The Second PUMaC 2009

This story begins long before Saturday, and in fact I'm writing this sentence on Wednesday. But while there's an interesting story that started far earlier, I'm going to start at the Saturday a week before the competition, when the power round was sent out. The power round this year was about lattices, meaning subsets satisfying the following properties:

• 


• If , then


• If and then

Those of you who like thinking of as an abelian group should immediately notice that a lattice is simply a subgroup of . Those of you who are reading my mind and thinking of it as a -module are noticing that it's a submodule (now of course abelian groups and -modules are the same thing, but some of the ideas later are better to think of in terms of modules).

The power round was basically then an excursion into basic results for -modules. It defined isomorphisms and asked for some isomorphism invariants, which were relatively simple. It then asked to show that all these submodules of are finitely generated, and then finally went into the canonical form of the submodules (which is better known for the quotient modules as the Structure Theorem for Finitely Generated Modules over a PID). So essentially I had made a post on this power round more than a month before it was released. Awesome.

So how did the power round go? Well basically I didn't want to do my homework on Sunday, so I did the power round again. By Sunday night, I had done every problem done except 3.5 and 5.6, both of which I knew how to do but it involved proving or citing the above mentioned Structure Theorem, or at least special cases (other teams went the citing route, I would have proven it but I didn't want to go through the entire proof). On Monday I came up with a clean proof of 3.5, but 5.6 still eluded me. Finally I gave up and just wrote up the proof...it started on page 5 and ended on page 10 of section 5 (well at the time it took up fewer pages, but we later changed it from 11 point to 12 point font and while the wording didn't change, the number of pages did).

So now that the power round was written up, I sent an email to the team telling them to check it for readability and correctness. So while Sam and Adam had checked the round fully before the Wednesday mandatory practice, the other five team members were made to read over the round, even if they didn't know linear algebra. By the time I left for school on Friday morning, this was the chart of who checked what (the initials of the team members are at the top):

Yes, I have a whiteboard in my room.

Okay, so every problem was checked by at least half of the team except for 5.7. Looks like we should be good to go for power. After printing the twenty-two pages that can be found on the wiki, I snapped the above photo and then went to school.

At lunch, the car arrangement was a source of some great entertainment. We set everything up, including a car with Grace, Allison, and Divya. Then, as a joke, we switched Divya with Lawrence to see Lawrence's reaction. Allison arrived first, and reacted with "WHAT THE HELL?" But after a few minutes, she said "Whatever, I'll have Grace's laptop with Asian dramas." XD

Then Lawrence arrived, and he also reacted with "WHAT THE HELL?" But instead of just switching himself with Divya (as we expected), he switched himself with Seung In. Somehow that stuck and the car ended up being Allison, Grace, and Seung In. I still have no idea how that actually happened.

My car consisted of Aviv, Sam, Renjie, Akshar, Jenny, and me, so we were a bit cramped in the van (Jenny had to sit in the middle of the back between Akshar and me). On the car ride, we played hearts for a bit and house for a bit. I sat out the first game of hearts, where Akshar got completely destroyed. Second game, Sam and Renjie decided to play as a team and I was in. I got to shoot once, having two runnable suits (I believe my distribution was 2056) and by leading the spades early, so that when I got the lead again I took the last 7 tricks, or something insane like that, which contained all of the points. Shortly afterward, I ended up getting the queen a few times and eventually lost, but that round of shooting was quite fun :).

Upon getting to the hotel, I was disappointed to find out that I had one of the small rooms, but it was fine. I played around with google wave for a few minutes (I had finally gotten my official invite just that morning) and realized that it was actually pretty laggy. I wouldn't want to use it for most communication in its current form. Especially big threads start to lag massively. I'm not sure what it is, but it reminds me of the lag you get when you're X forwarding. But really, if document viewers can display 200+ page pdfs without lag, you'd think that google wave would be able to handle waves with only 100 messages.

