Showing posts with label math. Show all posts
Showing posts with label math. Show all posts
Tuesday, September 13, 2011
A cute problem
Does there exist an algebraically closed field that is isomorphic to a proper subfield of itself?
Thursday, July 1, 2010
Contests
As many of you have probably heard by now, I did not make the IMO team. Although I was definitely sad for the hours after the TST and perhaps for a few days after that, I am not bitter about it. After all, I'm in no position to say that I deserve a spot any more than the six who got it.
There's something about contests that I've known for a while, but TST brought it up again. Contests aren't for deciding the best, as much as people would like to think that. No, the person who wins the USAMO is not necessarily the best mathematician, nor is the person who wins ARML, nor is the person who wins HMMT, nor the winner of any other competition. Math contests don't crown the best mathematician. They crown the winner.
Sure, a trip to Kazakhstan would have been great. Winning certainly does come with perks. But when it comes down to it, I know that the fact that I lost on the TST just means that I'm not on the IMO team. It doesn't mean I'm worse at math.
Look forward to a more complete post on MOP soon.
There's something about contests that I've known for a while, but TST brought it up again. Contests aren't for deciding the best, as much as people would like to think that. No, the person who wins the USAMO is not necessarily the best mathematician, nor is the person who wins ARML, nor is the person who wins HMMT, nor the winner of any other competition. Math contests don't crown the best mathematician. They crown the winner.
Sure, a trip to Kazakhstan would have been great. Winning certainly does come with perks. But when it comes down to it, I know that the fact that I lost on the TST just means that I'm not on the IMO team. It doesn't mean I'm worse at math.
Look forward to a more complete post on MOP soon.
Monday, March 22, 2010
A Mathematical Bridge Problem
Playing a spade contract, you reach trick 10 in your hand to see the following four card configurations:
Dummy holds: ♠ - ♥ AQJ ♦ - ♣ A
You hold: ♠ 2 ♥ 2 ♦ 2 ♣ 2
How do you play to maximize your chance of getting all of the last four tricks? Assume that the only point card left is the king of hearts and there is a diamond higher than the 2 in one of the opponents' hands.
Obviously it depends on your situation, so say that the following happened: your partnership started with 21 high card points between you and during play LHO has played 16 points and RHO has played none. Does this change your answer? What are the probabilities now?
Does your answers change depending on which of the following situations happened?
Dummy holds: ♠ - ♥ AQJ ♦ - ♣ A
You hold: ♠ 2 ♥ 2 ♦ 2 ♣ 2
How do you play to maximize your chance of getting all of the last four tricks? Assume that the only point card left is the king of hearts and there is a diamond higher than the 2 in one of the opponents' hands.
Obviously it depends on your situation, so say that the following happened: your partnership started with 21 high card points between you and during play LHO has played 16 points and RHO has played none. Does this change your answer? What are the probabilities now?
Does your answers change depending on which of the following situations happened?
- Neither opponent bid during the auction
- LHO opened an artificial 1♣ showing 16+ points
- LHO opened a standard bid showing 13-21 points. Does it matter what bid it was?
Tuesday, March 2, 2010
Medalia de Aur
As some of you know, I went to Târgu Mureş, Romania for the Central European Olympiad in Informatics. This year, I went to Bucharest, Romania for the Romanian Masters in Mathematics. The team consisted of Allen Yuan, Vlad Firoiu, Sam Keller, Tim Chu, Albert Gu, and myself, headed by coaches Po-Shen Loh and Yi Sun.
Two days before we were to leave, Po-Shen sent us an email that, among other things, notified us that Lufthansa was currently experiencing a strike and that if our flight out of DC was canceled, the entire trip would be also. Obviously, this did not sit well with us, as we were all strongly looking forward to the trip.
Luckily, the strike was called off before we left, although Lufthansa was still short pilots, so some of the flights got canceled, but ours wasn't one of them. The flights to Bucharest actually went pretty well, including the AMC B. I did worse on the B than the A, but it really doesn't matter. I took it mainly because I figured everyone else would also and I didn't want to be bored for those 75 minutes. Tim had gotten a 96 on the A and was worried that he didn't qualify for AIME, and he wasn't exactly relieved when he got a 96 on the B as well.
