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A Guide to Mathematics Competitions for University Students

A comprehensive guide to mathematics competitions at the university level, including the Putnam, IMC, and others, with advice on preparation, training, and why competitions matter beyond prizes.

ResourcesProblem Solving

Why Competitions Matter

Mathematics competitions are one of the most effective ways to develop problem-solving skills. They push you to think creatively, work under time pressure, and tackle problems that require insight rather than routine techniques.

But competitions are valuable even if you never win a prize. The preparation process — working through challenging problems, learning new techniques, and training your mathematical intuition — has lasting benefits for your mathematical career.

This guide covers the major university-level mathematics competitions, how to prepare for them, and where to find problems and resources.

Major University-Level Competitions

The William Lowell Putnam Mathematical Competition

The Putnam Competition

The Putnam is the most prestigious mathematics competition for university students in the United States and Canada. Run annually since 1938 by the Mathematical Association of America, it consists of:

  • 12 problems divided into two 3-hour sessions (A1–A6 in the morning, B1–B6 in the afternoon)
  • 60 points maximum (each problem is worth 10 points)
  • Problems cover a wide range of topics: algebra, analysis, combinatorics, number theory, and geometry

The Putnam is notoriously difficult. The median score in a typical year is 0 or 1 out of 60. Solving even one or two problems is a significant achievement.

Key fact: You do not need to be a math major to take the Putnam. Any enrolled undergraduate at a participating institution can compete. Talk to your mathematics department in the fall to register.

The International Mathematics Competition (IMC)

The IMC

The IMC is the premier international mathematics competition for university students. Held annually (typically in Blagoevgrad, Bulgaria), it brings together teams from universities worldwide. The competition features:

  • 10 problems over two days (5 problems per day, 5 hours per session)
  • Problems requiring creativity and deep mathematical knowledge
  • Gold, silver, and bronze medals for top performers

Mathematical Contest in Modeling (MCM/ICM)

COMAP MCM/ICM

The MCM/ICM is a team-based competition (teams of 3) where students spend several days working on an open-ended applied mathematics problem. This competition tests mathematical modeling, writing, and teamwork rather than speed and cleverness.

National and Regional Competitions

Many countries and regions have their own university-level competitions:

  • United Kingdom: British Mathematical Olympiad Senior Round
  • Germany: Bundeswettbewerb Mathematik
  • Romania: National Mathematics Olympiad (university sections)
  • China: Chinese Mathematical Olympiad (CMO) and Chinese Mathematics Competitions
  • India: Indian National Mathematical Olympiad (INMO)

Check with your national mathematical society for competitions in your country.

High School Competitions (Worth Knowing About)

Even as a university student, familiarity with high school competition problem styles is useful, as many techniques carry over:

International Mathematical Olympiad (IMO)

The IMO

The IMO is the world's most prestigious high school mathematics competition. Past IMO problems are an incredible resource for university students — the difficulty level of IMO problems overlaps with easier Putnam problems.

American Mathematics Competitions (AMC/AIME/USAMO)

The AMC pathway (AMC 10/12 → AIME → USAMO) produces thousands of problems ranging from easy to extremely difficult. The USAMO and AIME problems are excellent training for university-level competitions.

How to Prepare for Competitions

Start with Past Problems

The single most effective preparation strategy is working through past competition problems:

Essential Problem-Solving Techniques

University-level competition problems commonly require these techniques:

Algebra and inequalities:

  • AM-GM, Cauchy-Schwarz, and power mean inequalities
  • Polynomial tricks (roots, coefficients, Vieta's formulas)
  • Functional equations

Combinatorics:

  • Bijective proofs and counting arguments
  • Pigeonhole principle
  • Generating functions
  • Graph theory (coloring, Ramsey theory)

Number theory:

  • Modular arithmetic and the Chinese Remainder Theorem
  • Fermat's little theorem and Euler's theorem
  • Quadratic residues
  • p-adic valuations

Analysis and calculus:

  • Clever substitutions in integrals
  • Inequalities involving integrals and sums
  • Convergence and divergence arguments
  • The intermediate value theorem in creative settings

Recommended Books

The core library for Putnam preparation:

  1. Putnam and Beyond by Gelca and Andreescu — The definitive Putnam preparation book
  2. Problems from the Book by Andreescu and Dospinescu — Beautiful problems organized by topic
  3. The Art and Craft of Problem Solving by Zeitz — Excellent for building problem-solving strategies
  4. Problem-Solving Through Recreational Mathematics by Averbach and Chein — Gentler introduction

Join or Start a Problem-Solving Group

Working with others is one of the most effective ways to improve. Many universities have:

  • Putnam preparation seminars — Check if your math department offers one
  • Math clubs — Student-run organizations that meet to solve problems
  • Online communities — The AoPS forums and Math Stack Exchange have active competition communities

Training Schedule

A realistic training plan for the Putnam might look like:

TimelineActivity
SeptemberForm a study group, review past A1/B1 problems (the easiest)
OctoberWork through A2/B2 problems, study inequalities and combinatorics
NovemberAttempt full past exams under timed conditions
Early DecemberTake the Putnam, then review the solutions
OngoingSolve 2–3 competition problems per week throughout the year

Online Platforms for Competition Practice

Art of Problem Solving (AoPS)

AoPS hosts:

  • Forums where students discuss problems from every major competition
  • An extensive wiki with problem solutions and technique articles
  • Online courses specifically designed for competition preparation
  • Alcumus, an adaptive problem-solving system

Brilliant

Brilliant offers interactive problem-solving courses that build the kind of thinking needed for competitions.

Project Euler

Project Euler combines mathematical problem-solving with programming. Many problems are inspired by number theory and combinatorics.

Codeforces and AtCoder (for mathematical programming)

While primarily competitive programming platforms, Codeforces and AtCoder feature many problems that require deep mathematical insight. The "mathematics" and "number theory" tags on Codeforces are particularly relevant.

Competition Culture and Community

Important perspective: Competitions are a tool for growth, not a measure of your worth as a mathematician. Many outstanding mathematicians never competed or did not excel in competitions. The skills you develop — creative problem-solving, working under pressure, learning new techniques — are what matter, not the rankings.

Famous Mathematicians Who Competed

Many prominent mathematicians were competition participants:

  • Terence Tao won IMO gold at age 13 (the youngest gold medalist in history at the time)
  • Grigori Perelman won a perfect score at the 1982 IMO
  • Maryam Mirzakhani won two IMO gold medals
  • Peter Scholze won three IMO gold medals

But equally many great mathematicians — including Alexander Grothendieck and William Thurston — did not come from competition backgrounds. There are many paths to mathematical excellence.

The Putnam Fellows

Being named a Putnam Fellow (top 5 scorers nationally) is one of the most prestigious achievements in undergraduate mathematics. Past Fellows include Richard Feynman, John Milnor, and many other distinguished mathematicians and scientists. The list of past Putnam Fellows is published annually.

Benefits Beyond Competition Day

Even if you never score exceptionally well on the Putnam or IMO, competition training offers:

  1. Stronger problem-solving instincts. You become faster at identifying which techniques might work on a given problem.
  2. Better proof-writing skills. Competitions demand clear, complete arguments written under time pressure.
  3. A broader mathematical toolkit. You encounter techniques and ideas from across mathematics.
  4. Confidence. Solving hard problems builds the belief that you can tackle difficult mathematics.
  5. Community. The competition community is collegial and motivating.

Final Thoughts

Mathematics competitions are not for everyone, and that is perfectly fine. But if you enjoy the challenge of solving hard problems and want to push yourself mathematically, competitions offer an unmatched opportunity. Start with past problems, find a study group, and focus on the process of learning rather than the outcome.

References

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