Preparing for Math Competitions (Putnam)

The Putnam Competition is an annual undergraduate mathematics contest for students in the United States and Canada, designed to test problem-solving skills and foster mathematical creativity. Organized by the Mathematical Association of America, it consists of a six-hour exam split into two sessions, each featuring six proof-based problems. These problems span topics like calculus, algebra, and number theory, demanding both technical precision and inventive approaches. Roughly 3,600 students participate each year, competing individually and on behalf of their institutions.

This resource explains how to prepare effectively for the Putnam, whether you’re new to competitive math or aiming to improve your rank. You’ll learn strategies for tackling challenging problems, managing time under pressure, and building the mathematical toolkit required for success. The guide covers practice techniques, common problem types, and methods to analyze past solutions. It also addresses how online mathematics students can adapt their independent study habits to meet the competition’s demands.

For online learners, the Putnam offers a structured way to benchmark progress and engage with advanced material outside traditional coursework. Strong performance can strengthen graduate school applications or highlight analytical abilities for careers in tech, finance, or research. Even if you don’t score highly, the process sharpens critical thinking and exposes gaps in foundational knowledge. This article focuses on actionable steps: prioritizing high-yield topics, simulating test conditions, and leveraging digital resources to fill skill gaps. The goal is to help you approach the competition with clarity, turning abstract concepts into reliable problem-solving reflexes.

Understanding the Putnam Competition Structure

The Putnam Competition tests mathematical problem-solving skills under strict time constraints. To perform effectively, you need familiarity with its unique structure, scoring rules, and participation criteria. Below is a detailed breakdown of how the exam operates.

6-Hour Exam Format: Two 3-Hour Sessions

The exam occurs annually on the first Saturday of December. You’ll face two separate 3-hour sessions: one in the morning and one in the afternoon. Each session contains six problems, totaling 12 questions. The morning set (Problem Set A) and afternoon set (Problem Set B) are distinct in content but similar in difficulty.

Key details:

Timing: Sessions typically run from 10 AM–1 PM and 2–5 PM local time.
Materials: You can bring pens, pencils, erasers, and blank paper. No calculators, books, or prewritten notes are allowed.
Focus: The format prioritizes sustained concentration. You’ll need to manage fatigue while maintaining precision over six hours.

All problems require proofs or detailed explanations. Partial answers might earn points, but clear logical steps matter more than final answers alone.

Problem Types: 12 Questions Split Between A and B Sets

Problems range across undergraduate mathematics: calculus, linear algebra, abstract algebra, combinatorics, and geometry. While advanced coursework isn’t required, solving questions often demands creative application of fundamental concepts.

Structure of problem sets:

Set A (Morning): Six problems, numbered A1–A6. Early questions (A1–A2) are relatively accessible, while A5–A6 are more complex.
Set B (Afternoon): Another six problems (B1–B6), following the same difficulty gradient.

Each problem is designed to be solvable in 30 minutes or less under ideal conditions. However, most participants find only a subset fully answerable within the time limit. The exam emphasizes quality over quantity—a single well-solved problem often yields a higher score than multiple incomplete attempts.

Scoring: 10 Points per Problem, Team vs Individual Rankings

Every problem is worth 10 points, awarded for complete or partial solutions. Your individual score is the sum of points across all 12 problems (maximum 120). Team scores aggregate the top three individual scores from students at the same institution.

Critical scoring details:

Partial credit: Even incomplete solutions can earn 1–9 points if they demonstrate progress.
Averages: Historical data shows median scores often fall between 0–2 points. A score of 20+ typically places you in the top 50% of participants.
Rankings: Individual ranks are determined by raw scores. Teams are ranked by summed scores, with separate lists for U.S. and Canadian institutions.

Top performers receive cash prizes or honorary mentions. Roughly the top 20% of participants earn recognition, with the highest scorers named Putnam Fellows.

Team participation requires no separate registration—if three students from the same school take the exam, their scores automatically form a team entry. Individual participation is open to any eligible undergraduate.

Eligibility rules:

You must be an undergraduate enrolled in a U.S. or Canadian institution.
Graduate students or those holding bachelor’s degrees cannot participate.
Each school designates a faculty coordinator to handle registration and proctoring.

Preparing for the Putnam means practicing under timed conditions, analyzing past problems, and refining proof-writing clarity. Focus on building stamina for six hours of intense problem-solving while prioritizing problems that align with your strengths.

Core Mathematical Areas Tested

Preparing for the Putnam exam requires focused study on three mathematical domains that consistently appear in competition problems. Each area demands both technical precision and creative problem-solving strategies. Recognize these core subjects to prioritize your practice effectively.

