advanced-problems-in-mathematics-.pdf

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1、AdvancedProblemsinMathematicsS.T.C.SiklosAdvancedProblemsinMathematics43problemscompletewithfullsolutionsanddiscussionCONTENTSContentsiPrefaceiiiProblemsOdd-numberedpagesSolutionsEven-numberedpagesEachproblemhasbeengivenatitle(notusedlateron),toremindyouwhatitisaboutifyouhavealr

2、eadylookedatit,followedbyaroughindicationofthemathematicalcontentincaseyouwanttopickoutquestionsbytopic.Q1Anoldchestnut(generalmathematics)1Q2Findingageneralresult(elementaryalgebra)3Q3Trig.integrals(integration)5Q4Findinganothergeneralresult(moreelementaryalgebra)7Q550pencepiec

3、e(planegeometry)9Q6Bouncingdownstairs(projectiles,Newton’slawofelasticity)11Q7Snowballs(calculus)13Q8Choosinganoutfit(elementaryprobability)15Q9Achocolateorange(statics)17Q10TrianglesintheArgandplane(complexnumbersandalgebra)19Q11Integralsbysubstitution21Q12Solvingthecubic(algebr

4、aandtrig.)23Q13Randombodies(probability)25Q14Simultaneousequations(linearalgebra)27Q15Logandtanseries(integration)29Q16Ablastfromthepast(algebraicmanipulation)31Q17Lagrange’stheorem(functionsandcalculus)33iQ18Elastictowrope(mechanicsanddifferentialequations)35Q19Matrixmanipulatio

5、ns37Q20Fractionalarithmetic(inequalities)39Q21Approximatingintegrals(trig.functionsandintegration)41Q22Integralidentities(integralcalculus,areasandvolumes)43Q23Tennistournament(probability)45Q24Levitatinghoop(mechanicsandparticlemotion)47Q25Randomquadraticequation(probability)49

6、Q26Rodandball(momentofinertia,elasticimpact)51Q27Stirling’sformula(trapeziumrule)53Q28Setsandgroups(groupaxioms)55Q29Adifferential-differenceequation(calculus)57Q30Levelcrossing(probability)59Q31Snowploughing(applieddifferentialequations)61Q32Addinggroups(abstractgrouptheory)63Q33A

7、rchery(probability)65Q34Anintegralidentity(integrationbychangeofvariable)67Q35Bernoullipolynomials(recurrencerelationsandcalculus)69Q36Symmetry(rootsofquadraticandcubicequations)71Q37Windingroundapost(particlemotion)73Q38Vectorgeometry75Q39Matrixcalculus(matricesanddifferentialeq

8、uations)77Q40Complexloci(lociintheArgandplane)7