语言模型在高等数学推理方面的基准测试 Benchmarking LLMs on Advanced Mathematical Reasoning

BenchmarkingLLMsonAdvancedMathematicalReasoning

JonathanYueDanielKlein

ElectricalEngineeringandComputerSciencesUniversityofCalifornia,Berkeley

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May16,2025

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BenchmarkingLLMsonAdvancedMathematicalReasoning

by

JonathanYue

Athesissubmittedinpartialsatisfactionofthe

requirementsforthedegreeof

MasterofScience

in

ElectricalEngineeringandComputerSciences

inthe

GraduateDivision

ofthe

UniversityofCalifornia,Berkeley

Committeeincharge:

ProfessorDanKlein,ChairProfessorGireejaRanade

Spring2025

1

Abstract

BenchmarkingLLMsonAdvancedMathematicalReasoning

by

JonathanYue

MasterofScienceinElectricalEngineeringandComputerSciences

UniversityofCalifornia,Berkeley

ProfessorDanKlein,Chair

LargeLanguageModels(LLMs)haveimproveddramaticallyatmathematicalreasoning,progressingfrombasicarithmetictoolympiadlevelproofs.However,theexisting,short-answerbasedbenchmarkscansufferfromlimitedscopeforcomplexreasoningandthereforedonotsufficientlymeasurethereasoningcapabilitiesofLLMs.Formalproof-basedbench-marksexist,buttheneedtoconvertproblemstatementsintoformallanguageslimitsthescopeoftheproblems.Apotentialreasonforthissignificantgapincurrentliteratureisthedifficultyingradingproofproblemsatscale.Toaddressthis,wefirstproposeanLLM-as-a-judgeframeworktojudgemodel-generatedproofsandevaluateditsefficacy.Then,weproposeabenchmarkof77PhD-levelproofquestions,drawnfromRomanVershynin’s“High-DimensionalProbability:AnIntroductionwithApplicationsinDataScience”,andchallengedstate-of-the-artLLMswiththesequestions.WeevaluatedtheLLM-generatedsolutionsusingtheLLM-as-a-judgeframeworkandfoundthat,ingeneral,state-of-the-artLLMsarestillunabletoadequatelycompletetheseproofs.

i

Contents

Contents

i

1Introduction

1

2Relatedworks

3

2.1LLMformathematicalreasoning

3

2.2BenchmarkingMathematicalReasoninginLargeLanguageModels

7

2.3LLM-as-a-JudgeforEvaluatingMathematicalReasoning

10

3Datacollection

13

3.1Groundtruthhumanproofevaluation

13

3.2Mathematicaltextandgroundtruthsolutions

14

4LLM-as-a-judgeformathematicalproofs

16

4.1Evaluation

18

5LLMnaturallanguageproof-writing

21

5.1Methodology

21

5.2Results

23

6Discussion

25

6.1Limitations

26

7ConclusionandFutureDirections

27

Bibliography

29

ii

Acknowledgments

Iamdeeplygratefultomyadvisor,DanKlein,forhissupportandmentorshipoverthepasttwoyears,andtoProfessorGireejaRanadeforbeingmysecondreaderandofferingvaluableconversationsandfeedback.Ialsowishtothankmymentor,NicholasTomlin,forhisguidanceandthoughtfulinputthroughoutthisprocess.

IwouldliketothankallmyclassmatesintheCompSci294-271course,AIinEducation,taughtbyProfessorGireejaRanadeandProfessorNargesNorouzi,whereIengagedindis-cussionsthatinspiredtheconceptionofthisproject.Iespeciallythankmyteammembersinthecourse,NoraGuoandSveinungMyhre,whoIcollaboratedwithtoobtaintheIRBapprovalandconductarelatedstudy,forhelpingshapetheideasforthiswork.

Finally,Iextendmyheartfeltthankstomyfamilyandfriendsfortheirconstantencourage-mentandsupport,whichhasbeeninvaluableineveryfacetofmylife.