Algebra and Number Theory Fundamentals

Algebra forms the backbone of problem-solving techniques tested in the Putnam exam. You’ll encounter questions requiring manipulation of inequalities, polynomial identities, and systems of equations. Key topics include:

Linear and abstract algebra: Matrix operations, vector spaces, and eigenvalues appear in problems involving transformations or structured systems.
Number theory: Divisibility rules, modular arithmetic, and Diophantine equations are frequent themes. Expect questions asking for integer solutions to equations like x² + y² = z² or proofs involving prime numbers.
Complex numbers and roots of unity: These often arise in problems combining algebraic manipulation with geometric interpretations.

Focus on efficient symbolic manipulation and proof-based reasoning. For example, you might need to prove a number is irrational or derive a contradiction using modular congruences. Recurring problem patterns involve maximizing/minimizing expressions under constraints or identifying patterns in sequences defined by recursive relations.

Calculus and Analysis Techniques

Single-variable calculus dominates this category, but multivariable concepts occasionally appear. Problems test your ability to evaluate limits, integrals, and series while justifying each step rigorously. Key areas include:

Integral calculus: Compute integrals using substitutions, partial fractions, or symmetry arguments. Definite integrals often hide recursive patterns or require series expansions.
Sequences and series: Analyze convergence using tests like comparison, ratio, or root tests. Look for telescoping series or alternating series in summation problems.
Differential equations: Solve basic separable or linear equations, particularly in applied contexts like growth models or physics-inspired scenarios.

You’ll face problems asking for maximum/minimum values using derivatives or proving continuity/uniform convergence. A common pattern involves transforming a complex integral into a manageable form through substitutions like t = tan(x/2) or exploiting periodicity.

Combinatorics and Probability Patterns

Combinatorial reasoning separates high scorers from average performers. This section tests your ability to count efficiently, often under asymmetric constraints. Core concepts include:

Graph theory: Problems may involve proving properties of trees, calculating chromatic numbers, or applying Eulerian/Hamiltonian path theorems.
Generating functions: Use these to encode sequences and solve recurrence relations, especially in partition problems or counting restricted permutations.
Probability distributions: Expect questions on expected value calculations, conditional probability, or Markov chains, often disguised as real-world scenarios.

Typical problems ask you to count distinct configurations (e.g., seating arrangements with restrictions) or optimize combinatorial structures (e.g., maximizing edges in a graph without cycles). Recurring themes include the pigeonhole principle, inclusion-exclusion, and bijective proofs.

Sharpen your skills in pattern recognition and recursive thinking. For instance, you might derive a formula for the number of surjective functions from one set to another or compute the probability of a game ending in a finite number of steps.

By systematically practicing these areas, you’ll build the versatility needed to tackle the Putnam’s blend of foundational knowledge and inventive problem-solving.

Effective Study Strategies for Problem Solving

This section outlines methods to sharpen your problem-solving abilities for mathematical competitions. Focus on building intuition through targeted practice, refining technical skills, and optimizing test-day performance.

Analyzing Past Putnam Problems (1938-Present)

Prioritize solving actual Putnam problems over generic exercises. Exams from 1938 to today reveal patterns in topic distribution, difficulty curves, and stylistic preferences. Follow these steps:

Categorize problems by subject (e.g., combinatorics, number theory) and difficulty level. Track which areas appear most frequently in recent years.
Solve problems in timed intervals mirroring test conditions. Start with 30-minute blocks per problem, gradually reducing to 20 minutes as efficiency improves.
Identify recurring tactics, such as parity arguments in combinatorics or modular arithmetic in number theory. Note how these apply across different problem statements.
Review solutions line by line, even for problems you answered correctly. Compare your approach to official solutions to spot gaps in reasoning or missed shortcuts.

Focus on problems from the last 20 years first, as they reflect current trends, then work backward to older exams. Save the most challenging problems (e.g., B6 or A6) for later stages of preparation.

Building Proof-Writing Efficiency

Clear communication is as critical as solving the problem. Judges assess both correctness and clarity. Improve your proof-writing with these steps:

Write first drafts without editing, then revise for precision. Remove redundant steps and ensure each sentence logically follows the previous one.
Use standard notation and terminology unless the problem specifies otherwise. Define non-standard symbols if used.
Structure proofs vertically with line breaks between ideas. Highlight key conclusions with phrases like “therefore” or “hence,” but avoid excessive narrative.
Practice “defensive writing”: Anticipate where graders might question a step and preemptively justify it. For example, explicitly state why a function is bijective before using its inverse.

Simulate grading scenarios: Exchange proofs with peers for feedback. Identify instances where implicit assumptions or skipped calculations create confusion. Refactor proofs to address these issues while maintaining brevity.

