Getting Started with Olympiad Maths

This advice is specific to Ireland, but much of it is applicable to other countries too.

You may or may not have heard of the International Maths Olympiad, or IMO, maybe you’re here because you did well in round one of the Irish Maths Olympiad in School.

If you’re thinking about getting involved in olympiad maths, it can seem rather daunting. You might look at a few IMO questions and think to yourself ‘not a hope’. Nobody starts off able to solve IMO problems. They’re really, really hard.

What is the International Maths Olympiad?

The International Maths Olympiad, or IMO, is regarded as the most prestigious maths competition for second level students in the world. It was founded in 1959 and more than 100 countries take part each year. Each country sends a team of up to six students, chosen through a national selection test(or tests), who sit two 4.5 hour papers consisting of three difficult questions each, which they have to solve with proof(i.e. Giving the reason behind each step). The problems each day are in order of difficulty, 1 and 4 are the most approachable, 2 and 5 are harder, 3 and 6 are the hardest and solved by very few participants each year.

Ireland also sends a team of four girls to the European Girls Mathematical Olympiad, or EGMO, each year. Contestants qualify from a selection test in February. EGMO aims to decrease the gender gap in Olympiad Maths – each year about 90% of the contestants at IMO are male.

How do I get involved?

My first recommendation is to go to maths enrichment classes, if you aren’t already. There are weekly classes in UCC, NUIG, UL, Maynooth and UCD. Here you’ll learn some introductory theory and get experience solving problems. Even if you don’t want to compete, if you’re interested in maths I’d highly advise giving these classes a go. You will be challenged far beyond school and sharpen your problem solving skills. COVID-19 will likely affect these classes in 2020/21 so follow Science Angles for updates on what’s happening in each centre!

Should I take part? Am I ‘good enough’ to take part?

The only way to find this out is to give it a go! Worst case scenario: you discover that you don’t enjoy it, and you slightly improve your problem solving skills. Taking part doesn’t cost any money and there’s no commitment (unless you get on the team)  so you have nothing to lose! When you start, you’ll almost definitely feel like it’s too difficult, you can’t solve much and you’re way out of your depth. This is normal – if you’re not struggling, you’re not challenging yourself enough! 

The main things you need are creativity, logic and perseverance. Olympiad maths is really, really, really hard. Olympiad problems are designed so they can’t be solved straightaway, they usually require some creativity and outside the box thinking. Working on these kinds of problems greatly improves your general  creative problem solving skills – which is great as these are needed in so many different careers!

How does the Irish Maths Olympiad work?

The first round is taken in schools. If you score well in that your teacher will recommend that you go to your local maths enrichment classes. The round doesn’t really count for anything and you can go to maths enrichment classes without it. THe classes begin in January everywhere except UCC, which starts in November. In UCD there’s a selection test in February, after which the top 40 or so are invited back. The UCD selection test doubles as the EGMO selection test – the top four girls nationwide are selected to represent Ireland at the European Girls Mathematical Olympiad. The Irish Maths Olympiad takes place in May in all of the enrichment centres, and consists of two three hour papers, with five questions in each. The top six represent Ireland at the International Maths Olympiad.

In Ireland, the top 25 or so in the Irish Maths Olympiad are invited to attend training camps, usually one in June and one in August. These are a bit more advanced than the enrichment classes and are a really useful way to learn about different areas of maths. 

How do I get better at solving problems?

The only way to get better at solving maths problems is by trying to solve problems. It is a skill you’ll get better with over time. Some of the questions I found impossible at first now seem almost easy to me. Don’t just try a problem for a few minutes, then give up and look at the solution. Only look at the solution after you’ve spent ages on the problem, left it for a few days, then tried it again. Sometimes a hint (if available) or just looking at the start of a solution to get the main idea, before tackling the problem again, can be useful. Persistence is key!

In terms of solving problems, don’t start with ones from IMO. They’re rather hard, and if you have no familiarity with Olympiad maths they’re really not the best place to start. Past EGMO/UCD selection tests have a good selection of more accessible problems, and once you’re a bit more comfortable, the problems from the Irish Maths Olympiad are much more accessible than IMO ones, yet should still pose a challenge. EGMO Q1/Q4 and Q1/Q4 from older IMOs are easier than current IMO Q1s/4s. 