1

Chapter1

Introduction

TheadventofLargeLanguageModels(LLMs)hasresultedremarkableadvancementsinartificialintelligence,withmathematicalreasoningemergingasasignificantareaofinterest.LLMshaveevolvedfromperformingrudimentaryarithmetictoassistingwithmathematicaldiscover(Romera-Paredesetal.,

2024

),drivenbyincreasedmodelsizeandsophisticatedpromptingtechniquessuchasChain-of-Thought(CoT)prompting(Weietal.,

2023

)andTool-IntegratedReasoning(TIR)(Ahnetal.,

2024

).

Despitethesestrides,aconsiderablegappersistsinevaluatingthedeeperreasoningcapabil-itiesofLLMs.Existingbenchmarkspredominantlyfocusonshort-answerormultiple-choicequestions,suchasMATH(Hendrycksetal.,

2021

)andGSM8K(Cobbeetal.,

2021

).Whileuseful,thesebenchmarksoftenemphasizethefinalnumericaloutput,potentiallyneglectingtherigoroftheintermediatereasoningstepsandsufferingfromissuessuchasbenchmarksaturationanddatacontamination(Hongetal.,

2025

;Petrovetal.,

2025

).Furthermore,theirlimitedscopemaynotsufficientlyassesscomplexconceptualunderstandingortheabil-itytoconstructintricatearguments(K.Huangetal.,

2025

).ThepoorperformanceofLLMsonproblemsthatrequirerigorousproofgeneration,asopposedtonumericalanswers,sug-gestsapotential“reasoningillusion,”wheresuccessinsometasksmightstemfrompatternmatchingortoolassistanceratherthangenuinemathematicalinsight(Petrovetal.,

2025

).

Ontheotherhand,benchmarkscenteredonformalproofs,suchasMinif2f(K.Zhengetal.,

2022

)andPutnamBench(Tsoukalasetal.,

2024

),operatewithinthestrictconfinesofsymbolicsystemslikeLeanorIsabelle.Whilevaluableforassessingmechanicallyverifiablereasoning,thenecessityoftranslatingproblemstatementsintotheseformallanguages(aut-oformalization)canbeachallengingtaskitself,whichrestrictsthebreadthofproblemsthatcanbeaddressed(Gulatietal.,

2024

;J.Zhangetal.,

2025

).Thisleavesasignificantgapinevaluatingthegenerationofnovel,naturallanguagemathematicalproofs,whicharemoreakintohumanmathematicalpracticeandcrucialforinterpretability.Aprimaryobstacletodevelopingbenchmarksforsuchproofsistheinherentdifficultyingradingthemconsistentlyandatscale,asnaturallanguageproofslacktheimmediateverifiabilityofformalproofsandoftenrequireexperthumanevaluation(Friederetal.,

2023

).

2

CHAPTER1.INTRODUCTION

Toaddresstheselimitationsandbridgethegapinevaluatingadvancedmathematicalrea-soninginnaturallanguage,thispapermakestwoprimarycontributions.First,weproposeandevaluateanLLM-as-a-judgeframeworkspecificallydesignedforassessingthecorrect-nessandcoherenceofmathematicalproofs.Thisapproachoffersascalablealternativetomanualexpertevaluation.Second,leveragingthisevaluationframework,weintroduceanewbenchmarkcomprising77PhD-levelproof-basedquestions,designedtorigorouslytesttheadvancedreasoningcapabilitiesofLLMsonproblemsrequiringnovelandintricatenaturallanguageproofs.

Wefindthat,despiterecentprogress,currentleadingLLMsaregenerallystillunabletoadequatelycompletethesecomplexproof-basedtasks.Thisunderscoresthenecessityformorechallengingbenchmarksthatprobedeeperlevelsofmathematicalunderstandingandproofgeneration,therebyprovidingamoreaccuratemeasureofLLMreasoningabilitiesandguidingfutureresearchtowardsdevelopingmorecapableandrobustAIsystemsformathematics.

