonalis12.Intherstrow,wehave0+20+a=12andsoa=8.Onthetoplefttobottomright"diagonal,wehave0+4+b=12andsob=8.Inthethirdcolumn,wehaveentriesa=8andb=8whosesumis0.Thus,thethirdentrymustbe12.Finally,inthesecondrow,wehavec+4+12=1

7、2andsoc=4.Insummary,a=8,b=8,andc=4.Wecancompletethemagicsquaretoobtain:02084412161282017EuclidContestSolutionsPage3(c)(i)If1002n2=9559,thenn2=10029559=100009559=441.pSincen>0,thenn=441=21.(ii)From(i),9559=1002212.Factoringtherightsideasadierenceofsquares,we

8、seethat9559=(100+21)(10021)=121
79Therefore,(a;b)=(121;79)satisestheconditions.(Inaddition,thepairs(a;b)=(79;121);(869;11)