On the trip there, we were expecting to be housed at Hotel Moxa, a 4 star hotel in Bucharest. However, it turns out that it was actually Complex Moxa, which is used for college dorms and is just an annex of the hotel or something. The rooms were pretty unfortunately bad, but ours had a TV in it! (the others apparently didn't). Because of the 7 hour time difference, the Olympics were on after all of the events for a day ended, which was extremely convenient. I definitely watched more of the Olympics while in Romania than any other time.
Sam checking out the room
We also found out that the complex didn't have an open wireless access point....But Vlad had this USB thing that allowed him to get internet access in Romania. It's called Zapp or something. At least we had internet access, even though it was pretty bad.
The next day we still weren't competing. We got our first taste of Romanian breakfast, which included an interesting tea (I think it was purple) that tasted pretty good, as well as some cheese. Being American, we obviously thought the portions were way too small so we ate masses of bread with oil and vinegar.
Our first Romanian breakfast
After breakfast, we met our guides and went to the high school where we would be taking the contest in the following two days. After touring the school and dropping in on a ``superior algebra'' class, the guides asked us if we wanted to go into the gym to play some sports. Inside, there were lots of people from various teams playing volleyball, but the court was pretty full so we didn't join them. Instead, we saw a ping-pong table, but nobody had any paddles, so we started playing basketball while we waited for a guide to retrieve paddles from the complex.
For some reason, someone thought it would be a good idea to play outside, even though there were huge puddles of water on the ground and the court was not very even. There were also ping-pong tables outside, but they looked pretty bad. They were really low, weren't flat, and the nets were actually iron fences.
China plays on them anyway
Eventually we got some paddles and played some ping-pong, as did the Chinese. The Chinese team didn't know much English and the only Chinese speakers were on the US team as either a student or a coach, so they spent a lot of time with us (and also Allen and Tim were in a room with two of them).
At some point we went back to our room to hang out until dinner, after which would be the opening ceremony. But as we were just starting to chill in our room, our guides came up to inform us that the opening ceremony got moved from 2000 to 1600, and we had to go back to the school.
The opening ceremony was actually quite nice. Only a small part of it was dual-run in Romanian and English. All of the guest speakers spoke in English, so translation was unnecessary, and they also all kept it very short. It made the opening ceremony much shorter than what I expected.
The next day was competition day 1.
Go go go!
I read the problems and solved 1i on sight, as did the rest of the team except for Vlad, who apparently took 1.5 hours on it. I then spent a bit of time on 1ii, but wasn't quite getting the details. I figured it would be easy anyway and went to do number 2 before finishing.
Number 2 was dispatched rather readily, and at this point I had about 3 hours left, if I remember correctly. I drew the diagram for 3 (although I actually drew the wrong diagram, thinking ``external'' meant that the quadrilateral was external to the circle, rather than the circle is external to the quadrilateral), wrote down some random stuff, and went back to 1ii. After all, surely a number 1 number theory would be easier for me than a number 3 geometry, right?
So it turned out that I didn't solve 1ii, and didn't have anything worth partial on 3, whoops. In the last 5 minutes I wrote down some stuff for 1ii that I figured had no hope of working, but it turned out to be extremely close to the correct solution. I left the room thinking ``Man, I'm going to have to tell the rest of the team that I didn't solve number 1.''
So talking with the others after day 1, it seemed initially that most of them had solved two problems: either 1 and 2 or 1 and 3. The exceptions were Allen, who solved only 1i and 2, and Sam, who solved only 1. After talking a bit more, however, Albert determined that his 1ii was completely wrong, and so he had only solved 1.5 problems as well. After day 2, we would find out that during coordination the coordinators had thought that Albert's solution had worked too, and Yi and Po-Shen had to tell them it was wrong to keep the spirit of the contest.
Allen and I both had essentially identical progress on 1ii, and since it was so close to the correct solution, we came out of coordination with 6s...somehow. The graders were apparently pretty lenient with scoring.
After day 1, we just went back to our room to hang out, being exhausted from the competition. Nothing much interesting happened. We just watched the Olympics and played card games, mostly.
We woke up the next day for day 2 of the competition.
No geometry! Wooo!
So I read the day 2 problems and I thought ``YES! There's no geometry! Let's get a 21 on day 2! Oh wait, these problems look time consuming. 4.5 hours might not be enough...'' Anyway I looked at problem 4 and killed it in about 20 minutes. I start working on problem 5 and it dies in another 50 minutes or so. At this point it's about 1050 and I have two complete solutions written up and I'm starting to think maybe number 6 is really hard and they gave us two really easy problems to compensate (a la IOI day 1).
