Math Olympiad Guidelines (Basic to Intermediate)
Written by:
Rafif Abrar, BdMO National Camper, 2024, SSC-24, RUMC
Saaji Sehnai, BdMO National Camper, 2024, SSC-24, RUMC
Nahian Parin Ifa, EGMO National Camper, 2024, SSC-24, RUMC
Abdul Majid, BMTC National Champion, 2024, SSC-25, RUMC
Advised by:
Md. Sazzadur Rahman, BdMO Academic Team Member, HSC-24, RUMC
“Mathematics is the music of reason”
- James Joseph Sylvester
This guideline is designed for students of different reading levels eager to participate in the Bangladesh Mathematical Olympiad, from beginners to advanced. The books are roughly arranged from easier to more challenging. You don’t have to read all of them, just pick any one you like and you can start reading. If a book feels too difficult or the language doesn’t feel right for you, don’t worry—it’s okay to move on to another one that suits you better. Above all, reading should be fun, so keep reading and enjoy your journey.
Problem Solving Strategies:
Books:
1.BDMO প্রস্তুতি- মুনির হাসান, অভীক রায়, তুষার চক্রবর্তী, ফরহাদ মহসিন।
2.গণিতের স্বপ্নযাত্রা ১ ও ২: আর্ট অব প্রবলেম সলভিং- আহমেদ জাওয়াদ চৌধুরী, তামজীদ মোর্শেদ রুবাব।
3. বাংলাদেশ গণিত অলিম্পিয়াডের যত প্রশ্ন- মুনির হাসান (Contains all past BdMO problems with solutions)
4. Art and Craft of Problem Solving- Paul Zeitz.
5. Problem Solving Strategies- Arthur Engel.
Guidelines:
1. Make sure to have a thorough understanding of all the concepts and theories from textbooks up to the highest grade in your category.
2. After you’ve finished understanding the textbooks, you can start your math olympiad journey with any of the above mentioned books! Each of the last three books mentioned contains almost all the basic theory required to excel at the national level.
3. The 4th book contains amazing problems. It’s highly recommended to work on them.
Number Theory:
Books:
- সংখ্যাতত্ত্ব ও কম্বিনেটরিক্স – মুতাসিম মিম।
2. Modern Olympiad Number Theory- Aditya Khurmi - Elementary Number Theory- David M. Burton
- 104 Number Theory Problems-Titu Andreescu (For problem solving)
Guidelines:
1. Number theory is more theory-based than the other 3 subjects, so make sure to know your formulas properly.
- Give importance to LCM, HCF, divisibility, primes, modular arithmetic and Fermat’s little theorem.
- The language of the third book given above may be a bit complicated for new Math Olympiad aspirants, so if you find it hard then you should save it for later.
Algebra:
Books:
1.বীজগণিতের সূত্রপাত: ১ম ও ২য় খণ্ড – এম আহসান আল মাহীর, তাহমিদ হামীম চৌধুরী জারিফ, আফসানা আকতার
- Introduction to Algebra- Art of Problem Solving
- 101 Problems in Algebra-Titu Andreescu (For problem solving)
- Inequalities- A Mathematical Olympiad Approach
- An Introduction to Diophantine Equations- Titu Andreescu
Guidelines:
- Algebra relies very little on theory and focuses more on problem solving, so building up that skill will naturally improve your skill in algebra.
- Functional equations and inequalities are the trickiest topics to learn in Math Olympiad Algebra. It’s okay to skip these at the beginner level, but the faster you can train yourself in these, the better. For functional equations, reading “The Art and Craft of Problem Solving” by Paul Zeitz is enough.
- Having a thorough understanding of functions and their graphs is a very useful tool for solving algebraic equations in Math Olympiad.
- Master AM-GM inequality, basics of series-sequences and induction.
Geometry:
Books:
- জ্যামিতির যত কৌশল – দীপু সরকার
- প্রাণের মাঝে গণিত বাজে (জ্যামিতির জন্য ভালোবাসা) – সৌমিত্র চক্রবর্তী
- A Beautiful Journey Through Olympiad Geometry- Stefan Lozanovski
- Euclidean Geometry in Mathematical Olympiads- Evan Chen
- Geometry Revisited- H. S. M. Coxeter
Guidelines:
- Master all the theorems and lemmas of NCTB textbook up to class 10.