The most important thing is that you enjoy it. If you don’t, just don’t bother doing it. 

What about theory? Do I need to learn loads of stuff?

 Well, of course you need some level of knowledge. Trying to do number theory without familiarity with things like modular arithmetic, Fermat’s Little Theorem and Euclid’s Algorithm, or inequalities without AM-GM or Cauchy-Schwarz won’t go well. However, you don’t need to get too bogged down with theory. It doesn’t matter if you know loads of advanced theory if you can’t solve problems, so once you’ve got the basics down don’t worry too much about the rest.

Remember, it is not about getting on the IMO or EGMO team. Sure, those things are enjoyable, but they should not be your main motivation. You should be doing it because you love it, because you enjoy the challenge and beauty of the problems. Olympiad maths improves a range of skills, from problem solving to communicating complex ideas to persisting even when it’s hard, but it should also be fun! Good luck and enjoy!

Interview with Kieran Cooney: Part 2: Mathematical Olympiads and Self-directed learning

Welcome back to part 2 of our interview with Kieran Cooney! We discuss participating in Mathematical Olympiads and self directed learning. If you missed part one be sure to check it out here .

Continue reading Interview with Kieran Cooney: Part 2: Mathematical Olympiads and Self-directed learning

People in STEM Interview #1: Dr Anca Mustata and Dr Andrei Mustata

Welcome to the People in STEM Interview Series! Our first interview is with Dr Anca Mustata and Dr Andrei Mustata, lecturers and reasearchers in the field of algebraic geometry in University College Cork.

We discuss algebraic geometry, working as a mathematician and their experience taking part in maths Olympiads in Romania. Algebraic geometry is a fascinating branch of mathematics looking at the geometric properties of solutions to polynomials.

Continue reading People in STEM Interview #1: Dr Anca Mustata and Dr Andrei Mustata

EGMO Diary Part 2

9th of April

For preparation, I did exactly the same thing as yesterday, because that seemed to help me tremendously. We did nothing for 45 minutes, and then stretched for the last 15 minutes. 

The test opened with a geometry — great! However a slight scan at the other two gave me the impression that they are elaborate, complicated Number Theory + Combinatorics questions. That scared me a little bit, but anyways I have to focus on geometry first.

Continue reading EGMO Diary Part 2

IMO Diary Part 2: Contest

9th of July

Long story short, first day was not so well…

I didn’t sleep well yesterday and this morning I was quite a mess. I was on the bus and trying to wake my brain up by doing multiplications. Yet instead of waking up it just caused me to panic when I could not figure out what is 15 time 15 and what is 25 times 25.

After getting to the arena at a little past 8, I just sat there doing nothing. Can’t really do reviews now, as all our things are taken away from us except for some food and our pencil case. The contest was due to start at 9 and a half. I spent the hour trying to wake up and figure out what is 15 times 15. On my desk, there was a bottle of water, a banana, a few snacks, my name tag and the folder for the questions. In the folder, there was a sealed envelope containing today’s questions, in whatever language you like, and three smaller folders that you are supposed to put the answer sheets in. There are two types of paper, one yellow for scratch work, one green for a formal answer. There were 5 cards in the folder for “Water” “More paper” “Toilet” “Questions” and “Help”. If you need anything, raise the corresponding card. Before the test even started, I raised my white card to ask for more paper–10 pages that were given would not be enough, I knew.

4:30:00. The test would be 4 and a half hours long. I opened my envelope as the timer started ticking away, and was relieved to find Q1 to be a geometry–my strongest subject. Nice.

I scanned through the questions and quickly jotted down all thoughts that came to my mind. Geometry, use equal angles, similar triangles or perpendicular lines? The third one (Although the possibility of me even trying the third one is rare) is a combinatorics question—guessing, and induction? Sequences might be involved…  I have to use the first half an hour to make sure I understand the problems and have no doubts because, after the first half an hour, you are no longer allowed to ask questions.

I’m quite a beginner in Olympiad, so I am not expecting anything better than an honourable mention. Therefore I would focus all my energy in solving the first problem.

