HMMT 2024 Day 2

The HMMT 2024 Day 1 post was mainly about the Friday Night Events. This post is about competition day on Saturday, February 17th!

After what I considered a tolerable night’s sleep, I woke up at around 6:30 AM, after my alarm. I had trouble falling asleep last night because I’m not very comfortable sleeping with another guy on the same bed as me.

I did my usual morning routine. Me and my roommate packed up all our stuff within 30 minutes (thankfully I didn’t bring that much) and went down to the lobby around 7 AM, the time when everyone was supposed to show up. We got here only to find a debate team checking out and piling onto their buses. No one from our two teams had showed up.

Not knowing what to do, I gave the clerk the room keys and waited for the senior officers. As it turned out, the rest of the team was just starting to get ready. Thankfully, not long after our arrival, some of our friends showed up. Eric collected room keys and Pavan went upstairs to knock on sleepy rooms.
Meanwhile, we huddled up and talked like nerds, as usual. I joked along with my friends about how I wished for more geometry and noticed a sculpture that could be a tangent spheres problem on the contest. We confirmed how HMMT should be a no “orz” or “xonk” environment, although I wasn’t enough of an AoPS user to deduce exactly what the latter meant.

Eventually, we hurried everyone out of the hotel by around 7:30 AM, considerably late if we were to snatch served breakfast at MIT. It was funny how we were all terrifically confused about where Lobby 13 was until I realized that it was the same place where we ate lunch last year. I directed the parents to drive to Lobby 7 instead and noted to clear up location names for next year’s math club presidency.

When we got into the building, through the same entrance I went through last year, I spent my time relishing the intricate structure of the building. It’s a profound moment when you’re inside one of the lobbies of the world’s most prestigious schools, the school everyone would love to get into, the home of the world’s megaminds. My freshman friends who carpooled with me also took a moment to soak this all in.

We were late for breakfast because most of the team had already arrived and were waiting in line. Before seeing them I thought I had to wait at the back of the line, which would have been at least half an hour long considering its length. Breakfast at HMMT was very mediocre because all they served were a bunch of croissants, bagels, and muffins. It was too much carbohydrates and sugar to eat before a math contest. But regardless of its lackluster quality, I’m thankful that the HMMT committee had put in effort to feed us hungry children.

Slightly before we reported to our testing areas, I discussed with the other officers whether we needed some proctors during the competitions. Eric conceded that we had one proctor and we needed one more. I was glad that he knew 1+1=2, but I’m not sure what happened to this situation after this point.

We ventured into the main Lobby 10 and stalked the MIT hallways until we found our room 10-250. We were the first team there so we claimed the tables in front and played hangman on the blackboard until our proctor came. It was funny how our middle schoolers kept bugging each other about math concepts while the rest of us high schoolers kept silent. Being young feels great…

At around nine o’clock, one of the HMMT staff burst into our room signaling us to start. It was very funny because she was out of breath, which meant that she must have been running all over the building. Who knew that MIT students work out regularly in their halls? The other team had a No Show (similar to swimming!) so we got the room all to ourselves.

The team round started and right in front of me seemed like a simple problem 1 (I will abbreviate p1) algebra. Time to do the easy problems and make sure we get the points we should get. I also wanted to help my team for points, even if that meant solving an unchallenging problem.

I filled around with Cauchy which didn’t work for a while until I realized that the top sum of squares expression in the numerator and the direct sum in the denominator could yield yet another sum of squares! This yielded the answer and a valid construction for the maximum value of $a_1$

I turned to p3 and killed it by extending XM to meet line AB, which forms a whole ton of congruent triangles from equal lengths. Because one of my other teammates (also a fellow geometry addict) solved the problem and had more experience with the problem statement, I asked him to write the proof up.

We all collaborated on verifying the answers and proofreading our work. We had a couple of discussions here and there but not anything major because math involves more intuitive thinking that usually doesn’t require much talking. Eliana and Christina wrote up p4 and p2 respectively, and I read over their work, “checking” for mistakes. It was a little bit of a waste of time because I hadn’t seriously tried out each problem.

In the last five minutes, all of us who tried p5 resorted to asserting “No”, because it didn’t make sense after trying lots of examples. It turned out that the answer was yes, with the funky construction $(184, 345)$ . Huh, who could’ve thunk of that?

