2024數(shù)學(xué)競賽CHKMO中國新加坡合練訓(xùn)練題

FunctionsinNT

Yihan

Dec2024

1Introduction

Numbertheoryfunctionsandfunctionalequationsinintegers.WhilemostNTFEsfeelliketheyhavethesameboringideas(primes,induction…),wemustrememberthateveryFEisuniqueandmustbetreatedassuch.

2Problems

1.Foreachintegern≥2,letF(n)denotethegreatestprimefactorofn.Astrangepairisapairofdistinctprimespandqsuchthatthereisnointegern≥2forwhichF(n)F(n+1)=pq.Provethatthereexistinfinitelymanystrangepairs

2.Findalln∈Nsuchthatφ(n)+σ(n)=2n+8

3.Findallsurjectivefunctionsf:N→Nsuchthatforeverym,n∈Nandeveryprimep,thenumberf(m+n)isdivisiblebypifandonlyiff(m)+f(n)isdivisiblebyp.

4.Provethatifanunboundedfunctionf:N→Nsatisfies|f(mn)一f(m)f(n)|≤kforsomeconstantkVm,n∈N,thenfmustbemul-tiplicative.

5.Findallfunctionsf:N→Nsuchthatforanypositiveintegersx,y,

ff(x)+y(y)=f(x+y)+y.

6.DenotebyQ+thesetofallpositiverationalnumbers.Determineallfunctionsf:Q+Q+whichsatisfythefollowingequationforallx,y∈Q+:

f(f(x)2y)=x3f(xy).

7.Findallfunctionsf:Z→Nforwhich

f(x+f(y))2+f(y+f(x))2=f(f(x)+f(y))2+1

holdsforanyx,y∈Z.

8.Findallf:N→N,suchthatforanyx,y∈N,f(f(x)+y)|x+f(y).

1Introduction

CombiGeomproblemscanbesplitintotwomaincategories:thoserequiringacleverconstruction,orthosethatrequireaninsightfulproof.However,themaindifficultyusuallyliesindecidingwhethertoconstructortoprovesomethinginordertofinishtheproblem.

2Problems

1.FindallfinitesetsSofpointsintheplanewiththefollowingproperty:foranythreedistinctpointsA,B,andCinS,thereisafourthpointDinSsuchthatA,B,C,andDaretheverticesofaparallelogram(insomeorder).

2.FivepointsA?,A?,A?,A?,A?lieonaplaneinsuchawaythatnothreeamongthemlieonasamestraightline.Determinethemaximumpossiblevaluethattheminimumvaluefortheangles∠AiA;Akcantakewherei,j,karedistinctintegersbetween1and5.

3.Letn≥5beagiveninteger.Determinethegreatestintegerkforwhichthereexistsapolygonwithnvertices(convexornot,withnon-selfintersectingboundary)havingkinternalrightangles.

4.Letn≥5beaninteger.Considernsquareswithsidelengths1,2,...,n,respectively.Thesquaresarearrangedintheplanewiththeirsidesparalleltothexandyaxes.Supposethatnotwosquarestouch,exceptpossiblyattheirvertices.Showthatitispossibletoarrangethesesquaresinawaysuchthateverysquaretouchesexactlytwoothersquares.

5.WesaythatafinitesetSofpointsintheplaneis[i]balanced[/i]if,foranytwodifferentpointsAandBinS,thereisapointCinSsuchthatAC=BC.WesaythatSis[i]centre-free[/i]ifforanythreedifferentpointsA,BandCinS,thereisnopointsPinSsuchthatPA=PB=PC.

(a)Showthatforallintegersn≥3,thereexistsabalancedsetconsistingofnpoints.

(b)Determineallintegersn≥3forwhichthereexistsabalancedcentre-freesetconsistingofnpoints.

6.Thetoricdistancebetweentwopoints(x?,y?)and(x?,y?)isdefinedas√Ilr?-x?ll2+|ly?-y?1|2,

where||x||denotesthedistancebetweenxanditsnearestinteger.Findthelargestrealrsuchthatthereexistsfourpointsontheplanewhosepairwisetoricdistancesareallatleastr.

7.Findallintegersn≥3withthefollowingproperty:thereexistndistinctpointsontheplanesuchthateachpointisthecircumcenterofatriangleformedby3ofthepoints.

8.Findtheminimumpositiveintegern≥3,suchthatthereexistnpointsA?,A?,·,Ansatisfy