4:00:00 Geometry—get cracking! This one involves circles, isosceles triangles, and parallel lines. Not a common combination, I had to say. But the graph looked easy enough, and given the difficulty of the Q1 problem of the past few years were not high, I was fairly confident.

3:00:00 Wait, wait, wait… I wasn’t going anywhere! This keeps going round and round in circles! I assumed the wrong thing! I took a deep breath and asked for more paper.

2:30:00 I diverted my attention to Q2, an algebra question, hoping a ray of inspiration will hit me as I was busy thinking of something else.

However, my brain kept whispering: Algebra is not your strong suit! This is a Q2! Your best bet is on Q1 and I won’t have you not getting it because you wasted half an hour on a question that you won’t ever get out anyway. A wave of anxiety hit me as I was trying to figure out what should I do.

2:00:00 I was trembling. I raised the green card and escaped the room to go to the toilet. I ran to the toilet, locked myself in, raised my arms to form positive posture that according to Amy Cuddy (For anyone who have not seen her TED talk, super recommend!) will improve my confidence and overall performance. I splashed some water on my face—calmer now. I ran back to the exam hall. No time to lose!

I made up my mind now. Just do Q1— Even if I can’t get it out at least I can say I tried my best.

1:00:00 I drew my 5th graph of the problem and went over my data. Looking back, it’s amazing how fast time past when you’re stuck on a problem.

0:30:00 Although I know I still have hope I’m already accepting the possibility of not getting anything out of day 1. This was a cruel thought, my cheeks were burning and my head was throbbing and my hands are trembling. I told myself—relax, calm down, accept the fact and be happy. There’s nothing you can do about it, you tried your best, so be it. Walk out of here smiling.

0:20:00 I checked my answers, numbered them. I presented all I have, hoping, basically, to not get zero.  

0:10:00 I took one last frenzied look at the graph–ray of inspiration?

0:05:00 I wrote my name. Loudspeaker bellowed.

0:01:00 I put the answers into the folders. Arranged them neatly.

0:00:03. 0:00:02. 0:00:01. Time’s up. The timer turned red.

It was not a good feeling, walking out with a bad result. I was trying to be optimistic and telling myself that this is the first day, it was not over and also, it is the experience that matters, not the marks. But still, I did feel very disappointed with myself and it was still quite exhausting for me now, sitting here, half a year after, remembering it.

I stayed relatively calm and joyful after the exam and didn’t freak out. For the entire day afterwards I managed to push the thoughts of the exam into a box in the corner so that I could still focus on the present. We went shopping together…I thought it would be completely weird for us to, out of all the things, walk around a supermarket and not buying anything, but it turned out to be a fun and relaxing time.

After the hustle and turbulence of the day were finished, in the night lying by myself in the dark, I started reflecting. I comforted myself by thinking “I did my best” and of course internally I had doubts—did I actually try my best? Could I have done better? Could I have coped better and explore more and think more instead of just sticking that same point in the same question?

But the optimistic side, of course, was that there’s another day out there, and also—although it is a big cliche already— The marks weren’t everything and I just decided to try my best and go for another day.

10th of July

So this morning…

While walking into the big arena…

I had a eerie feeling that something BBBIIIGGG will happen.

I tribute this to the extra big mug of coffee I had this morning. But anyways, something did happen that morning, so who knows… sixth sense?

We arrived at the arena at around 8 o’clock, like yesterday. Less nervous I walked around the arena chatting to friends. I spent loads of time finding everybody from my team (when I found them, turns out everybody are at the table of the New Zealand team). I also managed to find my friend from Iran, friend from Ecuador, and friend from China.

Apparently, it wasn’t only me, everybody was much more relaxed than yesterday. A contestant was doing push-ups on the floor and his team members were around him counting. (I actually think this was the spark that inflamed everything that happened afterwards )

About 40 minutes to the exam start, I went to the bathroom and even before I re-entered the hall, I could hear roars of applause coming from the arena. Not wanting to miss anything, I speed-walked back and this is what I found (photo source: 2018 Cluj-Napoca facebook page)

Of course, I was shocked to find that a lot of my friends were lying down as if they were just having a relaxing day off in the warm sun on a Hawaiian beach. And the next thing I knew, I was lying down with them on the concrete floor, facing the bright fluorescent light. It really helped with the butterflies in my stomach.

