My INMO Journey for 2023

Some Context

International Mathematical Olympiad (IMO) is an annual contest held for pre-college students across the world. The top 6 students who are selected through an extensive procedure are chosen to represent their country at the international forum. 

The IMO team for India is selected through a procedure involving three stages of examination, namely, 

• Indian Olympiad Qualifier in Maths (IOQM), which selects the top 450-600 students in the country. The exam is held in a 3 hour session on a single day, consisting of 24 questions and 100 marks.
• Indian National Mathematics Olympiad (INMO), which selects around 60 students throughout the country. The exam is held in a 4 hour session on a single day, consisting of 6 subjective question, with 102 marks.
• International Mathematics Olympiad Training Camp (IMOTC), which selects the IMO team for India, consisting of six people. The selection is done through 4 TSTs, each for 4.5 hours, consisting of 3 problems each. 

I had qualified to write INMO multiple times previously, but despite my numerous attempts, two of which were fueled by days and days of work, I never made it to the IMOTC. Now I was in class 12th, trying out my last attempt at the Mathematical Olympiad. Let's see how it goes!

IOQM

I did not prepare for IOQM specifically this year. I had prepared for it extensively in class 8th, 9th and 10th, and I was banking on my previous knowledge to clear the cutoff. The exam was held on 30th October, 2022. I was feeling very calm at the exam hall, and came out solving about half the paper, expecting around 40-45 marks. Just out of the exam hall, I felt that the toughness was similar to the previous years. However, as I talked to my fellow participants, I came to know that the paper was tougher than the previous years. Then I checked on the internet, and indeed people thought that the paper was much tougher.

The Maths Olympiad Cell had decided to select 600 students for INMO this year. However, they also had set that the minimum score for selection to the INMO would be 30. As the paper was tougher this year, the cutoff ended up being 30, and they could only select 469 students by following all criterias. I scored 38 marks, and was lucky to get through the first stage this time. 

Time to think about INMO.

INMO

INMO had been a great barrier for me since the last 3 years. I had given INMO twice previously, having failed miserably both times. I had scored 2 in class 10th, 0 in class 11th. However, this time I felt that I could do better. This was just an intuition, and it was not like I had worked hard this time. Infact, I did not prepare at all for INMO, as my JEE exam was very close to the date of INMO. However, I decided to give the exam, and in the last day, I read some formulae and some other basic stuff, and went to the exam hall very cool and head on.

Reaching the Venue

The exam was scheduled to be held on 15th of January in Bhubaneshwar. Me and my father took a train from Sambalpur to Bhubaneshwar on the 14th of January. We reached there almost at midnight, and went to my aunt's home. We had a nice dinner of Roti and Paneer, and went to sleep. Next day, we woke up early, had a bath, ate breakfast (Dosa and Chutney), and went to the exam venue. I was very surprised to find that very few students were attempting the exam from Odisha region. This shows the lack of interest in Maths Olympiad in Odisha. Even those are qualified are not interested to give the exam. However, I was too worried about the exam to think about it at that time.

During the Exam

The exam started at 12 pm. As soon as I received the paper, I smiled, knowing that it was the last time I am writing INMO. I may or may not qualify, but I am not giving this exam again. After around 5 minutes, I started reading the questions. I randomly tried to progress on some problems, but could not do so. After around 30 minutes of trying out random things, I thought I had something on the third problem, and decided to try that out. The problem was the following, 

INMO 2023 P3: Let $\mathbb N$ denote the set of all positive integers. Find all real numbers $c$ for which there exists a function $f:\mathbb N\to \mathbb N$ satisfying:

• for any $x,a\in\mathbb N$, the quantity $\frac{f(x+a)-f(x)}{a}$ is an integer if and only if $a=1$;
• for all $x\in \mathbb N$, we have $|f(x)-cx|<2023$.

I tried out some ideas on this problem. Initially, I had thought that there must be no solutions. I tried to prove that for some time. First, I proved that if $c$ works, $c-k$ works for any integer $k$. Thus I was able to reduce it to solving for $c \in [0,1]$. Then I got an idea which shows that it is not possible for $c \in [0,\frac{1}{2})$. Just left with $c \in [\frac{1}{2}, 1]$. Then I got the crucial claim that if $x$ works, $1-x$ works. I had almost concluded that there were no solutions, when I realised that I was left with one case, i.e, $c=k+\frac{1}{2}$ for integers $k$. I was surprised to find that these indeed worked for the problem statement, and I had to rewrite the whole solution claiming that $c=k+\frac{1}{2}$ worked. My hands were literally shaking after I finished the proof, because I had solved an olympiad problem first time in contest. The whole procedure of writing took me around 30 minutes, and I was left with around 2 hours 30 minutes at that point of time. At the end, I was extremely happy, and for the first time, I believed that I had some chances to qualify. 

The next two hours were spent hopelessly trying to get my hands on something, but I failed horribly. I had some progress on problem 1 and 6, but they were not very solids and would worth very few points. With half an hour left on the clock, I decided to try out problem 4. The problem was, 

INMO 2023 P4: Let $k \geq 1$ and $N>1$ be two integers. On a circle are placed $2N+1$ coins all showing heads. Calvin and Hobbes play the following game. Calvin starts and on his move can turn any coin from heads to tails. Hobbes on his move can turn at most one coin that is next to the coin that Calvin turned just now from tails to heads. Calvin wins if at any moment there are $k$ coins showing tails after Hobbes has made his move. Determine all values of $k$ for which Calvin wins the game.

