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Mathematics guide

NMTC Guide

NMTC and IMO preparation guidance for Pakistan's selection pathway.

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2026-05-06
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NMTC Guide

Note: The guide has been reviewed by the Deputy Leader of Switzerland, Tanish Patel, and the Deputy Leader of Algeria, Maya Chouikrat.

(Hi, if you’re from the NSTC Alumni group)

Also note that this page exists to guide you from having no experience to being able to get selected for IMO. However, to aim for a medal, you will need to go beyond this guide and work out what works best you.

The stages of the contest

  1. Stage 1

You have an aggregate percentage of 60% or more in the core subjects (in the latest examination) for which the contest operates. This stage automatically completes if the registration process is successful.

  1. Stage 2

Once you've registered for Mathematics (and receive a confirmation on their site), you participate in the screening test.

The screening test is generally held around March for the competition next year. For example, the screening test for IMO 2024 was held in March 2023. You can tackle this test to a decent extent via the OL & AL syllabus of Mathematics. However, I would recommend extending your studies beyond these syllabi as they are not an accurate representation of the Olympiad Mathematics.

  1. Stage 3

Around 1000-1500 students participate in each subject's screening test, and around 50 of them get shortlisted for each subject's contest's third stage: the camp. Due to recent financial condition of the country and the covid situation, there has been varying amount of one-week "training" camp, both online and physical, allowing you to "feel" the rigor and essence of Olympiad Mathematics. Each camp generally consists of an exam at the end of the week which is used in shortlisting the number of contestants remaining. This test is as long as the actual Olympiad test. This process repeats until only 6 participants are remaining.

two things to look out for:

  1. You have to be less than 20 years of age on 30th June of the competition. (Therefore, for example, to participate in IMO 2025, you have to be less than 20 years old on 30th June 2025)
  2. Current students of 9th, 10th, 11th (Matric/O-Level) or FSC-1/AS can register for it, but A2 students can also register if they are eligible according to (i) and take a gap year

Click here to access their website

The guide

Before reading any further, you should know that Olympiad Mathematics is generally broken down in the following subjects:

  • Geometry
  • Algebra
  • Number Theory
  • Combinatorics

Please note that proof writing is a very important skill in Olympiad Mathematics and as such it is important to be able to write precise and accurate proofs. All of the questions in the IMO will require you to either prove a statement given in the problem, or given an to a question in the problem by proving your answer completely.

Also that any books recommended in this article are books I have come across on my own and there certainly may be better books out there. For a bigger list of recommendations of book, you can check that out at the AoPS (Art of Problem Solving) website. The website is the biggest online forum for Mathematics and you will find a lot of help from the website when you are preparing for the IMO journey.

All book mentioned in this article and some other books are available in this drive link.

O Level Mathematics, along with basic knowledge of Olympiad Mathematics is sufficient for the screening test. To get introduced to the field of Olympiad Mathematics, I recommend you to refer to the AoPS books, preferably in this order:

  1. To start with, use these introductory subject-specific books. It is highly likely that you will be familiar with most of the content in these books. However, there might be topics new to you, or topics you are not fluent in. Getting a fluency in such content is extremely important before going any further so that while doing an advanced problem, it is not the introductory content that causes you to get stuck.
  1. Once the basics are cleared, the following introductory books can be followed. Some of the content in them might be overlapping, that’s up to the reader to differentiate. These books cover all the four subjects of Olympiad Mathematics.
  1. Once you’re done with the introductory matter, the next step is to increase the difficulty. The following books will help you strengthen yourself in the respective subjects as they will teach you about the subjects in much more details and introducing many new techniques

Once you are done with these books, you will have a solid base in each of the 4 subjects and you can move on specialize in each of the subjects separately. However, it is important to note that this pathway may not be the best path out there for strengthening your bases, nor is a specific pathway best for all individuals. Depending on your existing level of expertise, a different route may work out better for you. In the end, you will often find your self using multiple books at a time, picking up specific content from each book. If you are not an absolute beginner, you may also refer to these books:

These books will cover a lot of basic ideas in Olympiad Mathematics, however they might contain topics not relevant to the screening tests. It is important to understand that the preparation for the screening test and the actual Olympiad Mathematics may differ vastly because the screening test is more focused on the school side of Mathematics to be able to shortlist students more appropriately because of the fact that Olympiad Mathematics is foreign to most people.