After just a few minutes of google wave and finding the latex bot (watexy@appspot.com for those of you who don't know about it yet) I went back to the lobby to wait until dinner time. We ate dinner at Quaker Ridge Mall, which is a horrible idea. There's basically nothing there to eat, except for an applebee's out in the parking lot, which we went to. The food was decent, but then Jenny tried to cheat us out of $1.50. We caught her when we ended up short a bit of money, and then she decided to complain that she didn't have any coins so she had to pay $.50 too much. I tried to give her the rest of my coins ($.03) but she wouldn't take it. This was after she decided to buy $.60 gravy...

Anyway after dinner I took my laptop up to Renjie's room where we were going to play mafia. While playing I read through the google wave API. It looks simple enough, but based on my googling javascript has unfortunately little support for dynamic graphics and I don't want to use flash, so I want to find a good way around that. The game of mafia went extraordinarily well. After the first day, where we killed Seung In for voting for no lynch, Sam and I saved Luke, who was also targetted by the mafia. Second day, we killed a mafia, and then Sam insisted on saving himself. I had no better ideas, so I shrugged and let him do it. But the best part was after the next night, when Sam wanted to save himself again. I was like man this is a pretty silly and pointed to someone else, but Sam eventually won the fight. Next day, Sam said "I think Brian is the mafia, because he and I are the medics and he wouldn't save me last night!" The game ended where, in the last night, I wanted to save Jenny, Sam wanted to save himself, and then after we agreed on Jenny, Sam said (in the middle of the night), "Wait! Do you think it's more likely that they killed one of them or one of us?" Turns out Sin had tried to kill Jenny, and I was right. Hah.
(On the other hand, if Sin believes that Sam and I are the medics, targetting Jenny ensures his loss. However, he says that he didn't think that Sam was telling the truth, so his decision is defensible.)

Anyway, it's time to talk about competition day. I skipped breakfast, as always, and we headed out shortly after our hoped departure time of 730 (Aviv was late, as always). We got to Princeton at some time which I don't know because I don't wear a watch, and then after Mrs. Gabriel registered us we started walking toward McCosh, during which I heard Jenny yell from behind me, "Hey Brian, look behind you!" I turned around and saw Sherry there, so I slowed down a little, but she stayed behind me by a bit. Oh well.

On the way into McCosh 50, I saw Amy Zhou just a few feet in front of me! But she didn't notice me. Anyway, once we got in, I decided I wanted to talk to some people, so I went back outside and went to the registration table where Damien and his team were standing. While I was there, several members of the Exeter team showed up and I talked to them. It was pretty much my first time talking to David Xiao since red mop. We talked about the power round, and I found out that both North Carolina and Exeter had failed to solve problem 5.6. Awesome, we looked to be in good shape. Good lead on all of the other teams. In the middle of the courtyard, Sherry was pacing around waiting for the rest of the AAST team, and I considered going over and talking to her, but her parents were there and I would have felt kinda awkward. So instead I just continued talking to Damien and the Exeter people. It was pretty good to catch up with them. Also that this time I said hi to Amy (she actually saw me this time), who I hadn't seen for over a year since I missed MathCamp '09 :(. Well anyway that was that for the outside talking. We then all went into McCosh 50 and continued talking. There I saw several more people that I knew, but AAST still wasn't there. When finally they walked in, I went up to the door and said "Way to be late, guys." (it was after the scheduled start of the opening ceremony I believe). The opening ceremony started shortly afterward, 5 minutes late I believe. They gave us our proctors and we headed off to our testing room. But when we got there, the door to the building was locked, so we had to go in the other entrance, up to the second floor, and then back down to get to our room. Door unlocking fail. We got to our room and sat down, informed our proctor that Sam and Seung In were switching some subject tests, and then started. Our proctor was Arthur Safira's roommate, and as he put it we "probably got the only non math or science guy" there. In any case, my first test was number theory. Many of the problems were classical, but two of them were pretty decent:

Then again, the second is pretty straightforward if you know continued fractions (and are able to compute that in the time limit, unlike Damien). Some of the problems were a bit too classical. #1 was essentially the same as a Math Prize problem, #2 was find the number of solutions to for positive integers a and b, and #4 is problem 103 in Engel's number theory section, according to Andre.