3

Chapter2

Relatedworks

2.1LLMformathematicalreasoning

TheintroductionofLargeLanguageModels(LLMs)hasmarkedasignificantturninartificialintelligence.Amongthemanyapplications,mathematicalreasoningemergesasakeyareaofexplorationandadvancement.ThissectionsurveysthedevelopmentofLLMsintacklingmathematicaltasks.

AdvancesandCoreTechniquesinLLMMathematicalReasoning

LLMshaverapidlyimprovedatmathematicalreasoning,progressingfromsimplearithmetictosolvingmorecomplexproblems(Liuetal.,

2025

).Akeydriverforthisprogressistheincreaseinmodelsize.Largermodelsfollowinstructionsbetterandreasonmoreeffectively.ResearchshowsthatLLMswithover100billionparameterscansolvedifficultmathproblemswiththerightprompts(Forootani,

2025

).

Thediscoveryofseveraltechniqueshaveaugmentedthisimprovementduetomodelscale.AmajorcornerstoneinthisareaisChain-of-Thought(CoT)prompting.ItguidesLLMstoliststepsintheirreasoningbeforegivingafinalanswer.Thisresultsinanimprovementinboththeperformanceoncomplextasksandtheinterpretabilityofthereasoningprocess.Weietal.,

2023

showedthatCoTpromptingallowsLLMstodeconstructproblemstheywouldusuallyfail.ResearchershavebuiltuponbasicCoTwithmoreadvancedversionsandnewtypesof“thinking”models.ExamplesincludeSoftCoT(Y.Xuetal.,

2025

),ContinuousCoT(Cheng&Durme,

2024

),andCoconut(Haoetal.,

2024

).SoftCoT,forinstance,utilizesasmallerassistantmodeltogenerate“softthoughttokens”whicharethenprojectedintotheprimaryLLM’srepresentationspacethroughatrainablemodule.Thisfine-tuningap-proachisparameter-efficientandhasdemonstratedenhancedperformanceonmathematicalreasoningbenchmarks.

Theemergenceofmodelsexplicitlydesignedfor“thinking,”suchasOpenAI’sO1and

4

CHAPTER2.RELATEDWORKS

DeepSeek-R1,representsanotableprogression.ThesemodelsoftentakemoretimetoreasonduringinferenceandaretrainedwithReinforcementLearning(RL)tobuildadvancedcogni-tiveskillslikeself-checkingandreflection.DeepSeek-R1,forexample,isreportedtodevelopCoTreasoningcapabilitiesautonomouslythroughapureRLtrainingparadigm(DeepSeek-AIetal.,

2025

).Multi-roundThinking,whereamodelusespreviousanswerstotryagainandimprove,hasalsoledtobetterresults,asseeninmodelslikeQwQ-32BandDeepSeek-R1ondifficultbenchmarkssuchasAIME2024(Tianetal.,

2025

).RL-basedreasoningmethodsarealsobeingusedinmultimodalLLMs.Vision-R1,forinstance,usesProgressiveThinkingSuppressionTraining(PTST)toimproveperformanceinsolvingmultimodalmathproblems(W.Huangetal.,

2025

).

ToaddressLLMs’weaknessinprecisenumericalcalculationandrigoroussymbolicmanip-ulation,Tool-IntegratedReasoning(TIR)hasgainedprominence(Ahnetal.,

2024

).TIRletsLLMshandoffsubpartsofaproblemtotoolslikePythonformathorsymbolicsolversforalgebra.Thisimprovesaccuracy,especiallywhenhighprecisionorstructuredmathisneeded.Forexample,thewinningsolutionintheAIMO-2competitionusedCoTfine-tuningfirst,thenfine-tunedagainonaTIRdataset.Thishelpedthemodelcombinenaturallan-guagereasoningwithstructuredcomputation(Moshkovetal.,

2025

).TIRframeworkslikeTATA(TeachingLLMsAccordingtoTheirAptitude)(X.Xuetal.,

2025

)helpmodelschoosebetweenCoTandTIRdependingontheproblem.