But the next thing that happened made the butterflies flutter even quicker. The committee apparently thought it was a good idea to put on climatic music on the second day of the contest so the arena began filling with dramatic Start War type of music. The music was really tempting for something big to happen, maybe it was just me and my butterflies, but I thought I could even feel the tension building up inside the room.

After a while, everybody started getting up… and started taking a stroll around the hall. I mean, why not? A stroll was very healthy, particularly in an international exam hall with several hundred people including child prodigies and possibly future Field medal winners from all across the world. Soon everybody was in a parade, pulling their friends and motioning to the volunteers to join in. When the atmosphere had built to this extent it was impossible not to start running, so everybody started to jog. Frankly, the speed was quite fast and I was panting and sweating after just a couple of lapses. (Were the people leading this a maths prodigy AND sports champions?) But I felt not at all tired, I felt exhilarated, running alongside my friends, old and new, running in the stream of brilliant minds from all across the world.

After a while, everybody was getting a bit tired and had to stop for a while. Just when the parade seemed to be ending, a boy in a yellow shirt picked up the flag of IMO from the row of flags at the side of the room and started sprinting with the flag flying behind him. Soon, the flags of different countries joined in. The teammates of each country started grabbing their flag and running after it, and soon the entire scene seemed far too surreal. My vision was soon filled with the moving colour and the moving stream of people waving their flags, and I couldn’t pick up all the information that’s flowing around me all at once. Almost if I was travelling the world, almost as if I’m joining a global community. We change posts in leading the flag whenever one is too tired in sprinting. And have I mentioned the music?

We moved to the back the room after a while, and raised the flags over our heads and gathered them together. We raised them on tiptoes, trying to reach as high as we can.

View from below, the flags are shining under the light of the big arena, the flag of IMO being held among them. We hugged our friends, shook hands with strangers and right here the young people from all around the world were connected through one single event. It was maths and knowledge that surpassed the boundaries of countries and brought all these people together under one roof.

We finally put the flags back where they belong and then slowly sat down. It was 10 minutes to the exam. The music slowly died down. But I was still panting and heaving from excitement, and I felt a strong surge of confidence as I looked at the folder with my challenge inside. I turned around, and smiled to the Indian contestant sitting to my left to whom I have never talked to before, and said: ”Good luck!”.

Am I over-romanticizing this? I must admit that I might be, as I thought that this will make world news but later nothing happened. But I was so very deeply affected by this event, I was almost on the brink of tears as I hugged my teammates and my friends from around the world.

Raising the flag of Ireland there in that Arena almost felt like raising the Olympic flame to me, and the resemblance is very strong in my opinion. One of my teammates later joked that this might be the closest we have ever come to world peace…

IMO Diary Part 1: training camp

Bit of introduction:

IMO (International  Maths Olympiad) is an international event hosted for pre-college students, it’s an annual event of 6 hard maths problems that required a lot of creativity to solve. The test is spread out over 2 days, each day with one paper, each paper consists of 3 questions and students are given 4.5 hours to solve them individually. Each participating country will send a team of 6, plus a leader and deputy leader. IMO 2018 (2nd of July-14 of July) took place in Cluj-Napoca, Romania.

2nd of July

12:00 p.m.

This is the big day! I have no idea what awaits in the next magic two weeks, and l’m sure I’m gonna love it!

On the plane: nothing much happened, we did problems and discussed together, and I finished reading the handouts from the training camp. In general no one disturbed us.

We got to the hotel, where we would be training with the team from Trinidad and Tobago, Everyone’s exhausted, yet I didn’t fall asleep immediately, I was stuck in this half-dream-half-awake state in which that I have to satisfy 3 equations in order to be allowed to sleep…

3nd of July

Very long day of training! We had our training session in the hotel dining room, the glass window above turned the room into a real greenhouse, and we’re really toasting while doing the math problems. We had three sessions, one on Number Theory, one on combinatorics and one on geometry. I got really stuck on a geometric question at the end of the day, and at one point I thought I got a solution. During supper, I was on the process of writing it down when I realized the solution doesn’t work and that I assumed something is true. I got really frustrated, as you can imagine, and I really could not taste the food in my mouth… It was the most surreal supper I ever had, and I wasn’t be able to sleep that night!

