IMO Practice Test

Welcome to our comprehensive Math Olympiad guide, designed to help you master the IMO Test and showcase your mathematical prowess in the International Mathematics Olympiad. Whether you're a seasoned math enthusiast or just starting your journey, our valuable strategies and tips will unlock your full potential and enable you to compete globally.

Free IMO Practice Test Online

The IMO holds immense significance as it promotes critical thinking, problem-solving skills, and collaboration among young mathematicians on a global scale. It provides a platform for participants to showcase their mathematical prowess, engage in friendly competition, and build lasting friendships with like-minded individuals who share their passion for mathematics.

By understanding the origins, structure, and purpose of the International Mathematics Olympiad, you can better appreciate the journey ahead as you prepare to compete in this prestigious event.

IMO Exam at a Glance

Free IMO - International Mathematics Olympiad Study Tools

Interactive IMO - International Mathematics Olympiad resources on PracticeTestGeeks, free to start:

• Free IMO Logical Reasoning Questions and Answers — 11 questions with instant scoring and answer explanations
• IMO - International Mathematics Olympiad flashcards — review key terms and concepts card by card
• IMO - International Mathematics Olympiad study guide — topic-by-topic overview of what the exam covers
• IMO - International Mathematics Olympiad cheat sheet — one-page printable summary for last-minute review

According to International Mathematical Olympiad (IMO Board / IMO Association); IMO 2027 is run by the HUN-REN Rényi Institute, Budap, “The competition consists of 6 problems, with each problem worth 7 points for a maximum total score of 42 points.”^[1]

Worked examples: how to reason through IMO questions

These are practice questions written for this site, not questions from the real exam. Each explanation shows why the right answer is right and why the others are wrong.

Example 1

Question: Which of the following numbers is a prime number?

1. 21
2. 27
3. 29
4. 33

Answer: C. 29

Explanation:

29 is the correct answer because its only divisors are 1 and itself, which is the definition of a prime. Each wrong option can be broken into two smaller whole-number factors. 21 is not prime because it divides evenly by a smaller number than itself. 27 and 33 fail for the same reason. Each has a divisor other than 1 and itself, so each is composite.

The rule to remember: to test a number for primality, try dividing it by the small primes in turn. You only need to go as far as the point where the divisor would be larger than the quotient. Odd numbers are not automatically prime, and all of the options here are odd, which is exactly the trap.

Example 2

Question: Ria is standing in a row. If she is 6th from the left and 20th from the right, how many people are standing in the row?

1. 27
2. 24
3. 26
4. 25

Answer: D. 25

Explanation:

The answer is 25. Ria is counted in the 6 people from the left and again in the 20 from the right, so adding those gives a total in which she appears twice. Take her out once to get 25.

The wrong options come from mistakes with that overlap. 26 is the plain sum of 6 and 20, which forgets that she is counted twice. 27 adds on a person instead of removing one. 24 removes too many people, treating her as extra on top of the double count.

The rule to remember: for a person in a line, total = position from one end + position from the other end, minus the person counted twice.

Example 3

Question: Aman is standing in a row. If he is 12th from the left and 16th from the right, how many people are standing in the row?

1. 26
2. 27
3. 28
4. 29

Answer: B. 27

Explanation:

The answer is 27. Aman is 12th from the left and 16th from the right. Adding 12 and 16 counts Aman in both positions, so you remove him once from the sum to get 27.

The wrong options show common slips. 28 is the plain sum, which forgets the double count. 26 takes away too many people. 29 adds people rather than removing the repeated one.

The rule to remember: when two counts from opposite ends both include the same person, add them and then correct for the overlap. A quick check is to picture the row split into the people before Aman, Aman himself, and the people after him, and confirm that the pieces add up to the same total.

How to use these practice tests

Start with the official material. Read the regulations and the about page so you know the format: proof problems, no calculators, written on paper. Then use our topic sets to find where you are weakest.

Work through each set in turn: Number Theory, Algebra and Equations, Inequalities, Geometry and Combinatorics and Counting. Use Patterns and Sequences, Logical Reasoning, Arithmetic and Operations and Integers and Data Handling as warm-ups for speed and accuracy.

1. Try each question on paper without a calculator, and write your reasoning out in full sentences as if it were a proof.
2. Only then check the answer, and read the explanation even when you were right.
3. Keep a list of the ideas you missed, such as a counting overlap or a divisibility test, and revisit that list each week.
4. Move on to past official problems from the IMO site, which are the real standard. Our questions are written by us and are not real exam questions.

Near the contest, practise under timed conditions on paper, one contest day at a time. Our practice sets are written with AI tools, so check unfamiliar methods against a textbook or your national olympiad training material.

Create an Effective Study Plan

Preparation Essentials for IMO Test

Proper preparation is essential to ace the IMO Test and showcase your mathematical abilities. In this section, we will guide you through the key components of an effective study plan, recommended study materials, and useful resources to maximize your preparation and build a strong foundation.