Unfortunately, time ran out as the rest of us attempted more problems. I was not working on any problem when time ran out, which was a bummer because I could’ve gotten another delicious one heated up before I went into the individual rounds. Sujay and Kevin were working on p6 and p8 respectively, and I’m sure they would’ve gotten them with a bit more time. Even on the B team that our district sent out, my teammates are a lot smarter than me. Kevin was a USA(J)MO qualifier while Sujay had a great chance for the USACO camp, and I could tell from their thought processes towards certain problems that they had cooked up a great intuitive mind towards competition math or computer science problems. To me, they are excellent examples of where one can get through patience, hard work, and struggle. And where I can be if I keep going.

I observed the team round concluded. Whenever I have a math test with a bunch of different categories, I always attempt the ones that seem the most pleasing to me (non-combinatorial, more algebraic, and geometric). In the end, I tend to attempt the very hard problems that involve concepts that I am more familiar with than the easier problems that involve unfamiliar ones.
We walked into our room for the individual rounds which I considered quite spacious until the majority of people poured in. We took our seats in the front of the auditorium.

Not much eventful happened during the approximately two-and-a-half hours of testing.

Here are my scores and remarks: (spoilers are contained below, try the problems posted here if you don’t want to see them)

Algebra: 2/10. Not much of an improvement from last year. I expected a 3 but I messed a problem up due to ambiguous wording and a faulty assumption. I guess it’s at least a nonzero score.

p1: This was not a bad problem and I got it within two minutes. Unfortunately, I was subconsciously aware that a strong eighth grader was sitting beside me, so I tried to finish problems before him, which destroyed my focus after the first problem. It was then that I realized that what you do in the first two problems often determines your outlook for the rest of the test!
p2: I got the meat of the problem but got stuck for about ten minutes trying to find the integers $a,b$ such that $a(a-1)b(b-1)=4800$ . I skipped and realized I could just list all the possible values of $x(x-1)$ where $x$ is a small positive integer. I did that and got the right answer.
p3: I messed this one up horribly. I did get the divisibility condition that $x | 999$ , but somehow I read the problem as if $\overline{bca}$ had $b \ne 0$ . I ended up putting $37$ instead of the correct $64$ . I was going to dispute this until I realized that $703$ worked for the $37$ case, thereby making the answer $0$ , if my interpretation was correct. Either way, I wasn’t getting my points.
p4-p10: I was a newbie enough to see problem 5.

Because of my atrocious performance in the first round(I considered algebra to be my strength, number theory not so much), I decided to push through it and be more focused in the geometry round.

Geometry: 4/10. This one was exhilarating to get some of the mid-range problems right, but painful to miss the top 50 rankings. However, this test taught me to stay calm and push through it till the end, even if time is up. Because if you’re tight on time and you know how to do the problem, if you stay organized and accurate, you will very likely finish the problem in time.

p1: This took me about five minutes, but basically you drop an altitude from the circumcircle of the big triangle to a side and find two different expressions to solve for $x$
p2: I thought this problem was overly hard for a number 2, but it turned out that the angle bisector also bisects the median and the altitude, which almost instantly kills the problem with the Pythagorean theorem. I don’t exactly remember what I did, but I used two variables and solved a system of equations that took a lot of time.
p3: I got this problem wrong, which cost me top 50 rankings in this round. I saw what I was missing at the last minute and rushed through the computations. It turns out that the 1:2:3 ratio constraint gives a midpoint of a chord (because 1+2=3), and then a bunch of Pythagorean stuff with radii finishes.
p4: I used a bunch of similar triangles and ratios here, but apparently you can extend lines from points and directly get the side length of the square with pythagorean… again… Pythagorean is everywhere!
p5: This one took me the most time, but I got the right answer at the end. I extended the lines of the trapezoid to form a big triangle and made some algebraic equations from the tangency conditions, to somehow get a quadratic equation and a few more substitutions.
p6-p10: didn’t even attempt this after the test…

Combinatorics: 2/10. I’m absolutely atrocious at combinatorics and this gave me an idea of how badly I underperformed in Algebra/NT. After problem 3 I didn’t have a clear understanding of any nice approach of the problems.

p1: thankfully I got this one right. At first, I thought it was a Fibonacci thing, but afterwards realized that 20 is divisible by 4 and 5, which then bijects the problem to find the number of ordered pairs of nonnegative integers $(x,y)$ such that $4x+5y=24$ . Clean and simple.
p2: This one was a tricky one that I almost messed up. You find the desired construction by going to the right.
p3: this one was just casework, but I messed up the cases. To this day I’m not sure where I went wrong…
p4: I drew the problem wrong. So much for spending twenty minutes on this.
p5-p10: didn’t look at them

I started getting hungry during the combinatorics round as I did last year. I was already pretty done with that test at the five-minute mark.