4th of July

I GOT IT!

I lay in bed thinking about the math problem last night, and I thought I had an insight, so this morning when I woke up, the first thing I did is to scribble down my solution. I completely abandoned my method from yesterday, but it took me very little time to get it out once I was on the right track. I was so happy, but at the same time bewildered that this question could take me so long.

We went to the Babes-Bolyai university today to train where our team leader managed to get a room. We trained the entire day, and in the evening we went to the national botanic garden to relax. In this process, we had loads of fun, and we got to know the Trinidad and Tobago members really well. They were all super nice, and I found a surprising shared passion with one of the teammates in our love of singing Disney songs and practicing Spanish. We would burst into spontaneous songs and dances and acting in the middle of the street, which caused everybody to stay away from us in order to save their faces.

I heard that the friendship between the two countries dated back a long time ago, when one of the IMOs was held in South America. We have been in touch since, and often trained together year after year. I think this kind of connection is really touching, how people from different time zones could be connected by a single event.

5th of July

Training! It was intense but I like it. At the same time though, it just taught me that I have a super long way to go. Ego losses occur.

Also, a very mind-opening conversation occurred between me and my teammates. We talked about what subject we are going to study at university. A lot of people’s are, without a single slight trace of doubt, mathematics. I, however, was still wavering between physics and maths. My interest in physics was peaked by a brilliant teacher, who was a deeply philosophical person and connected his teaching of laws of refraction with the philosophy of life and scientific methods. I was fascinated with the idea of exploring and figuring out the true law of nature. I never really considered maths a possibility, because it had a reputation of being hard and I never thought of exploring it any further than school work.

They shone some unique light into the difference between science and maths.”They have completely different ways of thinking, science is external coherent, and maths is internally coherent.”

I asked what this meant, and they explained that in science, you observe something that happens, and develop a theory to interpret it. Hence external coherent, you wrap your system around your observations. In maths, however, you build up you theories from the most basic axioms, and you never, never for the world contradicts what you already have with what you are developing, unless there is a desperate incoherency which forces you to break the limits, like what happened when people started to question the Euclid fifth postulate and developed projective geometry.

I found this an acute angle of looking at the difference between science and maths. This triggered great mental conflicts in me, and causes me to consider: is maths maybe for me?

6th of July

As a break, we decided on embarking on a great journey to Cheile Turzii (at least, that’s what Google map told us where we were).

We had a trip across the valley, looking at the beautiful mountains and the pairs of butterflies on the meadows. We crossed a brook barefooted to avoid having to walk all the way to use the bridge, and met up with a friendly shepherd and his flock of sheep. This picturesque country scene made me fall in love with the country immediately.

It was a great excursion, not to mention our great picnic under the trees and Romanian traditional dessert while doing math in the shade on the hilltop. (and feeding the shepherd dog that was always hungry) Oh, that moment was such a colourful moment, one that I would never forget.

And just after we finished our lesson on spiral similarity, however, there came the sound of thunder. A large clot of grey cloud hang heavy across the sky, and seemed to be coming at us. The shepherd started herding the flock of sheep over the hill back home. Some of us were a bit worried while others were unconcerned and insisted that staying under the trees will protect us very well. While we were debating, thunder can be heard rumbling down the hill and the Irish team would not stand it any longer. We raced down the hill, just when raindrop started to fall. We found a restaurant at the foot of the hill, and took cover under an umbrella. The minute we got settled, the rain started splashing down, blurring all our vision and even angrily invading the space under the umbrella. The rain covered the world in a thin sheet of downpouring water. About 10 minutes later, we saw the people who decided to stay slowly walking across the hill soaked in rain, a bitter smile on their face… We couldn’t stop laughing!

Looking back, it was a great adventure but I had to thank my luck. To think, I had my passport in my bag and everything. If that got soaked…

7th of July

We had a review session in the morning, we trained in a dark underground lobby, and because there weren’t any chairs everybody sat on bean bags, REALLY comfy!