Developing a well-structured study plan is crucial to ensure comprehensive preparation for the IMO Test. Here are some essential elements to consider:

1. Set Clear Goals: Define specific goals for each study session, focusing on the areas you need to improve.
2. Allocate Time Wisely: Plan your study sessions in advance, allocating sufficient time for each topic based on its importance and difficulty.
3. Vary Your Study Methods: Incorporate a mix of problem-solving, concept review, and practice tests to keep your study routine engaging and effective.

Recommended Study Materials

Choosing the right study materials is vital for comprehensive preparation. Here are some recommended resources to consider:

• Textbooks and Workbooks: Select renowned textbooks and workbooks specifically designed for the IMO Test, which cover a wide range of mathematical concepts and problem-solving strategies.
• Previous IMO Papers: Practice solving previous years' IMO papers to familiarize yourself with the test format, types of questions, and time constraints.
• Online Courses and Video Tutorials: Supplement your study plan with online courses and video tutorials that offer in-depth explanations of complex topics and provide additional practice exercises.

Algebraic Manipulations

Problem-Solving Techniques for IMO Test

The International Mathematics Olympiad (IMO) is renowned for its challenging math problems that test the problem-solving abilities of participants. To succeed in the IMO Test, it is essential to master various problem-solving techniques that are specifically tailored for this competition.

Algebra plays a significant role in the IMO Test, and mastering algebraic manipulations is crucial. By understanding and using algebraic techniques such as factoring, expanding, and simplifying expressions, you can unravel complex math problems and find elegant solutions.

Combinatorics Strategies

Combinatorics refers to the study of counting and arranging objects. In the IMO Test, combinatorics problems often involve permutations, combinations, and counting principles. Developing a solid understanding of combinatorics concepts and strategies will enable you to tackle these problems effectively.

Geometry Strategies

Geometry problems in the IMO Test test your spatial reasoning and logical thinking skills. Familiarize yourself with the fundamental principles of geometry, including the properties of triangles, circles, and polygons. Mastering geometry strategies such as drawing accurate diagrams, applying geometric transformations, and using coordinate geometry can significantly enhance your problem-solving capabilities.

Number Theory Approaches

Number theory focuses on the properties and relationships of numbers. Many IMO Test problems involve number theory concepts, such as prime numbers, divisibility rules, and modular arithmetic. Learning number theory approaches like the Euclidean algorithm, the Chinese remainder theorem, and the pigeonhole principle will equip you with the necessary tools to solve these types of problems.

Incorporating Multiple Techniques

While each problem-solving technique mentioned above has its strengths, it is essential to recognize that complex IMO Test problems often require a combination of approaches. Developing the ability to identify which techniques to use in different problem scenarios and applying them strategically will significantly improve your chances of success.

1. Plan your time wisely

Time Management Strategies for IMO Test

When it comes to the IMO Test, time management is essential. This highly competitive exam requires you to solve complex math problems within a limited timeframe. To help you excel, we have curated a list of effective time management strategies and techniques to optimize your performance and maximize your score.

Before diving into the exam, create a plan outlining how much time you will allocate to each problem. Read through the entire paper to gain an understanding of the difficulty level and prioritize questions accordingly. By setting a time limit for each problem, you can ensure that you do not spend too much time on a single question.

2. Practice timed exercises

One of the best ways to improve your time management skills is by practicing timed exercises. Set a timer and attempt problems from past IMO Tests or other math Olympiad resources. By practicing under time constraints, you can develop a sense of speed and maintain accuracy without compromising the quality of your solutions.

3. Break down complex problems

When faced with a challenging problem, it's important to break it down into smaller, manageable steps. This will help you avoid feeling overwhelmed and enable you to tackle the problem piece by piece. By breaking down complex problems, you can allocate your time more effectively and efficiently find a solution.

4. Use shortcuts and strategies

Learn and practice problem-solving shortcuts and strategies that can help you solve problems more efficiently. Familiarize yourself with common mathematical formulas, theorems, and patterns that can provide insights and save valuable time during the exam.

5. Stay focused and avoid distractions

During the IMO Test, it's crucial to stay focused and avoid distractions. Find a quiet and comfortable place to take the exam, free from interruptions. Switch off your phone and any other electronic devices that may divert your attention. By eliminating distractions, you can concentrate fully on the problems at hand.

6. Review and manage your time

As you progress through the exam, periodically review your time management. Check if you are on track according to your initial plan. If you find yourself spending too much time on a particular question, consider moving on and revisiting it later if time permits. Managing your time effectively throughout the exam will ensure that you have sufficient time for all the problems.

7. Practice mindfulness techniques

Mindfulness techniques can help improve your focus and reduce anxiety during the exam. Before starting the test, take a few deep breaths to calm your mind and center yourself. Throughout the exam, maintain a relaxed and confident mindset while staying present in the moment. This will enable you to approach each problem with clarity and concentration.

8. Assess and refine your strategies

After completing practice exams or mock tests, take the time to assess your time management strategies. Identify areas where you may have struggled or could have been more efficient. Refine your strategies accordingly and continue practicing until you feel confident in your ability to manage time effectively.