Still,LLMsstrugglewithhighlyadvancedmathproblems,especiallyonesneedingfullproofslikethoseinmatholympiads.Arecentworkevaluatedstate-of-the-artLLMson2025US-AMOproblems.Theresultsshowsignificantlimits(Petrovetal.,

2025

).EvenaleadingmodellikeGemini-2.5-PROachievedanaveragescoreofonly25%,withotherprominentmodelsscoringbelow5%.Commonfailuremodesincludedflawedlogicalsteps,introductionofunjustifiedassumptions,alackofcreativeproblem-solvingstrategies,andatendencytofalselyclaimthataproblemhadbeensolved.

TheobserveddiscrepancybetweenLLMperformanceonbenchmarksrequiringnumericalanswersandthoserequiringrigorousproofgenerationsuggestsapotential“reasoningillu-sion.”WhiletechniqueslikeCoTandTIRdemonstrateimprovementsoncertaintypesofmathematicalproblems,thepoorperformanceonproof-basedtaskssuggeststhattheseim-provementsmightnotalwaystranslatetogenuine,deepmathematicalunderstanding.LLMsmaybeleveragingpatternmatchingandtool-assistedcomputationtosucceedinnumericalanswer-orientedtasks,withoutnecessarilydevelopingdeeplogicaldeductivecapabilities.Thisillustratesthecriticalneedforbenchmarksthatspecificallyevaluatethesedeeperrea-soningandproof-generationabilities.

Formalvs.InformalMathematicalReasoningbyLLMs

MathematicalreasoningbyLLMscanbebroadlycategorizedintotwodomains:formalmath-ematicalreasoning,whichoperatesundertherigoroussyntaxofsymbolicsystemsandproof

5

CHAPTER2.RELATEDWORKS

assistants,andinformalmathematicalreasoning,whichexpressesmathematicsinnaturallanguage.

FormalMathematicalReasoning

Thissub-fieldischaracterizedbyitsemphasisonmechanicalverifiabilityandlogicalrigor.

LLMsforFormalProofGeneration(Lean,Isabelle,etc.):MuchresearchisdedicatedtotrainingLLMstogenerateproofsinformallanguagessuchasLean4(J.Zhangetal.,

2025

)andIsabelle(X.Zhaoetal.,

2024

).Thegoalofthislineofresearchistodevelopsystemsthatcanautonomouslyproducemachine-verifiablemathematicalproofs.

Severalnotablemodelsandsystemshavedemonstratedprogress.AlphaGeometrymadeheadlinesbysolvingOlympiad-levelgeometryproblems.Itcombinedalanguagemodel,whichsuggestspotentiallyusefulgeometricconstructions,withasymbolicdeductionenginethatformallyverifiesthesesteps.Akeycomponentofitsdevelopmentwasthegenerationof100millionsyntheticdataexamplesfortraining(Trinhetal.,

2024

).Animprovedver-sion,AlphaGeometry2extendeditsformallanguagetohandleawiderrangeofgeometricproblemsandleveragedtheGeminiarchitecture.AlphaGeometry2reportedlysurpassedtheaverageperformanceofhumangoldmedalistsonabenchmarkofIMOgeometryproblems(Chervonyietal.,

2025

).

TheSelf-playTheoremProver(STP)addressesthecriticalissueofdatascarcityinformalmathematicsbyemployingadual-agentsystem:aconjecturerthatproposesnewtheoremsandaproverthatattemptstoprovethem.Thisself-playloopallowsthesystemtoiterativelygenerateitsowntrainingdata,whichresultsinsignificantperformanceim-provementsonvariousreasoningbenchmarks(Dong&Ma,

2025

).Goedel-Proverisanopen-sourceLLMdesignedforformalproofgenerationinLean4.Itachievedstate-of-the-artresultsbyfirstformalizingalargedatasetofnaturallanguagemathproblemsintoLean4statementsandthenemployinganexpertiterationstrategytotraintheprover(Linetal.,

2025

).Kimina-ProverPreviewdevelopedareasoning-drivenexplorationparadigm,utilizinglarge-scaleRLbasedonQwen2.5-72BtogenerateproofsinLean4.ThisapproachhasalsosetnewSOTAperformanceontheminiF2Fbenchmark(H.Wangetal.,

2025

).