So I spend the next 3 hours trying various stuff on number 6, but I don't do the thing that actually leads to a solution because it looked stupidly messy. Oh well. I wrote up what I had (which wasn't exactly the cleanest thing in the first place), and then turned in the test. When I was leaving the room, I figured I probably had a pretty standard result on day 2.
However, when I talked to the rest of the team, I found out that I could hardly be more wrong. They had all solved problem 4 (except Albert, who got a 0 on day 2, unfortunately), but nobody else had solved problem 5. I was really surprised. Tim thought he solved problem 6, but none of us could really verify it since he was the only one who felt that he had made significant progress.
Later in the day, we found out (with our awesome Chinese-speaking skills) that CHN1 had been the only Chinese team member to solve either 5 or 6 (and he solved both (and CHN was really Shanghai, not all of China)). Apparently 5 was supposed to be very difficult. I still don't really see why.
After day 2, we went to the mall to play some laser tag! Except that the game was actually pretty lame. At first there was only like one person on the red team, so it was just walking around for a while until the person running the thing decided to restart it. Unfortunately, the respawn time was still around 3 seconds, so whenever you killed someone they could just follow you until they respawn and kill you immediately. It made for a pretty annoying game.
We got back to the complex pretty late, so we missed the normal dinner and had to order pizza, and our discussion of day 2 with Yi and Po-Shen was at around 2230, way later than we expected.
The awards ceremony was the day right after day 2. But before that, coordination had to happen. So to get rid of us pesky contestants for a while, they sent us to the village museum: a collection of traditional Romanian houses. It would have been a really cool experience, but the ground was extremely muddy and it was simply unpleasant to walk around.
When we got back it was time for the awards ceremony. Well, almost. It was actually delayed for half an hour. Anyway, the awards ceremony, just like the opening ceremony, was very quick. The speakers knew that we didn't want to listen to a bunch of long speeches (and it was hard to understand some of their English anyway), so they went straight to the awards. Albert was the first USA competitor called up for honorable mention (solving at least one problem perfectly).
Next up was the bronze medals. There were a lot of bronzes, and Sam was among them. I was actually pretty nervous during the bronzes because I wasn't sure if I had screwed up something on day 2, in which case I would probably be in the low end of silver. As the bronzes ended, I breathed a sigh of relief.
The bronze medalists
Silvers started getting called now, and I was preparing to go up. They called the other three, and after a bit I handed my camera to Albert, expecting to be called up at any point. but the number of silver medals remaining was very clearly diminishing, and then they stopped. Stunned, I almost missed taking a picture of the silver medalists. At this point, I was just amazed.
The silver medalists
The gold medals started being announced, starting with the Chinese perfect scorer. Then the other gold medalists, and finally ending with me. The suspense was incredible. After going up to receive my gold medal, my hands were incredibly shaky. I could barely take pictures of the remainder of the ceremony, where China handed the trophy over to Russia (RMM has one trophy that the winning team keeps until another team ousts them), and then a few more short words.
After the award ceremony, Po-Shen informed us that the reason the awards ceremony was delayed was because they had to argue for my solution to #5 for about an hour. There was a step that I thought was obvious and Po-Shen thought was obvious, but the graders disagreed. Apparently they had to call in a third party to give an impartial opinion. Eventually, though, they agreed to give me a 7. Lesson from this: write more on combo problems because other people don't have the same idea of obvious as I do for combo.
Mathcamp pride!
The team with our lovely (and camera shy) guides
Two days before we were to leave, Po-Shen sent us an email that, among other things, notified us that Lufthansa was currently experiencing a strike and that if our flight out of DC was canceled, the entire trip would be also. Obviously, this did not sit well with us, as we were all strongly looking forward to the trip.
Luckily, the strike was called off before we left, although Lufthansa was still short pilots, so some of the flights got canceled, but ours wasn't one of them. The flights to Bucharest actually went pretty well, including the AMC B. I did worse on the B than the A, but it really doesn't matter. I took it mainly because I figured everyone else would also and I didn't want to be bored for those 75 minutes. Tim had gotten a 96 on the A and was worried that he didn't qualify for AIME, and he wasn't exactly relieved when he got a 96 on the B as well.
