The End of Four Seasons

It’s finally over now. After four years of taking the American Math Competition (AMC) contest series, I’m finally done.

In a way, it feels liberating. No longer will I have to grind out test after test, fill countless notebooks with scrap work, or waste hundreds of sheets of paper printing out practice tests. I'll no longer have to count down the number of days until the next AMC test, waiting in anticipation to see if I can perform better than last year.

In another way, it feels bittersweet. I spent so long preparing for those contests, and I’ll probably never participate in another math contest after this year. I still have the American Regional Math League (ARML) competition later this year, but after that, it’s all over. My competitive math career will be over. As someone who wants to go into computer science in the future, many of the skills I've learned will be applicable in the future. But for many of the more niche topics or theorems that I learned about, they might never be used again.

I’m writing this post currently at 5:30 AM after not being able to fall asleep for a few hours. Just a month ago, I had finally achieved my dream of qualifying for the USA Math Olympiad (USAMO). However, when took the test just four days ago, it was nothing like I had dreamed of. I had worked so hard towards this goal for so long, but after taking the test, I just feel... normal. I'm not super excited or happy; I just feel like I normally do. When I first began dreaming about making USAMO back in middle school, I constantly envisioned what the moment would be like if I got in. Maybe I would cry, or at the very least, I would jump up in excitement and maybe knock over a few books. But none of that happened.

There’s a lot of reasons this happened, but a large one was the COVID-19 pandemic. The pandemic has almost entirely detached me from the real world. The version of myself right now is not the same one back in January of 2020 before the virus spread to the United States. The virus made me realize so many things about math competitions that I missed. I missed the way that everyone got together after the fact to discuss how we approached the problems.

I even miss the failures. I miss the times when I realized I made an algebraic mistake or didn’t see a certain way of approaching those problems. Those moments are the ones that pushed me to be better and made the victories all the more satisfying. Those moments of heartbreak kept me grounded in reality and made me realize that these contests were hard. But in a virtual environment, all of that is gone. Right after a test, we all say goodbye to our proctor and leave silently. None of us discuss anything. In a virtual era, the contests even seem to mean less. When I’m at home, the stakes seem so much lower, and the rewards seem much lower as well. The mistakes don’t hit as hard, and I normally shrug them off easily, which is something I never would have been able to do in the pre-COVID era.

Another reason this accomplishment feels more empty to me is the fact that I’m currently a senior in high school. I already received my college decisions, so this won't affect which college I get into. Additionally, senioritis is a very real thing that has full effect on me right now.

But senior year and COVID-19 aren't the only two factors that make me feel this way, and they certainly aren't the largest. During this post, I hope to explain why qualifying for USAMO this year felt more empty than I expected. To be honest, it's more of a summary of the past four years of my math career rather than a concise recap (I'm also skipping talking about ARML that much since I want to save that for another post after I take ARML this year).

Before High School

The first time I had heard of the AMCs was in 7th grade. My second cousin was a freshman then at what is now my current high school, and he had just taken the AMC 10 competition for 10th graders and below. After he took it, his mom told my mom about it, and my mom wondered if I had a chance at doing well. She printed out a test and handed it to me. Up until this point, I had very little math competition experience. I had taken a few small contests back in 4th and 5th grade, but nothing as major as the AMCs. As a result, I had only taken school math tests before, which I consistently got 95%+ on. When my mom printed out the AMC test, she didn’t tell me what it was, and I took it expecting it to be like a normal math test.

I still remember that night when I sat down to take it, and I got stumped on the 7th question. Panicking, I just skipped it, went on, and got stumped again. I skipped and skipped and skipped until I reached the end. After 30 minutes, I had only answered 6 questions out of 25, for a whopping 24% (Life Pro Tip: Never convert your math competition scores to percentages. It’s just not a good idea). I remember just breaking down crying at how poorly I did.

My mom was undeterred by my poor performance though. Despite how much I cried and said that I hated doing them, she just printed more and more and asked me to read the solutions to problems I didn’t know how to do. Gradually, I improved, and gradually, I grew to love the competitions. Up until that point, everything in math for me was formulaic. You either knew how to do a problem at first sight with a few formulas or you didn’t.

AMC problems weren’t like that. Instead, they were these puzzles that you had to fight with. You had no idea how to do the problem and you would keep poking and prodding until it finally cracked open. While there were some competition techniques you could learn that were akin to bringing a hammer into the ring to fight with, you could always work your way through the problem with the basics.

As I took more and more practice tests, I began scoring higher and higher. It was at this point that my mom told me about the AIME. If you score above the cutoff for the AMC test, then you can qualify for the 15 questions, 3-hour-long AIME contest. However, my eyes weren’t set on the AIME. They were set on USAMO. You can calculate your USAMO index by taking your AMC score which is on a scale of 0-150 and adding it to ten times your AIME score, which is on a scale of 0-150. If this combined index was above the cutoff, then you could take the incredibly prestigious USAMO exam, which was only open to around 250-300 of the top high school students.

There are further awards beyond USAMO, but for me, USAMO was the ultimate goal. At the time in 8th grade, I was only scoring around 110 on AMC 10 practice tests with the AIME cutoff typically hovering around 110-120. As a result, I made the following timeline for myself:

My target was making USAMO by 11th grade, with the primary reason being college applications. Making USAMO in 12th grade wouldn't affect my college applications, so I wanted to make it by 11th grade. During 9th and 10th grade, I set my goal to qualifying for the junior version of the USAMO, or USAJMO. Qualifying for it was easier than making it to USAMO, but I was only allowed to qualify for it until a max of 10th grade. Since qualifying for the USAJMO test was my priority for the first half of high school and involved scoring high on the AMC 10 test, I began grinding more AMC 10 practice tests. I would eventually have to switch to the harder AMC 12 test if I wanted to qualify for USAMO, but my focus was entirely set on going from AMC 10 to AIME to USAJMO at the time.

9th Grade

By the beginning of 9th grade, I had already done dozens of AMC 10 practice tests, and I was consistently scoring above 120 on them, making me feel like I was right on track with my schedule. At the beginning of the year, I was keen on joining my high school’s math team, as this would allow me to compete in various regional competitions and give me a chance to prove how much I had practiced over the last two years.

However, up until this point, I had never actually taken a real math contest before. I had only taken those elementary school ones and the practice tests at home. As a result, my first ever real math test was our high school’s math league tryout test. While everyone would be able to go to meetings and participate in larger contests, only 10 students would be selected for the team that competed at the CJML (Central Jersey Math League) contests. I was confident I would make the team. After all, I had done so many practice math competitions, and I was scoring pretty well.

The moment I sat down at the table for the tryout test and looked at the questions, I remember just instantly freezing up at the sight of the questions. They all seemed so... hard. 30 minutes to do 10 questions. Nerves kicked in, and rather than taking the slow disciplined approach that I had exercised throughout all of my recent practice AMC 10 exams, I took the approach that I did with my first one. I read the first one. I thought for a second and skipped the first one. I did the same again, and again. Upon reading the 5th one, I finally found one I could do and wrote an answer. My nerves began to settle. I thought to myself, “I can do this.” Then, I read #6, and my nerves spiked again. I began skipping around, jumping to random questions. In the end, I answered 5 questions, and only got 3 correct. The cutoff to make the team ended up being 4. I didn’t make it.

That day after getting home, I sat in my room and cried. Already, things were looking like a disaster. This first test was already a testament to just how nervous and poorly I performed under pressure. When I was taking the test, I immediately forgot everything I had practiced and panicked. Even worse, I wouldn’t get the chance to participate in more contests to help practice and relieve these nerves since I didn’t make the team. While there were still smaller contests for me to focus on, I did even worse on those and became more and more depressed. It was a negative feedback loop, and I felt hopeless since I just couldn’t perform well in a single contest and if I didn’t do well, I would never make it to the team. I might not even get selected to take the AMC 10 contest.

With these thoughts constantly plaguing my mind for the next 3 months, I finally got my lucky break. One of our 10 team members couldn’t make it to the next CJML competition, and my cousin volunteered my name as one who was available to go. And just like that, I was at my first CJML competition. I knew this was my chance to prove myself and overcome my nerves. And I ended up doing so. I only solved 6/10 questions correctly, but it was a hard test, and I tied for second place on our team. It was my best result of the year so far, and I had secured my spot on the team.

However, despite this positive uptick, the rest of the year didn't go well. While I performed okay at the next few competitions and retained my spot, I still constantly suffered from silly mistakes and struggled to solve some problems that I could easily do at home. Despite the brief positive upturn, my frustrations grew more and more. However, I still had one hope: the AMCs. I had practiced on so many AMC competitions that I was confident I could score well on it. I was convinced that I could excel in the AMC competitions.

The day I walked into the AMC 10 competition, I was confident. I had consistently scored 120-130s on my previous practice tests, meaning that I expected to make AIME pretty easily. Even if the first half of the year wasn’t great, I was sure I could turn it around for the second half. However, the moment I began taking the test, the nerves hit again. I began rushing through the problems, anxious to get a high score, and I hit a speed bump at question #3.

Upon seeing this question, I immediately set about bashing out the value of $10!$. I then bashed out how many seconds were in a day, and divided them. And of course, I made a mistake. Ideally, you wouldn't even need to bash this much on this question. You should keep both products expanded and divide the two, allowing you to easily cancel common factors between the top and bottom. However, in my rush, I completely ignored this and went straight for the brute force solution. One question wrong already, more to go.

I proceeded to get #4, #7, #11, #14, and #17 all wrong. And that's not even counting how many I skipped, which at this point I barely even remember. To guarantee qualifying for AIME, I needed approximately 17 correct to get a score of around 114. As time began to run out and without knowing that I had gotten so many wrong, I still only had 16 answered. Even if I got all of them correct, I wouldn't make it, as the cutoff was later revealed to be 111, while 16 correct would only give you 109.5.

But I didn't even get that 109.5, as I ended up receiving a 79.5 instead because of the 5 incorrect answers. I was absolutely devastated by the difference between this and my practice test scores. However, I still cautiously held onto hope. Even though I definitely wouldn't make AIME through the AMC 10A, my school still offered me a chance to take the AMC 12B a week later. It wasn't ideal because I had never practiced an AMC 12 before. They were harder than AMC 10s, but they also had lower cutoff scores. Additionally, taking the AMC 10 would set me up to qualify for the USAJMO, which was my goal for the first half of high school as it was much easier. Taking the AMC 12 would instead set me on the track of qualifying for USAMO, which was my ultimate goal, but extremely far from reach at the moment. However, this year, I just needed to qualify for AIME, not make USAJMO or USAMO. So there was still a chance.

The day I walked into the AMC 12B, I was already feeling much better than on the 10A. I breezed through the first couple of questions, fixated on the magic number: 14. I needed 14 correct answers to guarantee making AIME since that would give me a score of 100.5, and the cutoff was never above 100. And at the end of the hour and a half, I had done it. I answered exactly 14 questions.

That doesn't mean I got 14 answers correct though. As I nervously began checking my answers on the forums the next day, I was relieved to check each question and see that my answer was correct. Then, I checked #10.

I solved the question the intended way. The mode is the number that occurs the most amount of times, and for it to be unique, there must be only one number that occurs 10 times. Therefore, to get the minimum number of distinct values, you would want all of the other 2008 numbers to appear 9 times. So I did $2008 / 9 = 223 + 1/9$. Therefore, there had to be 224 other unique values in the set at least. So I wrote down 224 as my answer. However, I made a crucial mistake and forgot to include the mode back in the answer as a distinct value. So the answer was 225, and I was wrong.

I came out with a score of 94.5, but all hope wasn't lost. There was a chance that the cutoff was below 100, and maybe, by some great fortune, it would be below 95. A few weeks later, I was checking the AoPS forums, a famous site for discussing competitive math competitions. I saw that someone had posted that the cutoffs had been released. Nervously, I checked, and I saw a cutoff of 91 for the AMC 12B. At that moment, I instantly yelled out in excitement, which, to my embarrassment, attracted the attention of so many people on my bus. However, I was so happy because I had made it to AIME.

But I hadn't. The cutoff was a fake, a "troll post" on AoPS. The post was quickly removed, and just as my excitement had spiked, it declined again. When I checked the real cutoffs a few days later, it was 99. I would not be taking the AIME that year.

In retrospect, I wasn't too disappointed at not making AIME. I wasn't planning on applying to any competitive summer internships that year, so making AIME didn't affect many of my applications. However, I was most disappointed by my poor performance in various contests throughout the year. Nothing had gone well besides a few random math competitions here and there, and when it mattered the most, my performance suffered the most. What was the point of doing so much practice at home if it never translated to anything? And so my first AMC season ended in bitter disappointment and regret.

10th Grade

In 10th grade, things were looking up. I can't exactly pinpoint what changed. Maybe the several-month break between major contests helped refresh me. I also took several math classes, which helped give me extra practice with foreign problems and increase the amount of problem topics I was comfortable with. Either way, at regional math competitions, I started to perform much better. However, my goal that year was set on only one thing: doing well on the AMCs. Doing well in regional competitions definitely boosted my confidence though, so when I went in to take the AMC 10A that year, I was feeling much more prepared than in the previous year.

That day, I answered 21 questions, for a score of up to 132. After taking the test, I was relieved. Making AIME was almost assured, and if I didn't make any mistakes, qualifying for USAJMO was a very legitimate possibility. The index needed for making USAJMO was around 220-230 typically, so getting a score of 132 and then an 8 or 9 on AIME gave me a very high possibility to qualify.

However, of course, I didn't get 21 questions correct. The first question I realized was wrong was the following:

My thought process at the time was that $d$ was clearly less than both of the other values since it only considered the lower segment of the data. Therefore, I just had to compare $\mu$ and $M$. I knew from statistics that while the mean can be affected by outliers, the median is more resistant to outliers, so I thought the answer was $d < M < \mu$, as I considered the block of days from 1-28 as the dataset with 29-31 being "outliers". But the correct answer was $d < \mu < M$. There were enough values to significantly shift the median, as the days 29-31 were not actually outliers.

The next question was this:

For this question, I forgot to consider that negative integers could be used as well, so I calculated the wrong answer. However, the final question that I got wrong was the worst:

Yes, you read that number correctly, and no, I'm not joking. During the test, I simplified this to $2^0 + 1^1 = 1 + 1$. From here, this is basic arithmetic. And I wrote the answer as 1. And I moved on and never checked my work. With that, my score sunk to 118.5. A good score still, but qualifying for USAJMO had just become so much harder by these series of mistakes. While with a 132 I could've squeaked by with a 9 or maybe even an 8 in a good year, I would now need an 11 or 10. Since the questions on AIME got much harder later in the test, getting such a high score, especially with my incredible propensity for making silly mistakes, seemed impossible.

However, the thought that after doing so well so far in my sophomore year just to make the silliest of silly mistakes tormented me for weeks. Just a single check back to that #1 could've improved my score by 6 full points. Another few minuts could have saved me another 6 points from #5.

The AMC 12B the next week was rather uneventful. I don't even remember my score; it was ok and qualified for AIME through it, but my goals were set on USAJMO. I had no expectations of making USAMO that year, so I didn't pay much attention to the AMC 12B.

On the day of the AIME that year, I went in not expecting much. I had never scored above 9 on a practice test, so I had no expectations of doing so here. However, when I opened the test, I was pleasantly surprised. In the end, I answered 10 questions, skipping #6 and answering question #14. When I saw #14 on the test, I was shocked at how short it was:

With just a few short number theory formulas, one could find that the answer had to be 1 above a multiple of 16. After bashing out 17 and realizing it didn't work, I came across 97 as an answer. And it worked. For the first time, I had solved a "final five" on the AIME. I realized that I had a chance of making USAJMO. A score of 10 had a good chance of making it!

But alas, it wasn't meant to be. After the test, I realized I got #8 wrong:

By combining the given with the Pythagorean identity: $\sin^2(x) + \cos^2(x) = 1$, I bashed the problem and got an answer of 056. The correct answer was 067. I had gotten so close, but my answer was just a bit off due to a calculation error. As a result, I scored a 9 on the 2019 AIME.

Today, I'm still proud of that score. It was much higher than I expected going in, but the disappointment came from the hope I had when I first handed in my test. I thought I could get the 10 and qualify, but just that single small mistake this time cost me. With an index of only 208.5, I thought there was no way I would qualify for USAJMO, so for another year, I failed to reach my goal.

Therefore, the day the cutoffs came out for USAJMO, I didn't even check it at first. What was the point of just getting disappointed? Then, I got a text message from my classmate, who told me what the cutoff was. It was 209.5. Just one point higher than mine. The moment I saw that cutoff, I remembered the #1 I got wrong on the AMC 10. Just that single question that I could've checked in 5 seconds had cost me qualifying for USAJMO. It was a devastating blow. Once again, my AMC season ended in regrets and the constant thoughts of "what if". What if I had double-checked my work for the AIME #8 and the AMC #1? What if I had chosen to leave the question about the means, medians, and median of modes blank instead of answering it?

11th Grade

For the past 2 years, I had failed to reach the goals set out by my 8th grade self. However, I still had confidence that I could make USAMO in 11th grade. I could no longer make USAJMO because it was only available for 10th graders and below, but I knew that if I could just replicate the performance I had on my practice tests at home, I could make it. I was consistently scoring around 130 on AMC practice tests and around 10 on AIME practice tests, which would be enough. However, that was at home. During the contest was a whole other story.

For the AMC 12A that year, the first 10 or so problems went by smoothly. However, then, things started to get more difficult. I began getting stuck on questions in the 10-15 range, which was something that never happened in my practice exam. As I slowed down and lost momentum, I could feel the time pressure getting to me. My goal was to get over 120, so I began skipping every question I didn't know how to do in hopes that I could find something I did know how to do in the later questions. However, as I constantly skipped questions, I rushed through some and skipped ones that I could've solved with just a bit more experimentation. Before, I talked about how I loved doing AMC problems because of how you could just poke and prod at problems until finally finding the key. However, with the time pressure building, I gave up on that. I took a hammer, threw it as hard as I could at the problem, and when it didn't break on the first try, I moved on to the next problem.

In the end, I answered 18 questions for a potential score of 118.5. However, two of them were half guesses that I got wrong, and I made a calculation mistake on another one. As a result, my score dropped to 100.5. It would qualify for AIME, but it was so far from the score I needed to qualify for USAMO, as I would need a 14 now to qualify. However, I still had a chance with the AMC 12B. In the past, I had kinda disregarded the 12B since my focus was on USAJMO. However, this year, it was truly a second chance for me to make it to USAMO.

The 12B was a total flop. I got stuck at around question 6 and began skipping questions once again. I just kept on jumping around, and when I had only 25 min left, I realized I only had around 13 answered. As a result, I just began blitzing through questions, ruling out 1-2 options, and then just blind guessing. I made assumption after assumption and just put down answers. I put caution to the wind and went high risk, high reward.

I wish my story could end here with talking about how I got a 130 on the 12B. But such a fairytale ending didn't happen. Almost every single one of my guesses was wrong. My score plummeted, and I don't even remember what I ended up getting. Around 88.5? I don't remember, but the moment right after the test, I realized that my dream was over. Year after year, failure after failure, I had held out hope that the next one would be better. But that dream was all over now. Getting to USAMO would require a 14 on the AIME, which was a pipe dream at this point. That day was probably one of the worst days of my life, and I cried throughout the rest of my classes and during our robotics meeting afterward.

Most of all, I felt like all of my preparation had been for nothing. Up until this point, I had spent thousands of dollars on math classes and books. I had also spent hours of my life searching and doing math problems. I had notebooks and folders filled with old problems. But what did I have to show for it? Making it to USAMO was my ultimate goal, and that dream was over now.

A month later, I took the AIME at school, and once again, I scored a 9. This time, I made 2 mistakes, but when I realized that I made those mistakes, the pain was less. Since I was so far from qualifying from USAMO, it didn't hurt as much. While I was disappointed my score wasn't higher, it was still good, and I was proud of it.

At the end of the year, I was given a second chance: the online AIME (AOIME) which would be available for everyone to take, regardless of whether or not they already took the AIME that year. When I first heard about it, I wasn't excited about it. Getting a 14 was just so out of left field that I couldn't imagine it happening. When I ended up getting an 11 on it, I was pleasantly surprised by it, but not really excited, since it still wasn't close enough to qualify for USAMO. Everything was just numb after failing to qualify for USAMO.

12th Grade

However, after that competition season, it was liberating. I no longer had to grind math problem after math problem with the sole goal of qualifying for USAMO. Now, I could just do problems because I wanted to. For the first month after not getting into USAMO, I refused to do any competitive math problems. But after a while, I got back into them again, and I realized just how much fun these problems were. Before, I tediously grinded problem after problem, just wanting my score to go up, and I got incredibly frustrated when I couldn't solve easier problems that I needed to get the scores I wanted. Now, I no longer had to worry about that, and I could just focus on each problem individually and enjoy them without having to constantly worry about time pressure and stressing that I had to solve it. I still practiced a ton of math, even though it wasn't as much as I had done in previous years. On problems I got stuck on, I resisted the urge to look at the solution and instead worked through them slowly, wanting to understand them and really get to them.

When I went to competitions in 12th grade, as I mentioned before, the virtual environment made them feel a lot less impactful, which definitely helped with my nerves. However, the main factor was the change I just mentioned in the previous paragraph. I was just enjoying the problems without much pressure to do well, and this year was my best year in terms of math performance. When the AMC competition season rolled by, the same concept applied. I wanted to do well because I wanted to prove that I could make USAMO, but there was no looming threat that if I failed, my dream was over.

This change of mentality made the world of difference. I actually did really well on AMC and AIME, echoing my practice test performances at home. I even made it to USAMO. I'll talk more in-depth about this year's season in a different post since this one is just way too long right now, and I mostly wanted to focus on my reflections. In the one about this season, I can talk more about my thought processes and the problems.

But as I mentioned before, I didn't really feel too much about it. Yes, I was happy and excited to make it. But after spending years of visualizing the moment in my head, the actual thing was nothing like it. Part of it was because I took all of my tests at home. But another part of it was because of the fact that my dream had already been given up on, and part of me recognizes that it was because I failed my dream that I was able to move this far. Once my dreams and goals were over, they finally stopped putting pressure on me, and I was able to finally perform well. I could just focus on enjoying and doing the problems, and my performance was better than in the past where I constantly just strived towards this goal of qualifying for USAMO.

I wish I could have had this revelation in the past, but the sad part is that I'm not sure if it would have changed much. I feel like these words I'm saying are kind of things you hear all of the time but are really hard to take to heart. It was only after the soul-crushing destruction of my dream that I finally managed to put aside my feelings about my goals and instead just enjoy the problems. Additionally, even if I were able to put aside that constant goal, what would have motivated me to keep moving on after the failures? If I had just done math to enjoy the problems, then I wouldn't be here today either because I wouldn't have pushed myself so hard to study all of these concepts. It's a balance I constantly wonder about, but I can't change what happened in the past. I can just appreciate how it has transformed me into the person I am today and move forward in life.

To wrap up this super messy and long post, I wanted to say thank you for reading through all of this (or at least scrolling all of the way down here to the end). If anything, I would compare my four years of taking the AMC competitions to a roller coaster. There have been ups and downs. Moments of high highs and low lows. During those moments of plummeting, I've wondered why I got on this trip in the first place. But after having gone through it, I'm glad that I went on it, as it certainly was an exhilarating ride.