Iranian Combinatorics Olympiad 2025: Registration, Levels, and Exam Details

Iranian Combinatorics Olympiad 2025 popularly known as ICO 2025, is one of the most awaited international mathematics competitions dedicated to combinatorics enthusiasts. This unique Olympiad aims to nurture creativity, logical reasoning, and problem-solving skills among students worldwide while promoting a fun and challenging learning experience.

Organised by the ICO official team, the competition allows participants from any country and educational level to showcase their combinatorial thinking. The olympiad exam will be held online on October 30 and 31, 2025, and registration is now open at the official website, ico-official.com.

Iranian Combinatorics Olympiad 2025 Overview 

Combinatorics is one of the oldest and most vibrant fields of mathematics, tracing back to ancient India and Greece, where early scholars used counting techniques to solve complex problems. Have a glance of the ICO 2025 Olympiad Exam.

What is the Iranian Combinatorics Olympiad (ICO)?

The Iranian Combinatorics Olympiad (ICO) is a yearly international contest that aims to make combinatorics fun and easy to understand for students. It gives young math lovers a chance to test themselves with exciting problems that encourage creative thinking and logical skills.

What makes ICO 2025 special is that it is the first competition in the world to use an online system to judge all the theory problems. This new feature helps make the contest fair and open to people everywhere.

Why Participate in ICO 2025?

Participating in the Iranian Combinatorics Olympiad 2025 is more than just competing. It is about exploring mathematics as a creative adventure. Here is why students and enthusiasts should not miss it:

• Global Participation: Compete with math lovers from all around the world.

• Online Access: Take the exam from anywhere with ease.

• Creative Thinking: Tackle unique problems that test logic, reasoning, and imagination.

• Team Challenge: Work in teams of up to 3 members for a collaborative experience.

• Free Entry: Open for everyone without any registration fees.

Structure and Divisions of ICO 2025

The Iranian Combinatorics Olympiad 2025 is divided into four levels based on educational grade to ensure fair competition among students of similar ages and skills. Each level will include carefully curated combinatorics and problem-solving questions similar to those in the International Mathematical Olympiad (IMO) style:

Highlights of Iranian Combinatorics Olympiad 2025

• The first global competition focusing entirely on combinatorics.

• Includes an online scoring and ranking system for transparency.

• Two-round contest that challenges participants to solve advanced theoretical problems.

• Open to individuals and teams (max 3 members) of the same nationality.

• Features innovative and fun problem sets to inspire mathematical curiosity.

• Completely free — no hidden charges or registration fees.

Accommodations and Participation Guidelines

The ICO Olympiad encourages inclusivity by allowing students of all educational backgrounds to participate.
Participants can register individually or as a team, ensuring all team members share the same nationality.

To make the event more exciting, the organizers have requested participants to submit short introductory videos saying “Olamipiad Tarkibiyat Iran” (meaning “Iranian Combinatorics Olympiad”) in their native accent. This initiative celebrates diversity and connects participants across borders.

By participating in ICO 2025, students not only strengthen their mathematical foundation but also gain insight into how combinatorics connects with real-world applications.

How to Register for Iranian Combinatorics Olympiad 2025

Registering for the ICO 2025 is simple and quick. Here’s how to do it:

1. Visit the official website, https://ico-official.com.

2. Click on “Register Now” under the ICO 2025 section.

3. Fill in your details such as name, nationality, email, and education level.

4. (Optional) Form a team of up to three members from the same country.

5. Submit your registration and await a confirmation email.

You are now ready to participate in one of the most exciting global math contests of the year!

ICO 2025: Event Schedule and Exam Format

• Round 1: Introductory round with multiple combinatorial problems.

• Round 2: Advanced level with IMO-style theoretical questions.

• ICO Exam Date: 30th & 31st October 2025 (online format).

• Medium: English.

• Judging: Automated online evaluation through the ICO’s official judging system.

Benefits of Participating in the Iranian Combinatorics Olympiad

• Gain exposure to high-level international math competitions.

• Develop problem-solving and reasoning skills applicable in computer science and logic.

• Receive international recognition and connect with fellow math enthusiasts.

• Prepare for global Olympiads like IMO, IOI, and ICM.

• Experience a free, inclusive, and fun competition atmosphere.

Iranian Combinatorics Olympiad PYQs

The International Children's Olympiad (ICO) booklets for 2020 and 2021 provide a collection of challenging problems followed by detailed solutions designed to enhance problem-solving skills and inspire young minds. These sets contain questions spanning various subjects and difficulty levels, testing logical reasoning and conceptual understanding.

Below are selected problems from the ICO 2020 and 2021 editions, followed by their solutions as provided in the official documents.