On the trip there, we were expecting to be housed at Hotel Moxa, a 4 star hotel in Bucharest. However, it turns out that it was actually Complex Moxa, which is used for college dorms and is just an annex of the hotel or something. The rooms were pretty unfortunately bad, but ours had a TV in it! (the others apparently didn't). Because of the 7 hour time difference, the Olympics were on after all of the events for a day ended, which was extremely convenient. I definitely watched more of the Olympics while in Romania than any other time.
Sam checking out the roomWe also found out that the complex didn't have an open wireless access point....But Vlad had this USB thing that allowed him to get internet access in Romania. It's called Zapp or something. At least we had internet access, even though it was pretty bad.
The next day we still weren't competing. We got our first taste of Romanian breakfast, which included an interesting tea (I think it was purple) that tasted pretty good, as well as some cheese. Being American, we obviously thought the portions were way too small so we ate masses of bread with oil and vinegar.
Our first Romanian breakfastAfter breakfast, we met our guides and went to the high school where we would be taking the contest in the following two days. After touring the school and dropping in on a ``superior algebra'' class, the guides asked us if we wanted to go into the gym to play some sports. Inside, there were lots of people from various teams playing volleyball, but the court was pretty full so we didn't join them. Instead, we saw a ping-pong table, but nobody had any paddles, so we started playing basketball while we waited for a guide to retrieve paddles from the complex.
For some reason, someone thought it would be a good idea to play outside, even though there were huge puddles of water on the ground and the court was not very even. There were also ping-pong tables outside, but they looked pretty bad. They were really low, weren't flat, and the nets were actually iron fences.
China plays on them anywayEventually we got some paddles and played some ping-pong, as did the Chinese. The Chinese team didn't know much English and the only Chinese speakers were on the US team as either a student or a coach, so they spent a lot of time with us (and also Allen and Tim were in a room with two of them).
At some point we went back to our room to hang out until dinner, after which would be the opening ceremony. But as we were just starting to chill in our room, our guides came up to inform us that the opening ceremony got moved from 2000 to 1600, and we had to go back to the school.
The opening ceremony was actually quite nice. Only a small part of it was dual-run in Romanian and English. All of the guest speakers spoke in English, so translation was unnecessary, and they also all kept it very short. It made the opening ceremony much shorter than what I expected.
The next day was competition day 1.
Go go go!I read the problems and solved 1i on sight, as did the rest of the team except for Vlad, who apparently took 1.5 hours on it. I then spent a bit of time on 1ii, but wasn't quite getting the details. I figured it would be easy anyway and went to do number 2 before finishing.
Number 2 was dispatched rather readily, and at this point I had about 3 hours left, if I remember correctly. I drew the diagram for 3 (although I actually drew the wrong diagram, thinking ``external'' meant that the quadrilateral was external to the circle, rather than the circle is external to the quadrilateral), wrote down some random stuff, and went back to 1ii. After all, surely a number 1 number theory would be easier for me than a number 3 geometry, right?
So it turned out that I didn't solve 1ii, and didn't have anything worth partial on 3, whoops. In the last 5 minutes I wrote down some stuff for 1ii that I figured had no hope of working, but it turned out to be extremely close to the correct solution. I left the room thinking ``Man, I'm going to have to tell the rest of the team that I didn't solve number 1.''
So talking with the others after day 1, it seemed initially that most of them had solved two problems: either 1 and 2 or 1 and 3. The exceptions were Allen, who solved only 1i and 2, and Sam, who solved only 1. After talking a bit more, however, Albert determined that his 1ii was completely wrong, and so he had only solved 1.5 problems as well. After day 2, we would find out that during coordination the coordinators had thought that Albert's solution had worked too, and Yi and Po-Shen had to tell them it was wrong to keep the spirit of the contest.
Allen and I both had essentially identical progress on 1ii, and since it was so close to the correct solution, we came out of coordination with 6s...somehow. The graders were apparently pretty lenient with scoring.