Next up was combinatorics. Again the test was mostly straightforward, although I made a few mistakes that had to be corrected. And there was the now infamous problem:

As pointed out on AoPS, the fourth largest number of the set is 2, not 4. Many people interpreted it the other way (including the test writers and myself), so the answer 606 was considered correct, rather than 303. This could have been worded much better ("When these numbers are sorted in increasing order, what is the expected value of the fourth number?" or a variety of other ways), but regardless I got the points :). Unfortunately, Sam didn't, so our team suffered a bit because of the unclarity. Darn.

At this point PUMaC's amazing grading system that would announce individual finalists immediately after team round was revealed to us. All the answers were integers, so the proctor called us up one at a time to verify that he typed in our answers correctly to an online submission interface, which presumably handled grading automatically.

So next up was the team round. I forbade the team to discuss the individual round until after the team round, because they might become depressed about how badly they had done (if they had done badly), and also because I didn't want to answer questions about individual during the team round. Team round was quite interesting, with the answer sheet being a crossword puzzle. We got everything (either by solving or guessing) except for one problem, which we had 3 out of 6 digits for, but guessed all the other three wrong. That got us a total of 93.5 points out of 100. Not bad.

After team round, we were told that we had one individual finalist: me. It was a large disappointment that none of the rest of the team made finals, but I was happy that I had qualified. I decided to not bother waiting in line for lunch at that time, and went straight to the finals room where Peter Diao was the proctor. We both kinda partially recognized each other, and then he bothered me about giving him a weird look when he said hi. Then an AAST contingent including Sherry, who thought she had no chance of making finals, walked in. They certainly had more finalists than we did, but by no means did that necessarily mean that they did better on individual, since our scores were all reasonably high, just not high enough on any one test to qualify for finals.

Finals went okay. I looked at the problems and saw how to do problem 2 pretty quickly. It was a simple bounding argument and then a bit of cleanup at the end to eliminate two bad cases. After that problem was taken care of, I looked at the other two. Problem 3 looked obnoxious (47 46-gons? No thanks), so I worked on problem 1, which was a pretty annoying analysis proof that I wasn't completely satisfied with at the end, but I figured it was okay since it looked like 1 and 3 were both hard, so I probably had about as much as anyone else.

After individual finals, Sherry and I lagged behind everyone else, where I found out that Jenny had called her twice and texted her twice during individual finals XD. Way to go, Jenny. She tried calling back, but no answer, so we just went outside and started walking around. There was still free food for the individual finalists, so I decided to take some. I then offerred her some milkis, but she declined for the time being. We found Nassau street and went to the Panera, where I had a turkey artichoke panini and she just had some broccoli and cheese soup because she had eaten the PUMaC lunch before finals. It was a good lunch, though :). Afterward, we walked around Princeton campus for a bit, getting lost because her map from google maps had very few buildings marked on it. Eventually we ran into Divya, Grace, and Allison who were "totally not stalking" us. Of course not.

When I got back to McCosh 50, Greyson said "Congratulations on number theory." Apparently I had gotten all the problems right. I then asked to see the combinatorics answer sheet, and I had gotten all of those right too! With a double perfect score, some of the results were a bit less suspenseful.

The awards ceremony needed another 10 minutes of stalling after minievents to get a powerpoint made with results, so they had the math bowl finals, where AAST had a substantially better showing than last year. And one of the questions was StarCraft! But as Damien pointed out, StarCraft actually came out in 1996, BroodWar came out in 1998. Always recheck your facts. And apparently Damien was actually wrong, so he just fails.

The way they had the divisions split probably contributed a lot to the pumctuality of the awards, though, since we didn't have to listen to all the B division awards. First up was the subject tests. Geometry, Algebra, Number Theory, and finally Combinatorics. Vlad won geometry, so I was really happy off the bat, since he was my roommate at MOP last year. Then Chong won Algebra, who I knew from MOP 2008 (where he will be forever known as stomachache) with a perfect score. Interesting difficulty level, allowing perfect scores. It doesn't really appeal to me that much, since it potentially leaves people having nothing to do at the end of a test, but that's okay I suppose.