Time Management During Tests

The Putnam’s 6-hour format demands strategic pacing. You’ll face 12 problems split into two sessions, with most contestants solving only a handful completely. Optimize your approach:

Allocate 10 minutes per problem initially. Scan all questions in the first 30 minutes, marking targets based on your strengths. Classify problems as:
- Immediate: Concepts align with your preparation.
- Potential: Solvable with extended focus.
- Unlikely: Topics outside your current skill set.
Spend no more than 25 minutes on any single problem during the first 3 hours. If stuck, document partial progress and revisit later.
Divide the test into phases:
- Phase 1 (Hours 1-2): Solve all “immediate” problems.
- Phase 2 (Hours 3-4): Tackle “potential” problems with partial progress.
- Phase 3 (Hours 5-6): Review solutions for errors and expand on incomplete answers.

Track time rigorously using a wristwatch or timer. Abandon problems that show no progress after 15 minutes—return to them only if time permits. Prioritize problems with higher point yields first, but balance this with confidence in your answers.

Accept that incomplete solutions earn partial credit. A clear, structured argument for a single case often scores higher than a disorganized attempt at a full proof.

Practice Techniques and Mock Exams

Effective preparation for math competitions requires structured practice that mirrors actual testing conditions. This section outlines methods to build problem-solving stamina, refine your approach through feedback, and leverage historical materials strategically.

Weekly Timed Problem Sets

Consistent timed practice is the most reliable way to adapt to Putnam’s pace. Each week, block 3-6 hours for sessions that replicate the competition’s two-part format (3 hours per set). Use a timer set to strict 3-hour limits, and work in a distraction-free environment.

Key steps:

Select problems matching Putnam’s difficulty distribution: Include 2-3 easy, 4-5 medium, and 1-2 challenging problems per set.
Simulate test-day constraints: Write solutions by hand, using only permitted materials (e.g., pencils, blank paper).
Review immediately after each session: Identify patterns in errors—common issues include misinterpreting definitions, algebraic slips, or time spent on unproductive paths.
Track progress quantitatively: Record time per problem, success rates by topic, and solution clarity scores (1-5 scale).

Adjust problem selection every 4-6 weeks to target weak areas. Allocate 30 minutes post-session to rework failed problems without time pressure.

Peer Review Sessions for Solution Optimization

Collaborative analysis sharpens precision in mathematical communication. Organize weekly 2-hour meetings with 3-5 peers to dissect solutions. Focus on optimizing proofs for clarity, rigor, and brevity—critical for maximizing scoring in competitions.

Structure sessions as follows:

Each participant submits 1-2 original solutions to shared problems in advance.
Rotate roles: one person presents their solution step-by-step while others challenge gaps in logic or suggest alternative methods.
Vote on the most efficient valid solution, then rewrite it collectively to meet judging standards.

Common improvements identified in peer review:

Overly verbose explanations that obscure key ideas
Insufficient justification for non-trivial claims
Missing edge cases in induction or counterexamples
Inefficient algebraic manipulations

Maintain a shared document of optimized solutions and recurring errors. Revisit this document monthly to measure improvement in writing quality.

Utilizing MAA’s Historical Problem Archives

Past Putnam problems provide the closest approximation to actual competition material. Prioritize problems from the last 25 years, as older exams may reflect outdated styles.

Implementation strategy:

Categorize problems by topic and difficulty: Create custom problem sets mixing algebra, analysis, combinatorics, and geometry. Include at least one "A1" and "B1" problem (typically easier) in each set.
Build a cycle of exposure and recall: Solve each problem twice—first under timed conditions, then weeks later with fresh eyes. Compare solutions to identify persistent blind spots.
Reverse-engineer official solutions: Study the structure, notation, and pacing of high-scoring responses. Note how top solutions minimize extraneous computation while maintaining rigor.

For mock exams, use consecutive years’ Problem A and B sets (e.g., 2021 A with 2020 B). This prevents overfitting to a single year’s style. After grading, calculate a estimated score using historical Putnam percentile data to gauge readiness.

Integrate these techniques systematically: timed sets build speed and accuracy, peer reviews eliminate inefficiencies, and historical problems calibrate expectations. Adjust the balance between new problems and review based on performance data—dedicate 60% of weekly study time to new material and 40% to refining past work.

Digital Resources and Online Communities

Effective preparation for the Putnam exam requires combining structured practice with peer collaboration and visual learning. The following tools and platforms address these needs through problem-solving frameworks, discussion spaces, and visualization aids.