We went over what different types of questions could come up in different topics of olympiad maths, and what to do to meet the different challenges. The session is really structured and helpful. Next, we headed to our official hotels where we will be staying with the other teams. Our two teams arrived a week early because we would like to train together for a week before the actual contest starts, but actually, most of the countries arrive on this day. Turned out, though, all the countries are spread out across 4 different hotels, and our team and the Trinidad and Tobago team were separated cruelly. This was a pity, given that a huge part of IMO is socializing, and now we won’t get the opportunity to talk to a lot of other teams.

The best part, though, was the goody bag! A grey, neat and smart looking backpack that was stuffed full with stuff, two T-shirts, one of which have the “dance of the functions” on it (Which I LOVE! I want to dance every time I see someone wearing it) an earphone, a water bottle, a tray (yes, a small breakfast tray, to my great bewilderment…we still haven’t figured out what is the meaning the committee is trying to convey) and the Mascot, a cute little teddy bear with a small head and big body called ”MIMO”. We tried out the games room, which is a common feature of IMO, a room in which people can play games and play chess and relax and socialize. However, this year’s game room only had a handful of people in it, and they are from the US and Canadian team, so after 5 minutes mt teammate Anna and I went back to our room, feeling a bit intimidated (after all, these are IMO gold-medalists!). The boys stayed, though, and we were very curious about their result in their fearless contest against those amazing mathematical minds.

8th of July

The boys told us what was the result of yesterday. They did lose a few rounds of chess, but in the end they developed a strong bond with the New Zealand team by a game of Monopoly, which, as it turned out, later became a bit chaotic and no one knows who is the winner.

Today was the opening ceremony, so we changed into fancy clothes and went to the big arena which will be our exam room the very next day.

We looked around the find the Trinidad and Tobago team and found them seated very much to the back. We had like an hour before the actual thing starts, so we started chatting around. I chatted with the girl sitting in front of me who was from Iraq. This was her first year here and will be her last as she was in her final year in highschool. She was really nice and we exchanged contact numbers took a selfie together.

I was on the edge of my seat trying to get up and chat with the Chinese team. I come from China, and of course, I have long heard of maths olympiad team in China which held endless mystery to me. I was even strangely nervous when I saw them seated not too far from us. “Go!” my teammates urged me, and after hesitating a while, I went over to say hi.

They were a bit shocked when I approached and greeted them in Chinese. The two teachers started to chat with me while the 6 team members eyed me curiously at a distance. I felt a bit guilty, actually, to think that back in China I guess I won’t even be able to get in city-level selection, yet here I am, standing together with the Chinese team who have struggled for so long and trained for years to be here.

The teacher discussed the difference between western and Chinese education systems with me, and he thought that the western education system indeed holds many possibilities for students to explore their full possibilities. Like me, for example, never in the world will I go and poke around the realm of mathematics if I were in China. Whereas here, I was already starting to think of choosing maths as my future field of study. This is stuning change that I could never see coming.

Anyways back to the Chinese team. The 6 teammates looked the kind of quiet student you might think you could find anywhere in your district, but wait until you hear their history… In the 6 of them, 3 are accepted into Tsinghua University, 3 are accepted into Beijing University, the top universities of China! And by “accepted”, it was not the ordinary admission, it was the type that guarantees the university entrance, no exam results needed (ok, well, you’ll have to pass, but you really don’t need to bother about it much).

While we were doing all these socialising, the team leaders of different countries are sharing the problem sets from their countries. Turned out, this is a “ritual of exchange”, if you will, between countries. Each country would bring a booklet of the problems in their country and share them around with other countries. An exchange of knowledge and resources.

The opening ceremony itself was not all that exciting, loads of speeches, and we tried in secret to do maths problems. The parade of countries was interesting, though, we had our ”three minutes of fame” as we walked on stage with our Irish flag.

After the ceremony, we started taking pictures with other countries, and we also got to walk around a little and socialize with other countries. I found a lot of friends from EGMO (European Girls Maths Olympiad). Friends from Ecuador, friends from Ukraine…

In the afternoon, we had the choice of going to the arena again to see our seatings for tomorrow. However, we preferred instead to go to a street food festival just next to the arena. Yum!

After dinner, we went to the supermarket to buy food. The test would be long and people would get hungry. It was always recommended to buy food that you like and would give you energy, as long as the food was not, according to IMO instructions 2018, crunchy and noisy.