Despitetheseadvancements,akeychallengeremainsthescarcityoflarge-scale,high-qualitydatasetsofformalizedmathematicalstatementsandproofs.TherigoroussyntaxofformallanguagesoftenmakesproofgenerationinthesesystemsmoredifficultforLLMscomparedtonaturallanguage(J.Zhangetal.,

2025

).

Autoformalization:Thiscriticalsub-areafocusesontheautomatictranslationofinformalmathematicallanguageintoformallanguagesthatcanbeprocessedbyproofassistants.LLMslikeGPT-4haveconsiderablyadvancedautoformalizationcapabilitiesthroughin-contextlearning.Techniquessuchasback-translation,whereexistingformalstatementsareautomaticallyinformalizedintonaturallanguagetoresultinsyntheticpairsofformalandinformalmathematicalstatements,havebeenexploredtoaugmenttrainingdata(Yanget

6

CHAPTER2.RELATEDWORKS

al.,

2024

).

However,autoformalizationremainsadifficulttask,especiallyforcomplexorabstractmath-ematicalconcepts(Gulatietal.,

2024

).LLMscanstrugglewithselectingthecorrectpream-blesorlibraryimportsfromvastformalmathematicslibrariesandmayevengeneraterefer-encestonon-existentpreambles.Evaluatingautoformalizationfaithfulness—thattheformalstatementaccuratelycapturesthesemanticsoftheoriginalinformalinput—remainsanareaofongoingresearch.

Informal(NaturalLanguage)MathematicalProofs

Thisdomainconcernsthegenerationandcomprehensionofmathematicalproofs,intheformmostfamiliartohumans:innaturallanguage.

CurrentLandscape:WhileCoTtechniquesenabledLLMstogeneratenaturallanguagestep-by-stepreasoning,therehasbeenrelativelylittleresearchintothegenerationofrigorousandverifiablenaturallanguagemathematicalproofsforcomplexproblems.MuchoftheexistingworkonmathematicalreasoninginLLMsfocusesonsolvingproblemsthatyieldnumericalorshort-formanswers.

ThegenerationandevaluationofnaturallanguageproofsbyLLMspresentauniquesetofchallenges:

•Verifiability:Afundamentaldifficultyliesintheautomatedverificationofnaturallanguageproofs.Unlikeformalproofs,whichcanbemechanicallycheckedbyproofassistants,naturallanguageproofstypicallyrequirehumanexpertevaluationorhighlysophisticatedAI-drivenverifiers.Evenminorerrorsinlogicorcalculationcaninvali-dateanentireproof,andthesecanbedifficultforcurrentsystemstoreliablydetectinanaturallanguagecontext.

•Ambiguity:Naturallanguageisinherentlyambiguousandoftenreliesonimplicitcontextualknowledgeandcommonsense.Forinstance,whetheromittingcertainstepsinamathematicalargumentisacceptabledependsontheassumedleveloffamiliarityofboththeauthorandtheaudience.LLMsmightnotfullygrasptheseambiguitiesandmaintainthenecessarylevelofprecisionthroughoutaproof.AnotherconcernisthatanLLMmayinadvertentlyexploittheseambiguitiesandpretendasifithadcompletedaproof.

•DataScarcityforComplexProofs:High-quality,large-scaledatasetsofcomplexnaturallanguageproofs,particularlythosewithdetailedstep-by-stepannotationsorexplanationsofreasoning,arescarce.Thisisespeciallytruefornicheoradvancedmathematicaltopics.