Day 1 Scores
| ID | Name | P1 | P2 | P3 | Total |
|---|---|---|---|---|---|
| USA1 | Timothy Chu | 7 | 7 | 0 | 14 |
| USA2 | Vlad Firoiu | 7 | 3 | 7 | 17 |
| USA3 | Albert Gu | 3 | 0 | 7 | 10 |
| USA4 | Brian Hamrick | 6 | 7 | 0 | 13 |
| USA5 | Sam Keller | 7 | 0 | 0 | 7 |
| USA6 | Allen Yuan | 6 | 7 | 3 | 16 |
After day 1, we just went back to our room to hang out, being exhausted from the competition. Nothing much interesting happened. We just watched the Olympics and played card games, mostly.
We woke up the next day for day 2 of the competition.
No geometry! Wooo!So I read the day 2 problems and I thought ``YES! There's no geometry! Let's get a 21 on day 2! Oh wait, these problems look time consuming. 4.5 hours might not be enough...'' Anyway I looked at problem 4 and killed it in about 20 minutes. I start working on problem 5 and it dies in another 50 minutes or so. At this point it's about 1050 and I have two complete solutions written up and I'm starting to think maybe number 6 is really hard and they gave us two really easy problems to compensate (a la IOI day 1).
So I spend the next 3 hours trying various stuff on number 6, but I don't do the thing that actually leads to a solution because it looked stupidly messy. Oh well. I wrote up what I had (which wasn't exactly the cleanest thing in the first place), and then turned in the test. When I was leaving the room, I figured I probably had a pretty standard result on day 2.
However, when I talked to the rest of the team, I found out that I could hardly be more wrong. They had all solved problem 4 (except Albert, who got a 0 on day 2, unfortunately), but nobody else had solved problem 5. I was really surprised. Tim thought he solved problem 6, but none of us could really verify it since he was the only one who felt that he had made significant progress.
Later in the day, we found out (with our awesome Chinese-speaking skills) that CHN1 had been the only Chinese team member to solve either 5 or 6 (and he solved both (and CHN was really Shanghai, not all of China)). Apparently 5 was supposed to be very difficult. I still don't really see why.
After day 2, we went to the mall to play some laser tag! Except that the game was actually pretty lame. At first there was only like one person on the red team, so it was just walking around for a while until the person running the thing decided to restart it. Unfortunately, the respawn time was still around 3 seconds, so whenever you killed someone they could just follow you until they respawn and kill you immediately. It made for a pretty annoying game.
We got back to the complex pretty late, so we missed the normal dinner and had to order pizza, and our discussion of day 2 with Yi and Po-Shen was at around 2230, way later than we expected.
Day 2 Scores
| ID | Name | P4 | P5 | P6 | Total |
|---|---|---|---|---|---|
| USA1 | Timothy Chu | 7 | 2 | 5 | 14 |
| USA2 | Vlad Firoiu | 7 | 2 | 0 | 9 |
| USA3 | Albert Gu | 0 | 0 | 0 | 0 |
| USA4 | Brian Hamrick | 7 | 7 | 4 | 18 |
| USA5 | Sam Keller | 7 | 2 | 0 | 9 |
| USA6 | Allen Yuan | 7 | 2 | 0 | 9 |
The awards ceremony was the day right after day 2. But before that, coordination had to happen. So to get rid of us pesky contestants for a while, they sent us to the village museum: a collection of traditional Romanian houses. It would have been a really cool experience, but the ground was extremely muddy and it was simply unpleasant to walk around.
When we got back it was time for the awards ceremony. Well, almost. It was actually delayed for half an hour. Anyway, the awards ceremony, just like the opening ceremony, was very quick. The speakers knew that we didn't want to listen to a bunch of long speeches (and it was hard to understand some of their English anyway), so they went straight to the awards. Albert was the first USA competitor called up for honorable mention (solving at least one problem perfectly).
Next up was the bronze medals. There were a lot of bronzes, and Sam was among them. I was actually pretty nervous during the bronzes because I wasn't sure if I had screwed up something on day 2, in which case I would probably be in the low end of silver. As the bronzes ended, I breathed a sigh of relief.
The bronze medalistsSilvers started getting called now, and I was preparing to go up. They called the other three, and after a bit I handed my camera to Albert, expecting to be called up at any point. but the number of silver medals remaining was very clearly diminishing, and then they stopped. Stunned, I almost missed taking a picture of the silver medalists. At this point, I was just amazed.