If you develop a solid grip on Olympiad Mathematics (by this I mean, you are even able to do some IMO questions), I would advice you to pay a visit to Olympiad Training for Individual Study (OTIS). It is an online program for training for Olympiad Mathematics, and financial aid is available if it is an issue for you financially.

It is also important to note that, if you seriously want to get into the IMO team, or better yet, an IMO medal, you need to start dedicating multiple hours a day daily till the actual competition. This does not mean that you need to make Olympiad Mathematics your personality. It goes without saying that your best chances are if you work in mediation.

Before reading the specifics of each subject, it might be beneficial to be aware of what to expect when studying them. The following unofficial syllabus will be able to help with that:

Unofficial Syllabus in math olympiads

Geometry

For Geometry, the following book would be enough for pretty much anything you will need to know for Olympiad Mathematics. The first four chapters cover everything that might be tested on the initial screening test and any subsequent selection test, though not sufficient enough for IMO:

Euclidean Geometry in Mathematical Olympiads

However, do note that the above book might be challenging initially. Though being challenged is pretty normal and helpful in Olympiad Mathematics, you might want to be sure, this is not because your base in geometry in flawed.

Geometry is hackable in the sense that it will probably never require knowledge foreign to you if you’re well versed with the knowledge in the above book (Unless its a Problem 3 or 6). As such, this is also the subject that is focused on the most in Pakistan and the most tested upon. However, Geometry, like every other subject, requires a lot of practice, and it would be a good advice to follow this book by completing all the challenges and by following along the book with a notebook and pencil in hand to take notes. Moreover, it is commonly misinterpreted in Pakistan that it being hackable makes it easier than other subjects and hence students spend more time studying it. That is not the case, and is subjective on an individual basis.

Number Theory

There is a high chance that if you’re reading this article, you plan to do something involving Mathematics in University. In that case, you’re likely to study Number Theory a lot. As such, when you are studying Number Theory, you are not just studying for the Olympiad but something you’re going to likely be involved with in the future. For this reason, you can often find University lectures from universities on the Internet which may turn out to be extremely helpful. One way to find these lectures is by typing “pdf:{some number theory topic} {university name, you can leave it blank}” so for example “pdf:gcd and lcm MIT” or “pdf:Quadratic Residue”.

However, just like other subjects, you will need a proper track to follow to avoid any gaps in your knowledge. Some books I have been recommended a lot by others are:

  1. Modern Olympiad Number Theory by Aditya Khurmi
  2. NUMBER THEORY Structures, Examples, and Problems
  3. An Introduction to Diophantine Equations
  4. 104 Number Theory Problems

As mentioned earlier, these are just the books I have encountered, and some I even feel have a few flaws. However, it’s your job to find out which books work the best for you.

Combinatorics

Similar to Number Theory, Combinatorics is a field you will encounter a lot in your university life. Even more than Number Theory! If you choose to pursue something related to Computer Science you’re likely going to learn the same things in University because Combinatorics is used a lot in Algorithms and other places. Combinatorics is a subject you need to study extensively, arguably even more than for IMO, for IOI, the CS equivalent for IMO. Therefore just like Number Theory, University lectures online are going to come to help a lot when you are preparing for IMO.

Some of the recommended resources are:

  1. Olympiad Combinatorics
  2. A Path to Combinatorics for Undergraduates
  3. 102 Combinatorial Problems (Just for Problems relating to Combinatorics)
  4. And a lot of University Lectures notes

Combinatorics, in my personal opinion, is the subject that requires the most brain activity and even with very little subjective knowledge, you can crack difficult problems if you think long enough. In IMO 2022, I cracked Problem 1, which was arguably easy, after 4 hours of playing with pens, pencils and erasers and shuffling them according to the given prompt.

Algebra

Algebra in Olympiad Mathematics is very loosely divided into 3 different chapters:

  1. Functional Equations (FE)
  2. Inequalities and Equations
  3. Polynomials

The best FE resource I’ve come across is the little fermat playlist. This playlist covers most of the techniques used in FE and explains each question clearly. Another good resource for Algebra overall are the OTIS Excerpts. Inequalities and Functional Equation is covered extensively in these notes. You may use the handouts “Basic of Olympiad Inequalities” and “Olympiad Inequalities” for Inequalities. For Inequalities, you may also refer to the book “Advanced Olympiad Inequalities”. For Polynomials, you can read the book “117 Polynomial Problems”.