ThecomparativelysmallervolumeofresearchfocusedongeneratingandverifyingnaturallanguagemathematicalproofsbyLLMs,asopposedtoformalproofsorproblem-solving

7

CHAPTER2.RELATEDWORKS

withnumericalanswers,maybeattributedtoseveralfactors.FormalsystemsandproofassistantsmakeformalproofgenerationanattractivetargetforAIresearchasitprovidesaclear,objectivemeasureofcorrectness.Thisreliablefeedbackloopsformodeltrainingandevaluationstandsincontrasttotheinherentambiguityofnaturallanguage.Additionally,formalmathematicslibraries(e.g.,MathlibforLean)offeragrowing,structured,andverifiedcorpusofmathematicalknowledge.ThefieldofAutomatedTheoremProving(ATP)alsohasalong-standingtraditioninAI,historicallyfocusingonformallogic.

Despitethesechallenges,webelievethatthegenerationandevaluationofcomplexinformalmathematicalproofsisvitallyimportant.Naturallanguageproofsareoftenmoreinter-pretable,whichmakesthemmoredesirableinmanysettings.Forinstance,inaneducationsetting,capabilitiesincomplexmathematicalreasoningmayallowformoreefficientproofgradingorstudenthintgeneration.ThespaceofmathematicalreasoningcanalsoactasatestbedtoAIreasoninginless-definedreal-worldenvironments,whereformalizationwouldbenearlyimpossibleandwhereinterpretabilitywouldbeevermoreimportant.

2.2BenchmarkingMathematicalReasoninginLargeLanguageModels

TheevaluationofmathematicalreasoningcapabilitiesinLLMsreliesheavilyonbenchmarks.Thissectionreviewsexistingbenchmarks,categorizingthembytheirfocusandproblemtypes,andcriticallyexaminestheirlimitations.

Short-AnswerBenchmarks

ThesebenchmarkstypicallyassessLLMsonmathematicalproblemswheretheexpectedoutputisafinalnumericalansweroraconcisetextualresponse.Prominentexamplesinclude:

•MathQA(Aminietal.,2019):DerivedfromtheAQuAdataset,MathQApresentsmathwordproblemswithmultiple-choiceanswers.Itwasdesignedwithanempha-sisoninterpretabilitybymappingproblemstosequencesofpredefinedmathematicaloperations(Aminietal.,

2019

).

•MATH(Hendrycksetal.,2021b):Thisdatasetiscomprisedofchallengingprob-lemsfrommathematicscompetitionssuchastheAmericanMathematicsCompeti-tions(AMC10/12)andtheAmericanInvitationalMathematicsExamination(AIME).Whilesolutionsoftenrequiremultiplereasoningsteps,evaluationtypicallyfocusesonthecorrectnessofthefinalnumericalanswer(Hendrycksetal.,

2021

).

•GSM8K(Cobbeetal.,2021):Consistingofgradeschoolmathematicswordprob-lems,GSM8Kisdesignedtotestmulti-steparithmeticreasoning.Modelsareexpected

8

CHAPTER2.RELATEDWORKS

toproduceanumericalanswerderivedthroughasequenceofcalculations(Cobbeetal.,

2021

).

Despitetheirwidespreaduse,theseshort-answerbenchmarksfaceseveralcriticisms:

•EmphasisonFinalAnswers:Aprimarylimitationistheirpredominantfocusonthecorrectnessofthefinalanswer,oftenneglectingtheintermediatereasoningsteps.Thisevaluationapproachcaninadvertentlyrewardmodelsthatarriveatthecorrectsolutionthroughflawedreasoningprocesses(Petrovetal.,

2025

).

•BenchmarkSaturation:Manyofthesebenchmarksarerapidlyapproachingsatu-ration,withstate-of-the-artLLMsachievingveryhighaccuracyscores(e.g.,over97%onGSM8KasreportedinBudagametal.,

2024

).Thissaturationdiminishestheirefficacyindifferentiatingthecapabilitiesofadvancedmodels.

