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How to start with Math olympiads

Hi fellow people and cats :D.. Welcome to the first blogpost of the OMC blog. The Online Math Club is an initiative to reach high school students interested in math and give them a platform to learn more and interact with others!

By the way, I am Sunaina and you will be reading my thoughts :P

I hope you all are doing well and are enjoying math! You probably guessed what type of content today's post will have! So without further ado, lemme begin.

We all remember the first time we heard about the term "Olympiad math" ( Let's not consider competitions like SOF, simply consider HBCSE's Olympiads, USA(J)MO or the equivalents in other countries). What were your reactions? Lemme share my reactions when I heard about them for the first time, " WHAT?!!? HOW AM I SUPPOSED TO GIVE THE SAME EXAM AN 11th GRADE KID IS GIVING." (I found out about olys in grade 8 btw, but started seriously preparing in grade 9 :P) Haha anyways do comment what your first thoughts were!

So, I was so new to Olys and I did not have any idea how to start :(. I still remember, I didn't even know what any of AoPS or Quora or Math Stack were (Yes, I was from the stone age). 

The first thing you should do is make an account on AoPS. 

An unofficial simple guide to AoPS:

High School Math is for discussion of PRMO, AMC, and similar math contest problems. Harder problems are also discussed.

High School Olympiads is for Olympiad problems discussion. This is actually the most important forum. Here you can discuss olympiad problems of any level! 

Contest Collections is a treasure. Here, you can find past year's papers of contests throughout the world, the solutions discussions, and reviews of almost any paper!

Additionally, make sure to check some of the "other forums" such as Peer to Peer Programs, AoPS Mock Contests, India, and ofc Oly-PD. :P Although OLY-PD is dead, it does have nice problems of INMO level.

Moreover, do try to post solutions (it really helps in developing your confidence in writing solutions). And learning LaTeX is pretty useful, so do learn it!  If you are interested to learn, see this link.

How to write a nice solution?

"Can you please share the best books I should do
Now that I am in this community for 3 years, it feels just so nostalgic.. I still remember DMing so many seniors asking how they prepared and luckily, some of them were so kind to reply! 

But why take so much time to Direct Message people? Here's a link to Books, basic handouts, and resources for the newbies who are completely new to Math Olympiads or as we say " Math Olys" written by my favorite senior and the co-director of OMC Rohan Goyal. Again, we hope to make our recommendation list after some time :P

Cool, so now we have Books, handouts, and sufficient resources to start the "preparation" BUT BUT the most important question is How am I supposed to learn, how should I try problems, etc etc?

Firstly, have patience. Any problem should be tried for at least 1.5 hours. Do not give up before that. And in these 1.5 hours, think, think and think. LIKE REALLY THINK and do not be blank or simply stare at the problem.

"But I am  completely new Sunaina, I really do not know how to solve or even what to think?"
Fair question.

Let's take a few questions which really do not need any theory.

The first problem is from this year's IOQM! It's IOQM 2022 P7.

Problem: Find the number of maps $f: \{1,2,3\} \rightarrow \{1,2,3,4,5\}$ such that $f(i) \le f(j)$ whenever $i < j$.

First thought from a newbie perspective: OMG Functions!! So fancy.

Do not be intimidated by the problems. The statement I always tell myself is, "If this problem appeared in the test $\implies$ it is solvable and the proposer expects us to solve it."

Now, think about what the problem is actually asking... It's basically asking you to find three numbers $x,y,z$ such that $x\le y\le z.$

Cool! Now here's how you can proceed!  The first thing I will tell is the codomain is random. So try for $\{1,2,3\}$ then  $\{1,2,3,4\}.$ What do you observe?

Moreover, another way to think is, What will happen if $x=5?$ Can $y=4?$ No! Because $x\le y.$ So $y=5.$ Similarly $z=5.$

What happens when $x=4$ ? What possible values  $y$ can  take? Similarly, try for $x=3,2,1.$ 
Remember, you have a lot of time so try to observe and see patterns! Do comment below about your solutions to this problem!

We will also have another blog post dedicated to how to approach a problem! For now, here's a small tiny list of ideas you can try:
  • Check for small values. The first thing that strikes my mind is "Why did the proposer give that specific value"
  • Induction is a pretty handy tool
  • A good diagram is always handy, especially if you are a beginner. Not only true in geometry
  • Be greedy and think about what ways you can prove and try to simplify the problem
  • Think about various $\pmod n.$
  • Try relating the problem with some other problem
Just try the problem. Also, I have noticed this with myself, I tend to remember solutions and mostly ideas to those problems where I have really starved, died, and tried a lot.

I also have a small notebook, where I note down extra amazing clever ideas. It is really helpful since before any important exam, you can simply see through that notebook once :P

Some FAQs:

1. What do you do if stuck with a problem?

Ans: Did you try to simplify the problem or tried becoming greedy? I assume you tried your very best. Firstly, remember that it's okay not to solve all the problems you get. 'Cause, that implies that there is still room for improvement :D. 

It is a good idea to try the olympiad problem for maybe 5-6 hrs without assistance and then if you have no progress whatsoever try starting off with a few hints to push you in the right direction to a solution. 