They finally got to number theory, and I won (Did anyone not see that coming? If not, you need to read the entire post). When I went up to receive the medal, it wasn't large enough to go over my hair properly, so it got stuck on my hair and glasses, much to the amusement of the audience. Going back to my seat, the rest of TJ said I should just stay there. I decided not to take this piece of advice. So next up was Combinatorics, where I won again (seriously, if you didn't see this coming read the post). And again, the medal got stuck on my hair, again to the amusement of the audience. I now had two medals to show for my trip :) Laura got a picture of this event, which has some hilarious comments.

Then overall results. Here, because of lack of communication about the method for finals, I had no idea if I would place or not. I was especially worried when Vlad placed 8th, since he had gotten problems 2 and 3, while I had only gotten problem 2 and most of 1. But I ended up winning overall as well, completing my sweep of individual events.

Yay two medals an a trophy :)

Next was power results. We sat through places 10 through 2: Beijing STFX1, The Evil Geniuses for a Better Tomorrow, Montgomery Blair High School, AAST Mu B, North Carolina, PEARL, Albany Area Math Circle, Murph and the Magictones, and Lehigh Valley Fire. Now at this point we should probably be suspicious since AAST Mu A and TJ A are both missing, but we didn't think about it too much. As the announcer said "and in first place...", a shout from the audience came: "Brian Hamrick!" (This was from the North Carolina area, so I'm not sure exactly who said it. If anyone knows, I'd be happy to hear). "AAST Mu A"

Okay, at this point, all of us in the back right corner of the auditorium (where TJ always sits) are thinking "WHAT THE ****?" And this wasn't limited to just the TJ team: Peter Diao immediately ran off to the grading room to figure out what went wrong, since we were definitely supposed to place. We were thinking somehow one of our problems got lost, either in printing or by the sponsors, or by PUMaC themselves. We even considered that maybe they mixed up our solutions with another team, but every single page of our power round had TJ A on it, so that was pretty unlikely. Meanwhile, AAST, to our left, was ecstatic because we had apparently massively screwed up the power round.

What?!?

We sat listlessly through the rest of the ceremony, as we placed fifth on the team round and sixth overall. We all knew that we were supposed to place higher, but we didn't know how high. I know I barely could pay attention to the top rankings, as PEARL took third, Beijing STFX1 took second, and Lehigh Valley took first (who is on Lehigh Valley, anyway? They didn't do so well on team and power so they must have had pretty good individuals). After the conclusion of the official awards ceremony, we met Peter againat the front, where he informed us that our power round score had failed to be entered, and that we had actually won the power round and taken third overall. So we missed out on a huge trophy because of a PUMaC screwup. Way to go, guys. Peter said that we got 78/86 on the power round, and I asked him what the second place score was, to which he replied 74. He also said that we had only beaten the teams that were at the actual competition; the German IMO team had gotten a perfect score. (Note: I've now been informed by Peter that he was mistaken and AAST in fact got an 85 on power, so we were actually second on power and third overall. He also says he thinks we lost a substantial number of points on 5.6, which I think is completely correct, so my guess is that the graders simply didn't understand it and took off points, which has happed before, such as to Alex Zhai on USAMO. Also apparently our score was entered, but it was entered as an 11 because someone sucks at reading. Massively.)

Regardless, I was pretty happy with my individual results, and third overall wasn't too bad for the team. When we left, I realized that I still had all of the milkis in my backpack, I had forgotten to give Sherry one after we went to panera :(. I was going back home in my dad's car, rather than in the car I went up with, so I gave them six milkis because we had done pretty well and took the other six in our car. Hopefully Jenny got her milkis this time. She complained after Duke that I had given Sherry milkis and not her. Heh.

The trip back was pretty uneventful. I slept a bit but the way the seat was I had a really sore neck upon waking up. Bleh. Whatever. I got home pretty much satisfied from the trip. While PUMaC certainly still had some kinks, it was run much, much better than past years. I think the contests could have been a bit harder, since the finals cutoffs seemed a bit high (probably around 30) to have 7 and 8 point problems, and a lot of the later problems were actually pretty easy and/or guessable. But besides the screwup on power and the couple of mistakes on the tests, it was a good event, and I got some good stuff out of it.

The spoils