Feeling beaten up after more than two hours of testing, I asked Eliana (our club president) whether to wait and collect T-shirts or get lunch first. We eventually left her to wait for T-shirts and crossed the street to another MIT building for lunch. Again, we found most of the A team there, already gobbling down our precious pizzas! I stole a few slices before the boxes ran out. Lucky Ishan got a full box for himself.

I couldn’t get a seat where our team was sitting so I ended up standing and texting my parents about my results. They don’t care that much, but I was disappointed about my panic attack during that geometry round. However, I was glad that I had improved from that 2/0/1 during last year’s HMMT.

At around 3 PM, we went back into Lobby 7 and walked to the opposite side of the building to report to our Guts Round location. The testing auditorium was PACKED. Interestingly, there were tables set up in the front and in the back, so the most suboptimal place to sit for the runner would be the middle. It took some time to find where the rest of our team was sitting. We divided up into A team at the front and B team at the back.

After some urging from Eliana, I got an HMMT staff member to take a photo of our team. Good call, because that photo was needed for our history and club Instagram! Afterward, Eliana urged me to lead from the front and call everyone to stand for a group photo. I wasn’t sure if it was senioritis or an intent to train me to become president next year! (second one, tysm)

We chose her to become our runner for the B team. I had some experience with running for last year’s school team, so I know how anticipating the wait for answers can be. Rithik, probably the only guy besides myself somewhat serious about athletics, was the runner for the A.

During the Team Round, I dubbed myself as “The Paper Man”. He was no more when he ran out of paper for the Guts! This is a reminder to bring at least a hundred sheets for every math competition if you’re a grand paper consumer like me. (I’m sorry for killing so many trees…)

Before we knew it, the round started and we got two out of the first four wrong! The test was quite a bummer as we only got to the fourth packet when five minutes were left. Eliana gave up on us and suggested we skip straight to the estimation questions on the last slip. I was disappointed as we could’ve fought in the fire until the very end on those tricky problems, but I’m not the president of the club this year!

Afterward, we walked back outside the lobby and took group photos (hi, I take credit for the idea)

Before we left for home, I discussed with the other officers whether we should combine with North for future HMMTs. We decided that we should stick with the combination of our two high schools to optimize the quality of team members and increase our performance because if we don’t place at least top 30 overall our HMMT spot for even A team is not guaranteed.

After waiting for people to use the bathroom and drink water we said goodbye and separated into our separate carpools. So much for team bonding in our district. Our math club is quite competitive and doesn’t do much fun stuff, unlike the other clubs that also travel for contests.

Our carpool walked a bit into town and I spotted quite a lot of interesting nearby shops and restaurants. This is seriously my dream school, for real. Wouldn’t you want to walk in a corridor full of scientific posters? Wouldn’t you want to do homework in a room filled with dry-erase walls and chalkboards, filled with math formulas? Wouldn’t you like to live in that laboratory for your career, investigating the upper bound of an algorithm or designing the next NANO breakthrough? Wouldn’t you love to study on those cozy couches (instead of those crappy chairs) and sit in a lecture room with electronic-controlled blackboards (instead of a boring, unmotivated teacher blaring out useless information )? Wouldn’t you?

The first half of the carpool back home was just us napping and doing stuff on our phones, even though it was only six or seven. It was too dark to do math at all, so I wrote up a summary of everything I could remember that day (this is why this entry is filled with so many details!).

Interestingly, we stopped at the same Burger King that I stopped at during last year’s HMMT. I ordered a big Texas ranch burger, but it turned out way too spicy.

In the last hour, I had a great time with the freshmen and arrived home at around 10:30 filled to the brim with memories and math.

I loved the trip, and even though I didn’t get any awards, it was really fun.