Do ask for hints instead of dropping the problem! Feel free to ask in the OMC server (in the math-discuss channels).

Even taking a small break, trying other problems, etc isn't a bad idea. I personally do dancing or talk with my friends..

2. Is it necessary to complete all the chapters of a book?

Ans: No. Not at all. I am pretty sure completing all these books, and papers will take you a lot of time. Feel free to skip problems. Also if you feel a problem is ugly, just skip it

3. Is it okay to skip problems?

Ans: Yes. At least I think so. See, the whole essence of Olympiad math (at least Intermediate Olympiad math) is to make you love math and not force you to do the math. And doing problems, which don't seem ugly, is completely fine. 

4. How do you define ugly?

Ans: Self-explanatory. It's from person to person. I feel ugly problems are geometry inequalities, weird graph theory problems using high fancy stuff, and bashes. But this is my PoV, for you, it might differ. I know people who love to just bash a geo problem. So you see it varies.

5. How many hours should I study for Olympiads?

Ans: I never ever recommend this question to be asked. Not a good question. I used to ask too, but it is sometimes annoying. Here's the thing: it differs from person to person. Some of my friends just study for 3 hrs and clear INMO. But some study hrs and hrs. I read somewhere that on average people spend 48 hrs a week. But I think the time you spend on a problem depends on your priorities and how much time you are giving to maths in a day, other time constraints and things like that.

Do spend time though. In the end, think why are you doing Math Olys? You are doing it since you enjoy it! Don't take stress, simply enjoy the journey!

6. I just started out Oly but I am doing X too ( where X is NTSE, JEE, KVPY, etc). Is it okay?

Ans: Yes! In the worst case, when you look back you would still be happy that you spent time on some of the most beautiful problems in math oly :D..

7. Should I refer to some YT channels?

Ans: Yes. Sure! Here is a list of YT channels that have pretty good content ( There might be many more)

  • Shefs of problem solving
  • 3 Blue 1 brown
  • Art of problem solving
  • Venhance
  • Online Math Club

8. Should I make notes?

Ans: I make. But I am very careless, so I always misplace my notes or delete the pdf.  I am pretty sure most people don't.

9. Sometimes I don't feel like studying, what should I do?

Ans: Take a break and have a Kit-Kat. (Kit kat please sponsor)

Just take a break, your brain is asking you to take a break, that's it. And it's normal to have a break. When my EGMO TSTs were over, I took a break of 3 weeks. I simply didn't do any Olympiad math and rather focused on school math (which is very important).

10. Are teachers/Otis/Woot/Sophie Fellowship helpful?

Ans: Generally your parents and coaching teachers help but tbh it's your own hard work. But it's mostly self-study. If you get doubts, post it on AoPS or Math Stack Exchange.

Now, I'm not sure about Woot, you need to ask someone else. Do join OTIS! It is simply so good. Evan is also very kind with financial aid. Moreover, do not be scared of applying! And if there is some financial burden, do tell to Evan. 

Sophie Fellowship is the gem! It provides such a nice community and so many resources! 

11. Do you think gsolving is helpful?

Ans:  Not really in terms of Math. Although, it's very very very fun. So, do it when you are not really in the mood of doing math :D

12. Why should I be interactive with everyone else?

Ans: Because that is the whole Essence of Math Olys! To enjoy math as a subject and make similar friends! (Fun fact: Nobody studies in IMOTC)

13. I don't like cats, can I still do math?
Ans: No. Go do jee. (jk)

Here are some tips which I got after dming thousands of Seniors (Special thanks to Atul, Pranjal, Rohan, Aahan, Samarth and Gunjan)

#1. Do Not be obsessed with finding the right resources. Do whatever you find the best. Do not dm the seniors and start orzing them. Also, please do not care about the amount of time you spend on math olympiads. Simply enjoy!

#2. Don't hesitate to speak omc classes. Because the classes at the end are conducted just for you all! And the teachers love to hear doubts.

#3. Enjoy what you're doing and solve a lot of problems instead of cramming on theory.

#4.  Look at enough examples and solved problems when learning the theory/new concept. Moreover, You are not reading a storybook by, just looking at the problem and immediately checking the solution. Give enough time.

 #5. It's super important is that NOTHING WORKS FOR EVERYONE, no one has a magic procedure by which they became good at Olympiad, it's an important change to be made from doing other things to math olympiad that there are no "important books" which if you do you'll suddenly become insanely good.

#6.  Make sure you are able to prove any result in olympiads you want to use on the go. In other words, try to make your basics strong before learning any fancy theorems. Try to read the " Philosophy and not formalism" part of Evan's blog post.

#7. Try applying to various camps and do not be afraid of rejections.

Summary: Remember not to take the stress of the results and enjoy maths. Make Sure to make a lot of friends and be interactive in the OMC classes! Try to give problems sufficient time and do not be intimidated by hard problems. Also, be clear with your basics.

With this, I would like to end this post! Hope you all enjoyed reading it! Do comment your thoughts about the blog! Feel free to post feedback :D..Have a nice day and enjoy maths.

Sunaina 🍀


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