•DataContaminationandOverfitting:AsignificantconcernisthepotentialforbenchmarkdatatohavebeenincludedinthevasttrainingcorporaofLLMs.Thisdatacontaminationcanleadtoinflatedperformancemetricsthatreflectmemorizationratherthantruereasoningability.Studiesthathavecreatedperturbedversionsofthesebenchmarks,suchasGSM1k(acontamination-freeversionofGSM8K)(H.Zhangetal.,

2024

)andRV-Bench(whichintroducesrandomvariablesintoMATHproblems)(Hongetal.,

2025

),haveoftenobservednotableperformancedrops,suggestingthatmodelsmaybeoverfittingtotheoriginalbenchmarkdistributions.

•LimitedScopeforComplexReasoning:Thesebenchmarks,oftencenteredonarithmeticorbasicalgebraicmanipulation,maynotadequatelytestdeeperconcep-tualunderstanding,abstractreasoning,ortheabilitytoconstructcomplexproofs(K.Huangetal.,

2025

).MathQA,forexample,explicitlyomitsproblemsinvolvinghigher-orderpolynomialsorentirelynon-numericsolutions.

•SensitivitytoInputPerturbations:LLMperformanceonthesebenchmarkscanbesurprisinglyfragile,exhibitingsignificantdegradationinresponsetominoralterationsinproblemphrasing,numericalvalues,ortheintroductionofirrelevantinformation.Thissensitivitysuggestsalackofrobustunderstandingofunderlyingmathematicalprinciples.Forinstance,MATH-P-HardintroduceshardperturbationstoMATHprob-lemssothattheoriginalsolutionstepsdon’tapply,resultinginsignificantperformancedrops.Theresearchersfoundthatthemodelsblindlyapplylearnedproblem-solvingtacticseventowardsunsuitableproblems(K.Huangetal.,

2025

).

•LackofDiversityinProblemTypesandAssessedReasoningSkills:Existingbenchmarksmaynotencompassasufficientlybroadrangeofmathematicaltopicsorthediversecognitiveskillsrequiredforadvancedmathematicalthought.TheMaTTbenchmark,forexample,foundthatLLMperformancecanvarysignificantlyeven

9

CHAPTER2.RELATEDWORKS

acrosscloselyrelatedsubtopicswithinthesamegeneralmathematicalarea(Davoodietal.,

2025

).

Olympiad-LevelandProof-FocusedBenchmarks

Toaddressthelimitationsofshort-answerbenchmarksandtoprobemoreadvancedmathe-maticalreasoning,severalbenchmarksfocusingonOlympiad-levelproblemsandproofgen-erationhavebeendeveloped.

•Minif2f(K.Zhengetal.,

2022

):Thisbenchmarkprovidesacollectionof488formalproblemstatementsderivedfromOlympiad-levelmathematics(AIME,AMC,IMO)andundergraduatecourses,formalizedinmultipleinteractivetheoremprovingsystemslikeLean,Isabelle,andHOLLight.Itprimarilycoversalgebra,numbertheory,andinequalities,withformalizationsdonemanually.

•PutnamBench(Tsoukalasetal.,

2024

):SourcedfromthechallengingWilliam

LowellPutnamMathematicalCompetition,PutnamBenchfeatures1692hand-constructedformalizationsof640theoremsinLean4,Isabelle,andCoq.Currentmodelshave

demonstratedverylimitedsuccessonthisbenchmark,solvingonlyahandfuloftheproblems.

•Omni-math(Gaoetal.,

2025

):ThisbenchmarkaimstobeauniversalOlympiad-levelmathematicsevaluationsuite,comprising4,428problemsspanning33sub-domainsand10difficultylevels.ItutilizesGPT-4oasajudge(Omni-Judge)forevaluatingan-swers,whicharetypicallynumericalorSymPyobjects.

•FrontierMath(Glazeretal.,

2024

):Thisbenchmarkintroduceshundredsoforig-inal,exceptionallychallengingmathematicsproblemsattheadvancedundergraduate,graduate,andresearchlevels,craftedandvettedbyexpertmathematicia