1. Visualize Success

Mental Preparation and Test-Taking Strategies

In addition to academic preparation, mental fortitude and effective test-taking strategies are crucial for success in the IMO Test. Developing a positive mindset and overcoming exam anxiety can significantly enhance your performance. Here are some techniques to help you mentally prepare for the test and optimize your test-taking experience:

Before the IMO Test, take a moment to visualize yourself confidently solving challenging math problems and achieving your desired results. Visualizing success can help boost your confidence and motivation, allowing you to approach the test with a positive attitude.

2. Stay Calm and Manage Anxiety

Exam anxiety is natural, but it's important to manage it effectively. Practice deep breathing exercises or other relaxation techniques to calm your mind and body before the test. Remind yourself of your preparation and capabilities, and focus on staying present in the moment.

3. Time Management

Efficient time management is key during the IMO Test. Make sure to allocate your time wisely for each question, keeping in mind the difficulty level and marks assigned. Be mindful of the clock, but don't rush through the problems. Prioritize accuracy while maintaining a steady pace.

4. Read Carefully and Understand the Questions

Take your time to read each question thoroughly and understand the requirements. Pay attention to specific keywords or phrases that can guide your problem-solving approach. Break down complex problems into smaller, manageable steps to avoid confusion and ensure clarity.

5. Use Strategic Guessing

If you're unsure about the answer to a particular question, use strategic guessing. Eliminate obviously incorrect options and make an educated guess based on your mathematical intuition or any logical patterns you can identify. Avoid leaving any questions unanswered.

6. Practice Under Test-Like Conditions

Regularly practice solving math problems under test-like conditions to simulate the exam experience. Familiarize yourself with the format and time constraints of the IMO Test. This will help you build endurance, improve your accuracy, and develop a routine for tackling different types of questions.

By incorporating these mental preparation techniques and test-taking strategies into your study routine, you can optimize your performance in the IMO Test. Remember, it's not just about the mathematical knowledge you possess but also how effectively you apply it under pressure. Stay confident, stay focused, and give it your best effort!

Pros and Cons at a Glance

Sample IMO - International Mathematics Olympiad Practice Questions

Try these questions from our free IMO - International Mathematics Olympiad practice tests. The correct answer and an explanation follow each question.

1. If a train covers 3/5 of a journey in 3 hours, how long will the full journey take?

   • A. 4 hours
   • B. 5 hours
   • C. 6 hours
   • D. 7 hours

Answer: B. 5 hours

If 3/5 takes 3 hours, the full journey = 3 ÷ (3/5) = 3 × 5/3 = 5 hours.

Answer: A. 5 cm

By the Pythagorean theorem: hypotenuse = √(3² + 4²) = √(9 + 16) = √25 = 5 cm.

Answer: D. 15

15 = 3 × 5, so it is composite because it has factors other than 1 and itself.

Answer: D. 3055

Explanation: To find the least number of square stones required, divide the area of the rectangular courtyard by the area of one square stone. The dimensions of the courtyard are 325 cm by 235 cm, so its area is 76,375 square centimeters. Assuming each stone is x cm by x cm, we want to maximize x. The greatest common divisor (GCD) of 325 and 235 is 5, so each stone is 5 cm by 5 cm, covering 25 square centimeters. Dividing the area of the courtyard by the area of one stone gives us 3,055 stones.

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Sources & last verified

1. International Mathematical Olympiad (IMO Board / IMO Association); IMO 2027 is run by the HUN-REN Rényi Institute, Budap — number of questions (official source). “The competition consists of 6 problems, with each problem worth 7 points for a maximum total score of 42 points.” Verified Oct 24, 2026 · source checked Oct 24, 2026.
2. International Mathematical Olympiad (IMO Board / IMO Association); IMO 2027 is run by the HUN-REN Rényi Institute, Budap — passing score, delivery, eligibility (official source). “The total number of prizes (first, second and third) must be approved by the Jury and must not exceed half the total number of Contestants…” Verified Oct 24, 2026 · source checked Oct 24, 2026.
3. International Mathematical Olympiad (IMO Board / IMO Association); IMO 2027 is run by the HUN-REN Rényi Institute, Budap — testing window (official source). “The 68th IMO will be held in Budapest, Hungary (July 16-26, 2027).” Verified Oct 24, 2026 · source checked Oct 24, 2026.
4. imo2027.hu — next exam date (official source). “7 July 21 Wednesday Q&A · Excursion Contest Day 1 8 July 22 Thursday Q&A · Move to Main Site Contest Day 2” Verified Oct 21, 2026 · source checked Oct 21, 2026.
5. imo2027.hu — registration deadline (official source). “Registration of Leaders, Deputies, Observers and Student numbers March 2027 Submit problem proposals April 2027 Payment of fees May 2027 Registration of student details May…” Verified Oct 21, 2026 · source checked Oct 21, 2026.

How we verify: editorial & fact-checking policy. Report an error: contact us.