Introduction
The 60th International Mathematical Olympiad (IMO 2019) marked another remarkable chapter in the history of the world’s most prestigious mathematics competition for high school students. Hosted in the historic city of Bath, United Kingdom, the event brought together some of the brightest young mathematicians from around the globe to compete in a celebration of mathematical creativity, logical thinking, and problem-solving excellence.
Every year, the International Mathematical Olympiad challenges students with six carefully crafted problems that go far beyond standard classroom mathematics. Rather than testing memorization or routine calculations, the competition encourages participants to discover elegant ideas, construct rigorous proofs, and solve problems using originality and deep mathematical insight.
The IMO 2019 paper is widely appreciated for its balanced difficulty and beautiful collection of problems covering algebra, geometry, number theory, and combinatorics. Whether you are preparing for future Olympiads, teaching advanced mathematics, or simply enjoy solving challenging problems, the IMO 2019 problem set is an outstanding resource for learning higher-level mathematical thinking.
On this page, you can explore the complete IMO 2019 Problems and Solutions PDF, learn about the competition, and download the official problem set with detailed solutions for offline study.
IMO 2019 at a Glance
The 60th International Mathematical Olympiad was held in Bath, United Kingdom, from 11 July to 22 July 2019. The competition welcomed 112 countries and regions, making it one of the largest IMO editions ever organized.
A total of 621 contestants participated, each attempting six Olympiad-level problems over two examination days. Every problem carried a maximum of 7 points, giving contestants the opportunity to achieve a perfect score of 42 points.
The event celebrated not only mathematical excellence but also international friendship, cultural exchange, and the shared passion for solving beautiful mathematical problems.
Competition Format
IMO 2019 followed the traditional Olympiad format that has remained unchanged for decades.
- Two examination days
- Three problems each day
- Four hours and thirty minutes per session
- Maximum score of 42 points
The six problems represented the major branches of Olympiad mathematics:
- Algebra
- Geometry
- Number Theory
- Combinatorics
Success depended on creativity, logical reasoning, and the ability to present clear and complete mathematical proofs.
Team Results
The 2019 International Mathematical Olympiad featured exceptional performances from teams around the world.
Top Teams
🥇 China
🥈 United States
🥉 South Korea
These countries demonstrated outstanding consistency across all six problems and once again highlighted the strength of their mathematics training programs.
Why Study IMO 2019 Problems?
The IMO 2019 paper remains one of the best collections of Olympiad problems for students preparing for national and international competitions.
Studying these problems helps students:
- Improve logical reasoning
- Develop advanced proof-writing skills
- Learn elegant mathematical techniques
- Build confidence for Olympiad competitions
- Explore creative approaches to difficult problems
Unlike ordinary textbook exercises, IMO problems encourage independent thinking and reward elegant mathematical ideas.
Overview of the Problems
Problem 1
The opening problem introduced contestants to the competition with an elegant question that rewarded careful observation and systematic reasoning. Although approachable, it required a clever mathematical insight.
Problem 2
This problem challenged students to combine logical arguments with creative thinking. A successful solution depended on recognizing hidden relationships within the given conditions.
Problem 3
The final problem of Day One demanded deep mathematical understanding and persistence. It tested contestants’ ability to connect several advanced ideas into a complete proof.
Problem 4
The first problem of Day Two provided an opportunity for students to demonstrate careful reasoning. The problem rewarded those who identified the correct mathematical structure early in the solution.
Problem 5
This challenging question explored sophisticated mathematical ideas and required contestants to think beyond standard techniques. Precision and creativity played an equally important role.
Problem 6
The final problem of the competition represented the highest level of Olympiad mathematics. Solving it required originality, patience, and exceptional proof-writing ability.
Mathematical Ideas Explored in IMO 2019
The IMO 2019 problems introduce students to many important Olympiad techniques, including:
- Invariants and Monovariants
- Mathematical Induction
- Extremal Principle
- Functional Thinking
- Geometric Transformations
- Divisibility and Number Theory
- Combinatorial Arguments
Learning these techniques helps students become stronger mathematical thinkers and prepares them for future Olympiad competitions.
Download IMO 2019 Problems and Solutions PDF
The complete IMO 2019 Problems and Solutions PDF is available for students, teachers, and mathematics enthusiasts who wish to study the competition in detail.
The PDF includes:
- Official IMO 2019 problem statements
- Complete step-by-step solutions
- Clear mathematical notation
- Printable high-quality formatting
- An excellent resource for Olympiad preparation
Students are encouraged to attempt every problem independently before reading the solutions. This approach strengthens analytical thinking and develops confidence in solving unfamiliar mathematical challenges.
Why Previous IMO Papers Matter
One of the best ways to prepare for mathematics competitions is by studying previous International Mathematical Olympiad papers.
Past problems help students:
- Understand Olympiad difficulty levels
- Discover new proof techniques
- Improve mathematical creativity
- Learn time management
- Build confidence before competitions
Many successful Olympiad participants regularly solve previous IMO papers as part of their preparation.
The International Mathematical Olympiad 2019 is remembered as one of the most inspiring editions of the competition. Its carefully designed problems continue to challenge and motivate students around the world, demonstrating that mathematics is not only about finding answers but also about exploring ideas, recognizing patterns, and creating elegant proofs.
Whether you are preparing for national Olympiads, aiming for the International Mathematical Olympiad, or simply enjoy solving beautiful mathematical problems, the IMO 2019 problem set is an invaluable learning resource. Take your time with each question, think creatively, and enjoy the journey of discovering the elegance of mathematics.