Ok, I was almost at the end of the exam, and I needed to get something to work. It was obvious that $\max{k} \ge N$. After around 10 minutes of trying, I found that $\max{k} \ge N+1$. I tried another 10 minutes and convinced myself that $N+2$ doesn't work. 10 minutes left. I tried with all my might to prove that $N+2$ doesn't work. However, I could not do it, and in the end, decided to write the construction for $k=N+1$. I finished the construction just as the exam ended, and handed over my paper. I told dad that I had done better than the last time, but still there was very little chance of me qualifying.

Marks Declaration and Anticipation for Results

I was expecting around 20 marks. I thought that I would get full marks on the 3rd problem, and maybe 2-3 marks on the 4th one. Finally, HBCSE declared the INMO marks on 14th February (the final merit list was still pending). I went up and checked the results, not really with much expectation. I typed in my roll number, and up came the result. I had got 26 marks. The distribution was 0 0 14 12 0 0. 12 for the fourth problem!! I could not believe it. I thought that they must have made some mistake with my correction. I had wrote some rubbish claiming that $N+2$ doesn't work, maybe they fell for it! (although I later found out that they hadn't fallen for it, rather construction for $N+1$ was worth 12 points). At that point of time, I was overjoyed, and could not control my hapiness. It was one of the best moments in my life. Along with that, I got overconfident, and thought that my selection to IMOTC was pretty much guaranteed. Little did I know that I was wrong!

I talked with some of the past INMO awardees, and tried to predict what the cutoff would be. Some people said that doing one problem was enough to get to IMOTC. I was not too sure about it. I had checked the solution, and realised that problem 1 was way easier than I had thought. Also, many people also started to claim very high cutoffs like 30, 35 or even 40. I did not believe that either, and after careful consideration, I finally concluded that if I even have a chance to get selected, it would be by the barest of margins. 

Then the anticipation started. I was literally checking the HBCSE page like 5 times a day from 14th February onwards. Although they had said that they will release the final merit list on 28th February, I thought that there were good chances that they might release it earlier. I must tell that my board exams were just starting at that point of time, and I had to prepare for that as well. Also, JEE Mains second attempt was also planned to be held in April, and I also had to prepare for that. All this while I was checking the HBCSE web page constantly. 

    

Declaration of the Result

In the last few days before announcement of the merit list, I was literally checking the HBCSE web page every single hour. On the 28th of February, I had my chemistry paper(that was the day of the result announcement day). I checked the webpage early morning, and found a strange notice. It said something like, We are extremely sorry that we had published the wrong merit list yesterday evening. The correct merit list will be out today. This meant that the merit list had already once been released! I quickly called up my father and asked him if he had checked the merit list yesterday evening. He said that he had, and the cutoff for class 12th was 24 marks. I was very confused at that moment, had they taken back the result because they had to increase the cutoff. But I had an exam that day, and my father assured me that it must be some technical glitch, and that changes to the cutoff will not be made. Also I had an exam that day, and he said that I should focus on that right now rather than wasting my time on something beyound my control.

As soon as my chemistry paper was over, I rushed out and immediately took a phone and checked the HBCSE web page. Unfortunately, the internet was slow at that place, and I was unable to access the page. I called up Bitu Dada, and asked him to keep checking the web page after intervals. 

Finally, came his call. I was talking with a friend of mine at that time, and I hung up his call without even telling him (sorry Siddhartha), and accepted Bitu Dada's call. He said that the cutoff has changed to 26 for class 12, and said that I might have qualified. Again a confusion came to my mind. I thought whether the cutoff is the $\ge$ marks needed or $>$ marks. He was not aware of my roll number, so I was not sure. I then called up my father, and asked him for the roll number, and when I checked........Bingo!! I had qualified. I cannot express the joy that I felt at the time. I called up everyone I knew of and told them that I had qualified. Ruzul Dada was very happy, and he gave me many congratulations. So did all the other people.

I had finally achieved my biggest dream of life yet, to qualify INMO. I had finally broken the barrier that had stood before me for 2 years.

A Philosophical Viewpoint

I don't know if I have conveyed it yet, but clearing INMO was my biggest and only major target when I was in class 9th and 10th. In 9th, I tried very hard, but I was simply not aware of the level needed to clear INMO. I just did some basic RMO type problems, and I had not even heard of any olympiad held outside India like USAMO or PUTNAM or anything like that. I could not even clear RMO in class 9th.

However, I slowly started getting aware of these Olympiad, and the level of competition there is, in class 10th. I then tried to raise my competence level to those levels. I did work hard to raise my level to that level, but failed miserably (I will share my experiences in another article). So, although I wrote INMO in class 10th, I could not make it to the camp. 

After my miserable experiences in class 10th, I gave up on Math Olympiad, and instead focussed on doing JEE and KVPY. I did not take Olympiad seriously in class 11th, and as expected, I failed in INMO 2022.

Interestingly, a sudden change came upon me in class 12th. I practically still did not prepare for Math Olympiad in 12th, but atleast I started something, I did like 1-2 problems in a month. I do not know how, but I saw some change this time. Even with practically no touch with Olympiad style content (and 90% theory forgotten), I was able to solve some problems. There was a change in the way I tackled problems, and some stuff clicked in my mind. With this newbuild confidence, I saw a small glimmer of hope that maybe, just maybe I would be able to clear INMO this time. 

And it happened!! With almost no gearing up, I qualified INMO. It is so funny. When I was in class 10th, I was apparently trying with my full strength to clear INMO, and could not do it. But this time, under unfavorable circumstances, I did it. 

I would like to thank all people who always supported me. I thank my family members, my mother and my grandma for cheering me on. My grandfather was always interested in my academics, and helped me at many points, so thanks to him. Bitu Dada and Ruzul Dada deserve special thanks. I would finally like to express my deep gratitude to my father, for his immense help and guidance, even at the cost of his own sake. Thanks everyone, thank you very much.