MAA's Online Problem-Solving Workshops

Structured workshops focus on building the analytical skills needed for competition-level mathematics. These sessions emphasize dissecting complex problems into manageable steps, identifying patterns in abstract scenarios, and applying proof techniques common in Olympiad-style exams.

Weekly problem sets replicate the difficulty curve of Putnam questions, starting with foundational exercises and progressing to multi-layered proofs. Collaborative group work forms a core component, mirroring the brainstorming process used by successful teams. Participants submit written solutions for detailed feedback, which highlights logical gaps and suggests alternative approaches.

Workshops often incorporate timed problem-solving drills to simulate test conditions. Regular participation strengthens time management under pressure and exposes you to recurring themes in number theory, combinatorics, and real analysis.

Art of Problem Solving Forum Strategies

Active engagement in problem-solving forums sharpens your ability to analyze and articulate mathematical arguments. Start by reviewing existing threads on past Putnam questions to observe solution structures. Focus on posts that deconstruct proofs step-by-step, noting how conclusions are justified from initial premises.

Create a systematic approach to posting your own solutions:

Clarify the problem statement in your own words to confirm understanding
Outline preliminary ideas before drafting a full solution
Request specific feedback on areas like induction steps or combinatorial reasoning
Revise solutions based on critiques to eliminate ambiguities

Use the forum’s search function to find discussions on specific topics (e.g., generating functions, graph theory). Bookmark threads that compile advanced techniques or common pitfalls. Participate in mock exam threads where users attempt problems under timed conditions and compare results.

Interactive Math Software for Visualization

Dynamic visualization tools help internalize abstract concepts central to the Putnam exam. For geometry and calculus problems, software like GeoGebra lets you manipulate figures to test conjectures about angles, areas, or limits. Plot parametric equations or 3D surfaces to identify symmetries and constraints.

Desmos is particularly effective for exploring inequalities or trigonometric identities. Input both sides of an equation and adjust variables to observe convergence/divergence points. For number theory, use Wolfram Mathematica to generate prime factorizations, modulo sequences, or recursive function outputs.

When tackling optimization problems, visualize constraint boundaries using graphing tools. Overlay multiple functions to pinpoint maxima/minima or feasible regions. For linear algebra questions, matrix manipulation features help verify properties like invertibility or eigenvalues.

Integrate these tools into your study routine:

Pre-solve exploration: Use graphs to form hypotheses before attempting proofs
Post-solve verification: Check algebraic results against software-generated outputs
Counterexample generation: Test the validity of statements by plugging edge cases

Prioritize platforms with screenshot or animation capabilities to document your visual reasoning process. This habit builds intuition for recognizing patterns during time-constrained exams.

Combine these digital resources with disciplined practice to develop the speed, accuracy, and creativity demanded by the Putnam competition.

100-Hour Preparation Plan

This structured six-week plan allocates 100 hours to build skills for the Putnam exam. Each phase targets specific goals, progressing from core problem-solving techniques to full exam simulations. The schedule assumes you commit 15-18 hours weekly, adjusted based on individual progress.

Week 1-2: Foundation Building with Basic Problems

Spend 8-10 hours per week reviewing fundamental concepts and solving entry-level problems. Focus on:

Core topics: Algebra, single-variable calculus, basic number theory, and introductory combinatorics
Problem types: Questions labeled A1 or B1 from past Putnam exams, standard textbook exercises
Skill development: Clear written explanations, error-checking, and time management for 10-minute problems

Start each session with 5-10 minutes of mental math drills (prime factorizations, rapid derivatives, modular arithmetic). Work through 3-5 problems daily, prioritizing ones you initially find challenging. Use this phase to:

Identify gaps in foundational knowledge
Standardize your problem-solving process: read, restate, strategize, execute, verify
Build a reference notebook for frequently used formulas and theorems

Allocate 2 hours weekly to reattempt problems you solved incorrectly. Track recurring error types—these define your focus areas for later weeks.

Week 3-4: Advanced Concept Drills

Increase weekly study time to 10-12 hours, shifting to complex problems requiring layered solutions. Prioritize:

Advanced topics: Abstract algebra, real analysis, generating functions, combinatorial geometry
Problem types: A3/B3-level Putnam questions and Olympiad-style problems
Strategy refinement: Multi-step proof construction, non-routine substitutions, exploiting symmetries

Complete timed problem sets: 3 questions in 90 minutes, mimicking the exam’s pace. Focus on:

Pattern recognition: Classify problems by solution archetypes (invariants, extremal principles, etc.)
Efficiency: Compare multiple solution paths for the same problem
Proof rigor: Eliminate hand-waving arguments; formalize intuitive ideas

Spend 30% of your time analyzing published solutions. Reverse-engineer them to isolate key insights, then apply those techniques to new problems. Dedicate one session weekly to writing “perfect” solutions—polished, concise, and fully justified.