1. Question: Consider a 9×9 table with a bear in the middle cell of the top row and a beehive in the middle cell of the bottom row. We want to dig a hole in 6 cells (other than initial cells containing the bear and the beehive) of the table. After digging the holes, the bear starts moving towards the beehive. In each step the bear moves from a cell to a cell with no holes that has an adjacent edge to it. The holes must be chosen such that there is at least a valid path that the bear can take to reach the beehive. The bear always chooses the shortest possible path. In all possible ways of digging the holes, what is the maximum number of steps that the bear takes to reach the beehive?

Answer: The maximum number of steps the bear takes to reach the beehive under these conditions is 16.

2. Question: What is the number of sequences of letters a, b and c of length 7 such that no two adjacent letters are the same and no two sub-sequences are equal? A sub-sequence is a consecutive set of letters from the sequence.

Answer: The number of such sequences is 18.

3. Question: Consider the ordered pairs (0,0), (0,1), …, (9,8), and (9,9). Assign a card to each of these 100 pairs. We have a subset of these 100 cards and a device that takes two cards like (a,b) and (c,d), then other than returning these cards, it also gives us two more cards with ordered pairs of (min{a,c}, min{b,d}) and (max{a,c}, max{b,d}). Find the minimum number of cards needed to obtain all 100 cards using the device.

Answer: The minimum number of cards needed is 10.

4. Question: Football league organization of Abolfistan has announced that if the Corona pandemic does not end until December, due to shortage of time in the next season every two teams will play against each other exactly once. This league has 2048 teams. To choose the host, the organization uses the following algorithm: For each match in week i, if the teams who have to play each other have not hosted the same number of games in the past i−1 weeks, the host will be the team with lower number of hosting, otherwise the host will be chosen randomly. What is the maximum possible number of times a team can be a host?

Answer: The maximum possible number of times a team can be a host is 1029.

5. Question: Find the total number of possible ways to partition the set {1, 2, …, 99} into some subsets such that the average of elements in each subset equals the total number of subsets.

Answer: There is only 1 such way to partition the set.

6. Question: For each non-empty subset S of the set A = {10, 12, …, 26}, consider a number cS defined as multiplication of all elements in S divided by 2 raised to the power |S|, where |S| is the number of elements of S. For example, for the subset S = {10, 14, 18}, find cS.

Answer: cS equals 726,485,759.

7. Question: A frog is at the origin point of the Cartesian plane. Each time, it jumps one or two units to the right. Let a/b be the possibility of the frog getting to the point (10, 0) after a few jumps, where a and b are positive integers and gcd(a,b) = 1. Determine a + b.

Answer: The value of a + b is 1707.

8. Question: Each diagonal parallel to one of the main diagonals is called a tape. Consider a 1399×1399 table. We want to put some pins in some of the cells of this rectangle such that each tape covers an odd number of pins. Let a and b respectively be the minimum and maximum number of pins to achieve this. Find the value of a + b.

Answer: The value of a + b is 1,958,602.

9. Question: Determine the maximum number of L-shaped tetromino tiles that can be placed in a 10×10 table such that no two of them share a vertex.

Answer: The maximum number of such tiles is 12.

10. Question: Consider a graph where the value of an edge is defined as the number of edges intersecting it (other than the ending points). The maximum value assigned to edges of a graph is called the ugliness of the graph. Asad wants to redraw the given graph to minimize its ugliness. What is the minimum ugliness that he can achieve?

Answer: The minimum ugliness achievable is 1.

11. Question: Some bids are placed in the cells of a 1399×2020 table. In each turn, choose a cell with more than one bid, take two of them and place one of them in the cell above and the other in the cell to the right. If the current cell is the highest cell in its column, the bid is placed in the lowest cell of that column, and similarly for the rightmost cell in its row, the bid moves to the leftmost cell. Determine the minimum number of bids needed such that this process can continue forever.

Answer: The minimum number of bids needed is 3,419.

12. Question: We call a set of rectangles as a good set if all sides are either horizontal or vertical, lengths from {1, 2, …,10}, at least one side length 6, and total area less than 100. Determine the minimum value of k such that all rectangles in an arbitrary good set can be fit into a 10×k rectangle without overlapping.

Answer: The minimum value of k is 18.

13. Question: For graphs with 10 vertices and A ⊆ vertices where |A|=4, for how many initial graphs G do we have f(G,A) = n - 1?

Answer: The number of such graphs G is 33,868,800.

14. Question: Let G be a cycle graph with 19 vertices. Find the total sum of f(G, A) for every subset A of G.

Answer: The total sum is 2,028,478.

15. Question: For a given graph G, find the number of subsets A of vertices for which f(G, A) = 9.

Answer: The number of such subsets A is 16,208.