- New to Geometry? Start with the first/third book!
- To perform well in the BDMO National Level (Secondary), studying only the first four chapters of the 4th book will be enough.
- Key concepts- angle chasing, similar triangles, congruence, area, cyclic quadrilaterals, circles, different centers in a triangle, basic trigonometry, radical axis and power of point. Coordinate geometry is also a situational but very useful tool to solve problems.
Familiarize yourself with these topics. By solving problems, you will gradually learn various techniques that demonstrate how these fundamental ideas can be applied to solve harder problems.
Combinatorics:
Books:
1.কম্বিনেটরিক্সে হাতেখড়ি: ১ম ও ২য় খণ্ড – আহমেদ জাওয়াদ চৌধুরী, আদীব হাসান, জয়দীপ সাহা
- Principles and Techniques in Combinatorics- Chen Chuan-Chong
- 102 Combinatorial Problems-Titu Andreescu (For problem solving)
- Problem-Solving Methods in Combinatorics- Pablo Soberón Bravo
Guidelines:
- Pay huge importance to permutations and combinations for regionals.
- For other topics, the fourth book listed above is good, but it’s also very advanced, so it’s advised to skip it until you can at least solve regional level problems with ease. Instead, you should firstly read the first/second book listed above which are great introductions to this subject.
- Solving a lot of problems and thus learning many problem solving techniques is important to be skillful in this subject.
For problem solving:
AMC easier problems (For regional)
AMC harder and AIME problems. (For national)
USAMO, ISL problems. (For further problem solving)
An example of the problem solving process:
Consider the following famous problem:
“A census-taker knocks on a door, and asks the woman inside how many children she has and how old they are.
“I have three daughters, their ages are whole numbers, and the product of their ages is 36,” says the mother.
“That’s not enough information,” responds the census-taker.
“If I told you the sum of their ages, you still wouldn’t be able to determine their ages,” says the mother again.
“I wish you’d tell me something more,” replies the census-taker.
“Okay, my oldest daughter Annie likes dogs.”
What are the ages of the three daughters?
The first step is to keep a clear mind and extract any piece of information you can get. Pay attention to each sentence. The first pieces of crucial information are in the following sentence:
“I have three daughters, their ages are whole numbers, and the product of their ages is 36,” says the mother.
Right now, they’re given in a non-mathematical format, but it’s easier to work with mathematical language. To convert, let x,y and z represent the ages of the three daughters. We immediately get that x,y and z are positive whole numbers (age can’t be negative), and xyz=36. The next pieces of information are
“If I told you the sum of their ages, you still wouldn’t be able to determine their ages,” says the mother again.
“Okay, my oldest daughter Annie likes dogs.”
Currently, it’s hard to understand what information they give us, but what we do understand is that the sum of their ages is somehow important in this problem. Finding no other strategy, let’s make a table, consisting of all possible values of x, y and z (knowing that they are positive integers with a product of 36 is enough to determine all possible values), and their sums. The upper row represents their ages and the lower row represents their sum.
(x,y,z) | (1,1,36) | (1,2,18) | (1,3,12) | (1,4,9) | (1,6,6) | (2,2,9) | (2,3,6) | (3,3,4) |
x+y+z | 38 | 21 | 16 | 14 | 13 | 13 | 11 | 10 |
Now, pondering the third info given makes sense: if the sum of their ages were 13, then the census-taker wouldn’t be able to determine their ages, since there are 2 possible options! We’ve narrowed the ages down to two options now: (1,6,6) and (2,2,9). The final information given seems unrelated to mathematics at all, so we should pay close attention to its implications. Note that the statement “My oldest daughter Annie likes dogs” implies that she has an oldest daughter i.e only one daughter is the oldest. This means that their ages can’t be 1,6 and 6 years! Thus, their ages must be 2,2 and 9 years, and the problem is solved.