Week 5-6: Full-Length Simulation Exams

Simulate actual exam conditions twice weekly using:

Past Putnam papers (2010-2022 recommended)
Original 6-hour sessions with strict timing
Graded scoring: 10 points per problem, no partial credit for incomplete answers

Exam day protocol:

Scan all 12 problems in the first 20 minutes
Rank problems by perceived solvability
Allocate 45-minute blocks to high-priority questions
Reserve final hour for cleanup and alternative approaches

Post-exam analysis:

Categorize missed problems by topic and error type
Calculate time wasted on unproductive approaches
Identify 2-3 strategic pivots for future attempts

Between simulations, revisit the 10 most impactful problems from earlier weeks. Solve them again under altered constraints: different time limits, verbal explanations aloud, or restricted scratch paper. Use the final 3 days to review your error log and solidify high-yield strategies.

Adjust the plan’s intensity based on performance data: if simulations show consistent weakness in analysis problems, reallocate time to targeted proof drills. Maintain a 1:1 ratio between problem-solving and focused review—every hour spent on new questions requires an hour analyzing past work.

Avoiding Common Preparation Errors

Success in math competitions depends on avoiding predictable errors that hinder progress. Many participants focus narrowly on technical skill development while ignoring strategic preparation gaps. Below are three critical mistakes to eliminate from your preparation process.

Overlooking Historical Problem Trends

Math competitions like the Putnam Exam repeat patterns in problem design, even when specific questions change. Failing to analyze historical problems leaves you unprepared for recurring themes in algebra, combinatorics, and calculus.

Identify topic frequency by categorizing past problems into major areas:

Linear algebra techniques in geometry proofs
Generating functions in combinatorial sequences
Inequalities with calculus-based optimization

Track shifts in emphasis over decades. For example, number theory problems dominated the 1980s Putnam but decreased in frequency after 2000, while abstract algebra questions became more common. Adjust your study plan to reflect these trends without overfitting to a single era.

Avoid assuming repetition. While patterns exist, each exam introduces novel problem structures. Balance historical analysis with adaptive problem-solving by:

Solving at least 20 years of past problems
Grouping similar problems to detect underlying methods
Creating "trend reports" summarizing shifts in question types

Neglecting Time Constraints During Practice

The Putnam’s six-hour format demands sustained focus under pressure. Practicing without time limits builds technical competence but fails to develop exam-specific stamina.

Simulate real testing conditions:

Use a timer for every practice session
Allocate 30 minutes maximum per problem initially
Gradually increase session length to match the full six-hour exam

Track time distribution to identify bottlenecks. Common issues include:

Spending 45+ minutes on early problems, leaving insufficient time for later sections
Getting stuck on complex proofs without switching to more approachable questions
Failing to leave time for reviewing partial solutions

Prioritize problems strategically during practice. Solve high-value questions first based on your strengths, and skip problems requiring unfamiliar techniques until after addressing solvable ones.

Underutilizing Team Collaboration Opportunities

While the Putnam is an individual competition, group study accelerates skill development. Isolating yourself limits exposure to alternative problem-solving methods and feedback.

Form or join a study group focused on:

Collaborative problem-solving sessions
Peer reviews of proof clarity and rigor
Sharing resources like uncommon theorems or problem-solving heuristics

Assign roles during group work to maximize efficiency:

One member presents a solution step-by-step
Others identify gaps in logic or suggest optimizations
Rotate roles to develop both presentation and critical analysis skills

Balance group sessions with individual practice. Over-reliance on collaborative work can mask gaps in personal understanding, while excessive solo practice limits perspective. Aim for a 2:1 ratio of individual to group study time.

Avoid unproductive collaboration by setting clear goals for each session. Groups that devolve into social gatherings or unstructured discussions waste time. Use agendas like "Solve three A1-A3 problems in 90 minutes" to maintain focus.

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This section avoids vague advice in favor of concrete strategies. Apply these corrections systematically to convert preparation effort into measurable performance gains.

Key Takeaways

Here's what you need to remember about Putnam preparation:

Study past problems weekly to identify common themes and solution structures
Practice solving problems under timed conditions (90-120 minute blocks) to build speed and accuracy
Alternate focused solo practice with group discussions to expose gaps in reasoning and learn new approaches

Next steps: Create a weekly schedule allocating fixed time for timed drills, pattern analysis, and collaborative sessions. Start with recent Putnam exams and track recurring problem types.

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Preparing for Math Competitions (Putnam)