General

This section contains websites, handouts, and books which are not focused towards a specific subject but focus on multiple subjects or Olympiad Mathematics as a whole.

And other resources, many of which you can find listed here. Remember that the resources can only help so much, and ultimately it’s your effort in the end which will lead you to a medal in IMO. Don’t cheat yourself in regards to preparation and give it its desired time.

The contents of the screening test

The generic layout of the Mathematics section is similar to the other subjects however there is some pattern to it.

The MCQ Section is there mostly to test your academic knowledge. It is of 50 marks. Having a strong grasp over Additional Mathematics (or A Level Mathematics) would allow you to correct answer most of the MCQs. However a few questions might include topic foreign to Additional Mathematics (and A Level Mathematics). This may include famous sequences and functions specific to this field. Besides this, elementary knowledge regarding topics exclusive to Olympiad Mathematics might be tested. The initial cutdown is based on the number of MCQs correct and the Subjective part is assessed of the students who manage to clear the initial shortlisting. The initial shortlist is around 100-200 students.

The Subject Section is more focused on topics specific to Olympiad Mathematics. It is of 25 marks with no fixed number of questions. According to the trend, the most common tested subject is Geometry followed by Algebra (mostly involving manipulation of numbers). The initial chapters in Euclidean Geometry for Mathematical Olympiads will be enough for the Geometry questions.

At this point I am assuming that you have at least covered AoPS Introductory Books and have revised OL & AS Mathematics (and OL Additional Mathematics). The reason for AS Mathematics being a necessity is two-fold: it provides you a head start for your A Levels whilst polishing your fundamentals. I would emphasize covering OL & AS before starting AoPS. If you haven't completed any of these, continue in a linear fashion: OL, AS, AoPS.

Regardless of the fact that matric education is not fully polished, 9th, 10th, 11th FSC Mathematics books for preparation of screening test should also suffice. The content in these books is better than the O & A Level syllabus and extends to the AoPS books mentioned above. During the screening test, focus mostly on the MCQs but do give at least 90 minutes to the descriptive question (or less depending on the difficulty of the questions).

If you have been shortlisted in the top 50 category, congratulations! If not, then do not worry, it is just the beginning of a journey and overburdening yourself with negative thoughts will only hamper your progress.

If you have cleared the screening test, it is time to move on to the books. I would recommend starting a subject and then rotating it every week or two. Eventually, there may be a subject specific camp, in which case you should just study that subject until the camp concludes, because the shortlisting will depend on the expertise on that specific subject. Do not leave any subject untouched for long as you will start getting rusty and will waste precious time getting back to the expertise you were previously on. When solving any problem, remember to not at the official solution until you complete it. If you reach a dead-end and unable to make any progress for over half an hour, leave that question and come back to it a few days later. If even after doing that multiple times, you cannot figure out the question, look at a solution one by one, and try to solve it with just the hints. This is my advice because when you solve a question by yourself, you knew how to build up that thought process and how you found out vital points. As soon as you peaked at the solution, you will be unable to think of another solution and you will learn nearly nothing since every other question in Olympiad Mathematics is meant to be unique.

From this point forward, it is important to self-study 100% of the syllabus. You will be given assignments during the camp takes. Completing these assignments will help you develop your skills in those specific chapters and you'll get used to solving "proper" problems in case you were still out of practice.

I would advice you to not rely on the camps. This is because the camps are designed keeping absolute beginners in mind, and if you are aiming for a place in the IMO team, the content there will mostly be easy for you. In a situation where you discovered about the preparation of IMO later, don’t fret and just note the topics discussed in class and spend hours on them by yourself after you are done with classes each day.

Eventually you will have a lot on your plate: university applications, college applications, extracurriculars, personal commitments, and so on. However, even two to three hours a day (when the camp is not taking place) would suffice. The key is to develop persistence and creative thinking. Unnecessarily overloading yourself with content won't do anything except tire you prematurely.

The Exam Format

The selection tests at the end of the camp in Pakistan will mostly follow this pattern: