Turtle Math : CATEGORY THEORY (1/2)

My room was a mess --- papers overflowing from my desk to the floor, collapsing stacks of books here and there, stuffed animal turtles everywhere... you name it. My mother gave me the arduous task of cleaning it up before I moved back to Pittsburgh for college.

As I reluctantly dug through the piles of papers and books, I discovered an old, dusty notebook. I didn't recognize it --- what could it be? I curiously opened the notebook... and was horrified by what I found.

Math. Endless math. Pages and pages cluttered with cryptic diagrams, systems of equations, cyclic quadrilaterals, modular arithmetic, and countless columns of answers to past math contests. Surely, whoever owned this book must have been definitively deranged. Indeed, what sort of sane person would be able to fill this entire book with math?

There was no mistake: this notebook must have belonged to me, during a past life long ago.

How could I have forgotten? Slowly, the memories came flooding back: Hours of pouring through Mock AIMEs and high school contests. Plane rides and amusement park trips where I worked through Korean Olympiads instead of anything remotely fun. A period of blind insanity where I craved one thing: A USAMO qualification.

Of course. That's why I forgot. That year was all a blur, a distorted smear of events that began when I first tasted hope that I could make it, and, most importantly, earn her validation.

I shut the notebook and focused on remembering. I remembered that it was like a dream, a fairy tale come true filled with unrealistic miracles that propelled me forward time and time again. I remembered anxiety and worry, worry that I wouldn't make it, worry that the AIME wouldn't be held, worry that I would fail to earn her attention. I remembered the journeys, those great adventures at math contests across the country, from Princeton to Carnegie Mellon to MIT. And above all, I remembered the victories: Victories beyond my wildest imagination in great twists of fate; victories by the smallest of margins, snatched from the fingers of the devil of misfortune. I remember now. I remembered it all. I knew that sooner or later, I should write down the countless stories that I remembered... before they are lost to time.

A year or so later, here we are. The chronicles of my mathematical journey, from middle school to college. I do not expect you to read it all, dear reader. Though if you made it this far, all I ask of you is to bear witness to a story or two. These stories, while written partly for my own records, are also a way for me to give back to you. After all, you too play a part in these stories, for Category Theory deems it so.

Some number of months after I was born, I said my first word: "Circle!" My parents were right to be concerned that their baby would become a mathematician instead of an electrical engineer, like everyone else in the family.

Throughout elementary school and middle school, I was very apparently the best at math. But this was not in my favor, since I thought I was much more competent than I actually was. I knew my powers of 2 up to $2^{16} = 65536$ , memorized $100$ digits of $\pi$ , could "do" some basic calculus, and knew some "college math" (e.g. $e^{i\pi} = -1$ , despite me not knowing what that meant at all). From this, I concluded that I was a math genius with a very high IQ. On the other hand, I was definitely a lazy mess that didn't do their homework and got pretty meh grades (i.e. B's and the occasional C). My parents accepted that I was doomed to go to a local community college.

These were certainly pretty bad signs for my future, but I won't deny that I had some talent. In 8th grade, I managed to score a 23/25, which I'd definitely consider to be a great score. Part of that may be attributable to the charity of a math tutor I had for about a year or two, though it's quite hard to remember the details.

My delusions were finally shattered when I entered high school and heard about W, a girl in the grade above me who had already completed Calculus BC. I was stupefied and distraught --- how could I not be the best? I was never able to skip a grade in math, how could she have skipped two? Soon, I learned that she was truly no ordinary student: By then, she had qualified for MOP, effectively placing her in the top $30$ high school students for math in the entire country.

My self esteem dropped to rock bottom. I lost my overbearing confidence as I realized that I had much to learn. That year, at the Nassau Math Tournament, I "only" got 5th place, finally hammering home the truth that I was not the best. I accepted that I would probably live a life of mediocrity. It's not like my grades were going to get me far.

The resentment and jealousy I felt toward W slowly morphed into reverence. I practically worshipped her. After all, she was, in my eyes, the god of math. I kept thinking that I would never in my life be able to match even half of what she had already accomplished in high school.

I got an alright score on the AMC 10 score that year, but it wasn't enough to make AIME. Disappointed, I figured that I didn't have a great future in math competitions, much less get anywhere near W's level.

The summer after 9th grade, I attended AMSP, taking Algebra 2.5 and Combo 2.5. The Combo 2.5 instructor had a thick accent and was difficult to understand. But I did learn some cool things in Algebra 2.5 like equivalence relations, even if they weren't very related to contests.

In 10th grade, my grades still weren't amazing, but I started getting a bit more involved in math contests, not expecting much. Thanks to W's exploits, I was actually able to get on a team to HMMT November, which was fun but I didn't do that well. I also attended a weekly AMC class, run by an eccentric teacher named H. It was a good opportunity to keep my math from rusting.

In spring, at the 2017 Nassau Math Tournament, my life began to take a turn. Given last year's performance, I had fully accepted that I wouldn't do well. I figured that I might get top ten, but that's about it. I did my best on the individual rounds, ate lunch, and went to the bleachers to await results. Sitting next to me on my right was a girl named M, though I wouldn't learn her name for another two years.

The awards ceremony began.

"10th place goes to... Brandon Weiss!"
"9th place goes to... Guanming Lin!"

So far so good.

"Tied for 7th place are... Jennifer Luo and Fayfay Ning!"
"In 6th place... is Ike Qian!"

M spoke. "Hey, is your name Thomas Lam?"

Hm?

"5th place goes to... Saajid Chowdhury!"

"Idunno, maybe. Why do you ask?" I was a bit of a troll.
"Oh, I was one of the runners and saw on one of the sheets that someone named Thomas from our school won the tournament."

I laughed. They were going to call my name any second now, right?

"4th place goes to... Vedant Singh!"

...Right???

"So, are you Thomas?"
"I guess, but there's no way I won."

It was strange that my name wasn't called yet. Maybe I didn't get top ten? But that was unlikely...

"In 3rd place... is Christohper Lo!!"

"Yeah I think you won."
"There's no way."

"In 2nd place... is Brandon Zhu!!"

"There's no way!"

"And in 1st place... is Thomas Lam!!!"

That day, I got my first taste of hope. Sure, I was still nothing compared to W, the equivalent of god. But I wasn't nothing. I won something, I actually won! Maybe I wasn't bound to become a failure. Even if I had a long journey ahead of me, I had potential.

Maybe, just maybe, if I worked hard, I could earn the attention and the blessing of god... by making it to the USAMO.

The year was 2017, when W was the current president of Syosset's mathletes. Although it was not quite carved in stone, one of the president's responsibilities was to make handouts for every meeting we had. As a seasoned Olympiad, W had seen countless Olympiad problems, some of which would end up on our handouts. I fondly remember one day in which she put up an additional problem on the board in Olympiad geometry. While I do not quite remember the statement, it made use of the following configuration.

$[asy] size(7cm); import olympiad; pair A,B,C,D,E,F,M,O,H,HA; A = dir(105); B = dir(205); C = dir(-25); draw(A--B--C--cycle); D = foot(A,B,C); E = foot(B,A,C); F = foot(C,A,B); O = (0,0); M = (B+C)/2; H = orthocenter(A,B,C); HA = (A+H)/2; draw(circle(O,1)); draw(circle(HA,length(HA-A))); draw(A--D); draw(B--E); draw(C--F); draw(O--M--H); draw(O--HA); dot("$A$",A,dir(A)); dot("$B$",B,dir(B)); dot("$C$",C,dir(C)); dot("$D$",D,dir(D)); dot("$E$",E,dir(E)); dot("$F$",F,dir(F)); dot("$H$",H,dir(H)); dot("$H_A$",HA,dir((1,0))); dot("$O$",O,NE); [/asy]$

The key claim that the problem required was that $H_AHOM$ is a parallelogram. But why? Since $AH \parallel OM$ , it suffices to show that $H_AH = OM$ , or that $AH = 2OM$ . Though this is not very obvious... is it?

The proof is practically one line: Use the Euler line (!!). The homothety which sends the medial triangle to $\triangle ABC$ sends $O$ to $H$ and $M$ to $A$ . Since the dilation factor of the homothety is $-2$ , it must follow that $2OM = AH$ . I was pleasantly surprised that there was such a tight relation between the orthocenter and the circumcenter!

Just 11 days later was my first PUMaC competition, in Division B. Since my strength was in geometry, I naturally chose geometry as one of the two individual round tests.

The first four problems were fine, but I hit a bit of a roadblock with Problem 5.

Quote:

A right regular hexagonal prism has bases $ABCDEF$ , $A'B'C'D'E'F'$ and edges $AA',BB',CC',DD',EE',FF'$ , each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$ . The plane $P$ passes through points $A$ , $C'$ , and $E$ . The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$ where $n$ is not divisible by the square of any prime. Find $m + n$ .

There's a "simple" solution, but I had the idea to use something I learned in AoPS Volume II: 3D coordbashing. I set coordinates for everything, found the equation for the plane using a cross product, found the relevant intersections, used them to find the relevant lengths, and concluded. It was a mess, but I somehow didn't mess up!

Problem 6 was not notable. Then came Problem 7.

Quote:

Rectangle $HOMF$ has $HO = 11$ and $OM = 5$ . Triangle $ABC$ has orthocenter $H$ and circumcenter $O$ . $M$ is the midpoint of $BC$ and altitude $AF$ meets $BC$ at $F$ . Find the length of $BC$ .

I couldn't believe my eyes --- the $AH = 2OM$ fact that I had learned last week was immediately applicable and completely destroyed the problem! I swiftly finished the problem and moved on to Problem 8, which unfortunately I could not solve. But solving 7 of 8 problems was enough to outscore the vast majority of the competitors in the room!

And so I secured a 2nd place in Geometry and an 8th place in Individuals, because W felt like putting a problem on a board 11 days ago.

I was a high school junior back when that coincidental miracle occurred. That academic year was the pinnacle of my accomplishments as a competitor in competition math, a career which was full of similar miracles.

The summer before, I was at AMSP again, where I became much more knowledgeable about how the AMC competitions functioned --- particularly, what exactly the USAMO was and how qualification was based on an index formed by combining the AMC and AIME scores. In an insane bid to earn W's attention, I threw my entire life at studying competition math with the goal of qualifying for the USAMO. If I managed it, then Syosset would have 2 of the 300 qualifiers: Me and her.

It truly was a deranged period of my life. I gave up videogames for half a year. I did math everywhere I went, whether I was at lunch, class, or a themepark. I did every competition I could find, dubiously acquired every textbook and article that I could scrape off the internet, and enrolled in various classes like WOOT. My work ethic improved, and as a bizarre side effect my grades went up.

At H's AMC class, the week before the AMC 12A date, we were working through some problems as usual, and one such problem required the simplification of a nested radical such as
$\sqrt{6+\sqrt{35}}.$ I was stumped, but H had a trick: First pull a $4$ out of the $35$ to write the radicand as
$6+2\sqrt{35/4}.$ Then, if there is a simpler form for the square root, it would have to be of the form $\sqrt{a}+\sqrt{b}$ , and in this case $a,b$ satisfy the system
$\begin{cases}a+b=6 \\ ab = 35/4\end{cases}!$ Solving gives $a = 5/2$ and $b = 7/2$ , so the simplified form is $\sqrt{\frac52}+\sqrt{\frac72} = \frac{\sqrt{10}+\sqrt{14}}{2}$ . It's quite a clever trick!

Then came the 2018 AMC 12A. I was extremely nervous, since in my head, this test was going to determine my future and the course of my life. Now I'm wiser --- things aren't that extreme. But I can certainly sympathize with those students today that feel similarly.

I open up the test booklet and work on Problem 1. I can't solve it. I keep getting an answer that isn't any of the other choices. I'm taking forever to crack it. I hear everyone else around me bubble an answer and move on. How could this be happening? I check my work again and again. At least 3 minutes have gone down the drain. I knew that it was just my nerves getting to me. I took a deep breath, mulled the problem over again, found my error, and finally circled an answer and moved on.

Problem 2 seemed like an easy one to mess up, but I confidently answered it. Problems 3 through 8 fell quickly. Problem 9 was intimidating, but I cheesed it via process of elimination. Problem 10 fell via "proof by picture", and 11 and 12 were routine.

At last I reached the middle of the exam, where the difficulty ramped up. Problem 13 took a bit. Problem 14 was nuked by my "convert to base $e$ " trick, which consistently nukes all log problems. 15 and 16 were similarly easy. When I read Problem 17, I somehow immediately saw the auxillary lines to draw. That's when you know you've fallen deep into the math competition rabbit hole.

Problem 18 was routine, and I cracked 19 with divine inspiration. Problem 20 was difficult, but it was a fun tour of various computational methods. At this point, I likely had a good score if I didn't mess up. It remained to peck at the last 5 problems.

I didn't have a good intuition from Problem 21, so I skipped it. Moreover, Problem 22 caught my eye... It asked me to compute two square roots of complex numbers, such as
$\sqrt{4+4\sqrt{15}i}.$ No way. Did I luck out again by learning the key idea a week before? Sure, the scenario isn't quite the same... but could I make it work?

I rewrote the radicand as
$4+2\sqrt{60}i.$ Then, I surmised that if there were to be a decent closed form, it surely would have to look like $\sqrt{a}+\sqrt{b}i$ . When squared, this entails attempting to solve the following system:
$\begin{cases}a-b=4\\ab=60\end{cases}$ Oh my god, this is really happening isn't it? $a=10$ , $b=6$ was the painfully obvious solution, so that the solutions to $z^2=4+4\sqrt{15}i$ were $\pm(\sqrt{10}+\sqrt{6}i)$ . Doing the same with the other complex number, I finished quickly by applying the Shoelace Formula.

Despite my expectations, I managed to solve one more problem: Problem 24. After time was called and we handed in the bubble sheets, W and I rushed to compare answers. Every. Single. Answer. Matched. For the very first time, I managed to get nothing wrong, securing a score of $136.5$ . I cried in relief --- a major hurdle was over.

The informed reader would know that the journey was far from over. The two miracles I experienced during the AMC 12 merely won me half the battle, with the AIME looming on the horizon. Fortunately, for a reason that I cannot recall, I was scheduled to take the AIME II rather than the AIME I, giving me a bit more time to prepare. Thus, the next competition for me (after HMMT) was the NYSML, on March 17th, where yet another crazy miracle occurred.

I understand that numerous readers may be unfamiliar with NYSML, so here are some details. First, some hilarious history: NYSML was established as a math competition for teams in New York. After gaining some repute, a Massachusetts math team wished to compete. NYSML made an exception for them, and, much to the chagrin of New York mathletes, Massachusetts won that year. Naturally, NYSML threw a bit of a hissy fit and vowed to never allow any outside-state teams to compete again. Massachusetts mathletes responded by creating their own competition. They called it ARML.

As you can guess, the structure of NYSML is exactly identical to ARML, except it's easier and there's no super relay. Historically, the competition hasn't been very interesting in terms of the results: New York City's "Tinman" team always gets first place by a long shot, so the competition is really just NYC's annual flex and a fight for second place. As long as I remember, second place tends to go to Nassau County A (my team!).

In 2018, NYSML was being held at a school that was around 4-5 hours away from my county, so our All-Stars team had to take a bus there and stay at a hotel overnight. Sadly, due to a lack of space, I was burdened to sleep on the floor, and due to snoring I only got $4$ hours of sleep. Fortunately, I had a clever tool up my sleeve for counteracting the sleep loss: Calories! Minutes before the individual round began, I downed an entire chocolate bar and an entire water bottle. I was ready!

Problems 1 and 2 were fine. W somehow misread 1, and consequently I heard a positive amount of cussing. (I can't quite tell you what any of these problems were unfortunately. NYSML does not publicly release their problems, which is a real annoying shame.)

Problems 3 and 4 on the other hand... well, I barely had enough time to finish 3. I had essentially no time to think about 4. I was screwed. I guessed $100$ , with my hopes of qualifying for tiebreakers vanishing into thin air.

$100$ is what I historically tend to guess for computational short-answer problems that I cannot solve. For me, it strikes the perfect balance between "a completely random number" and "a number with some patterns to it". Moreover, I figured that if I kept guessing $100$ to problems I could not solve, there was a chance that it would be correct eventually, right?

And so when time was called, I held up my problem sheet in the air, with a confident answer to 3 and a completely uneducated guess for 4. The guy at the front announced the answers:

"For Problem 3, you drew this and that line, did some trigonometry, and got <correct answer>!" Ok cool, so far so good.

"For Problem 4, you did some counting, added up this and that card, blah blah, and you got... $100$ !" What.

You've got to be joking.

I walked out of the individual round with a score of $9$ out of $10$ . At lunch, my head was spinning. Did that actually just happen? Am I making tiebreakers? Did I really guess $100$ or am I misremembering? W assured me that I probably indeed guessed $100$ and that it was going to be fine. She was right! The tiebreakers cutoff was exactly a $9$ . I made it by a hair.

The AIME II was scheduled to take place in just four days, on March 21st. But there was bad news.

On the 19th, weather forecasts predicted a major snowstorm that would pummel much of the east coast on the 21st. There was virtually no chance that I would be able to take the AIME that day. My USAMO qualification, my hopes and dreams, my future, the validation I desperately desired... it was all in jeopardy because of something I couldn't control.

The one who wanted my USAMO qualification the most was me. The one who wanted it second-most was my mother. She called school after school, searching for a location that was safe from the storm, was hosting the AIME, and was willing to accommodate other students. On the 20th, she finally found a school upstate in Shenendehowa that fit the bill! Some change of venue forms were signed and we went on a very last-minute road trip to the school. (Aren't parents amazing?)

The school staff was very friendly, and they got us a quiet room, plenty of scrap paper, and several nice proctors. Unfortunately, my performance on this AIME was not so nice.

Problem 1 was a disgusting rates problem, which I got through all right. Problem 2 was standard "find the pattern", so it fell quickly. Problems 3 and 4 weren't so bad either.

Problem 5 was where the trouble began. It just seemed incredibly nasty for some reason. Any approach I tried devolved into a mess. I was a bit deterred, but I had good test-taking habits and skipped the problem for now.

Problem 6 was easy. I got an answer for Problem 7, though I wasn't too certain about it. Problem 8 was a simple bash, and Problem 9 was not too bad, albeit annoying to compute even if you found the simplifying trick.

Problem 10 was a disaster --- I gave an answer, but I had little certainty in it. Problem 11 was not much better --- the only solution I could think of was a horrifying PIE bash, which I followed through with. I got an answer, but I had very little confidence in it. Among the last four problems, only Problem 13 seemed doable, and I did some work and wrote an answer for it. I went back and eventually managed to solve Problem 5, and that was it. I walked out having answered $12$ problems, and I guessed $100$ for the remaining three.

On the way home, I had time to mull over what I had just done.

Oh no, I missed a case for Problem 11.

Didn't I forget to account for something in Problem 10?

I started to frantically text W.

"if I don't make usamo I might be depressed"

"oh no i think i missed a case for one of them"

She told me to stop worrying about it and to go have fun instead of thinking about math. It's some of the best advice I've ever received. It's what I tell people in math competitions today, since a lot of people need to hear it.

She predicted that I was safe unless I got another wrong. Lo and behold, I soon realized that I screwed up Problem 7. I was looking at a max score of $8$ if everything else went well. My score was more or less confirmed once AoPS unlocked, at which point the forums were flooded with exclamations of the AIME's difficulty, with "hardest AIME in history" claims being thrown around. I was panicking and extremely distressed, but there was nothing I could do about my index anymore. I took W's advice to heart and focused on other things for the time being.

On April 11th, the cutoffs were finally released. The USAMO cutoff was $216$ . My index was $216.5$ . I finally achieved my dream, with a margin of half a point. Syosset finally had two qualifiers: W and me.

If I sillied anything on the AMC, it wouldn't have happened.

If I didn't happen to learn that last-minute trick the week before the AMC, it wouldn't have happened.

If W never moved to my county, it definitely had no chance of happening.

I was living in a miracle.

For the first time in an eternity, I opened up a videogame: Undertale. I played it through to the finish. It was fun.

With the USAMO days away, I wasn't studying math like I had been doing constantly for months. I think I finally earned my break.

The USAMO itself was great. I got to cut my first three morning classes for both days. I got to talk with W, and I even managed to solve two problems. I had everything I wanted, thanks to the countless hours I put into math and all the people that helped me along the way to get every last fraction of a point.

I got a score of $15$ , with the extra point coming from guessing the correct answer to Problem 2. Apparently I missed the MOP cutoff by "only" two points, but I didn't care too much. USAMO was already everything to me.

As you can infer, I often feel as if much of my accomplishments hung by the flimsiest of threads. Perhaps I did not deserve these successes, and indeed I may have completely lucked out on at least one occasion. However, it cannot be denied that the slimness of the margins within which I prevailed is all the more redeeming of my labors. Every single bit of preparation counted, from awkward analytical techniques to random geometrical properties to methodologies that help me get even half a point. Any less, and the foundations of my track record could have very well crumbled.

Turtle Math

CATEGORY THEORY (1/2)

by greenturtle3141, Aug 13, 2023, 7:42 PM

2 Comments