Introduction
The 62nd International Mathematical Olympiad (IMO 2021) stands as one of the most memorable editions in the history of mathematical competitions. Organized during a challenging period for the world, the event brought together the brightest young mathematicians from across the globe through a carefully coordinated remote format. Despite the unique circumstances, the spirit of competition, creativity, and collaboration remained stronger than ever.
For aspiring Olympiad students, teachers, and mathematics enthusiasts, the IMO 2021 problem set offers an exceptional collection of problems that combine elegant ideas with deep mathematical reasoning. From clever algebraic manipulations to sophisticated geometry, combinatorics, and number theory, every problem encourages participants to think beyond standard formulas and develop original solutions.
On this page, you will find the complete IMO 2021 Problems and Solutions PDF, along with an overview of the competition, team rankings, and insights into why these problems continue to inspire students worldwide.
IMO 2021 at a Glance
The 62nd International Mathematical Olympiad (IMO 2021) was hosted by Saint Petersburg, Russia, and took place from July 14 to July 24, 2021. Although the official host city was Saint Petersburg, most contestants participated from their home countries under local supervision due to international travel restrictions.
The competition featured 107 countries and regions, making it one of the largest IMO editions ever organized. A total of 619 contestants competed over two examination days, solving six challenging problems designed to test creativity, logical reasoning, and advanced mathematical thinking.
Each contestant attempted three problems per day, with every problem carrying a maximum of 7 points, resulting in a perfect possible score of 42 points.
Competition Format
The IMO follows a traditional format that has remained unchanged for decades.
- Day 1: Problems 1, 2, and 3
- Day 2: Problems 4, 5, and 6
- Time Allowed: 4 hours and 30 minutes each day
- Maximum Score: 42 points
The problems cover four major areas of Olympiad mathematics:
- Algebra
- Geometry
- Number Theory
- Combinatorics
Rather thIMO 2021 Team Results
The 2021 competition witnessed an outstanding performance from several countries, with traditional mathematical powerhouses dominating the leaderboard.
Top 10 Teams
🥇 China – 208 Points
🥈 Russia – 193 Points
🥉 South Korea – 190 Points
- United States – 183 Points
- Poland – 173 Points
- Vietnam – 170 Points
- Japan – 167 Points
- Taiwan – 159 Points
- Romania – 158 Points
- India – 158 Points
China once again demonstrated remarkable consistency, securing the overall championship with an impressive team performance.an testing memorization, the competition rewards originality, elegant proofs, and creative problem-solving.
Participating Countries
The IMO 2021 welcomed 107 countries and regions, including:
China, United States, India, Russia, United Kingdom, Germany, Japan, South Korea, Vietnam, Singapore, Canada, Australia, France, Italy, Brazil, Romania, Poland, Iran, Turkey, Ukraine, Kazakhstan, Mexico, Thailand, Indonesia, Malaysia, South Africa, Spain, Switzerland, Sweden, Norway, Netherlands, Israel, Belgium and many other nations representing every continent.
The diversity of participants highlights the truly global nature of the International Mathematical Olympiad.
Why Study IMO 2021 Problems?
The IMO 2021 problem set is widely appreciated for its balance between accessibility and depth.
Students studying these problems will encounter powerful mathematical techniques such as:
- Invariants and Monovariants
- Functional Equations
- Extremal Principles
- Geometric Transformations
- Number Theoretic Arguments
- Combinatorial Constructions
Working through these problems develops analytical thinking that extends far beyond mathematics competitions and is valuable in computer science, engineering, and research.
Overview of the Six Problems
Problem 1
A beautiful introductory problem that rewards careful observation and logical reasoning rather than complicated calculations.
Problem 2
An elegant geometry problem requiring visualization, construction, and proof techniques.
Problem 3
One of the most demanding problems of Day 1, combining deep insight with creative algebraic thinking.
Problem 4
A cleverly designed combinatorial challenge where identifying the right strategy is more important than computation.
Problem 5
An advanced number theory problem that explores hidden patterns and surprising relationships.
Problem 6
The final problem presents a sophisticated mathematical structure that tests persistence, originality, and proof-writing skills at the highest level.
Download IMO 2021 Problems and Solutions PDF
The complete IMO 2021 Problems and Solutions PDF is available for students, teachers, and Olympiad enthusiasts who wish to practice offline.
The PDF includes:
- Official problem statements
- Complete detailed solutions
- Clear mathematical notation
- Printable high-quality formatting
- Ideal material for Olympiad preparation and classroom discussion
Whether you are preparing for national Olympiads or simply enjoy challenging mathematics, this resource provides an excellent opportunity to strengthen your problem-solving abilities.