The silver medalistsThe gold medals started being announced, starting with the Chinese perfect scorer. Then the other gold medalists, and finally ending with me. The suspense was incredible. After going up to receive my gold medal, my hands were incredibly shaky. I could barely take pictures of the remainder of the ceremony, where China handed the trophy over to Russia (RMM has one trophy that the winning team keeps until another team ousts them), and then a few more short words.
After the award ceremony, Po-Shen informed us that the reason the awards ceremony was delayed was because they had to argue for my solution to #5 for about an hour. There was a step that I thought was obvious and Po-Shen thought was obvious, but the graders disagreed. Apparently they had to call in a third party to give an impartial opinion. Eventually, though, they agreed to give me a 7. Lesson from this: write more on combo problems because other people don't have the same idea of obvious as I do for combo.
Mathcamp pride!Final USA Results
| ID | Name | P1 | P2 | P3 | Day 1 | P4 | P5 | P6 | Day 2 | Total | Award |
|---|---|---|---|---|---|---|---|---|---|---|---|
| USA1 | Timothy Chu | 7 | 7 | 0 | 14 | 7 | 2 | 5 | 14 | 28 | Silver Medal |
| USA2 | Vlad Firoiu | 7 | 3 | 7 | 17 | 7 | 2 | 0 | 9 | 26 | Silver Medal |
| USA3 | Albert Gu | 3 | 0 | 7 | 10 | 0 | 0 | 0 | 0 | 10 | Honorable Mention |
| USA4 | Brian Hamrick | 6 | 7 | 0 | 13 | 7 | 7 | 4 | 18 | 31 | Gold Medal |
| USA5 | Sam Keller | 7 | 0 | 0 | 7 | 7 | 2 | 0 | 9 | 16 | Bronze Medal |
| USA6 | Allen Yuan | 6 | 7 | 3 | 16 | 7 | 2 | 0 | 9 | 25 | Silver Medal |
The team with our lovely (and camera shy) guides
Tuesday, February 23, 2010
Thoughts on HMMT
Overall, HMMT was well run. However, some of the tests could definitely have been better written. I'm going to just talk about the Combinatorics and Calculus subject tests from individual, since those were the two I took, and I'll also talk about team and guts.
First up is Calculus. I think everyone should realize that a 4 way tie for first at 29 is a problem with the test. The problems that I liked on calculus were 1, 2, 3, and 8. The rest of them have some issues.
Problem 4: Everyone who thinks about this problem can probably get it, but I think it's not exactly kosher to assume that people that people know the equidistribution theorem.
Problem 5: Just differentiate 4 times...seriously? I mean there's the nicer approach where you can notice that you only get 4 copies of
when you differentiate the 
term 4 times, so you can directly pull out the coefficient by looking at just that term. By the time problem 5 rolls around I think you should be moving away from the stupidly straightforward problems.
Problem 6: I didn't actually solve this problem, although I had enough intuition that I could have finished it rigorously somewhat quickly. I just said, ``Let's put the line through the inflection point'', which is exactly what you want to do as cubics are symmetric about the inflection point.
Problem 7: This problem shares the same issue as many of the problems on the test. The answer (set two equal and imaginary and the third one real) is guessable (although I don't think anyone did), but it's completely unreasonable to expect students to prove it in 50 minutes when there are 9 other problems to work on.
Problem 9: Nice solution, but do you really expect anyone to get it?
Problem 10: This one is definitely doable...but it basically has seeing it before as a prerequisite. I thought that was what we were trying to avoid after last year's #10. It's a nice technique, but I don't think anyone would be able to come up with it during the test.
Overall, calculus had relatively easy problems #1-#6, a doable #8, and impossible #7, #9, and #10. 29 was getting all the doable problems. It really doesn't help the test to put a bunch of impossible problems on. The difficulty just has such a huge jump between 6 and 7, with 8 in between somewhere. I'd not be surprised if there is not only a huge tie at 29, but also a huge tie at 23. Perfect scores aren't a problem; ties are.
Next up: Combinatorics. Most of this test was actually good. I only really have complaints about problems 7 and 10.
Problem 7: This problem is just so out of place at HMMT. Looking at the rest of the problems, there is absolutely no strenuous computation. This problem, in contrast, is a complete computation-fest, after a moderately silly manipulation with expected values.
Problem 10: Same issue as Calculus #7. It's somewhat possible (although I doubt anyone did) to guess the optimal configuration, but not reasonable to expect students to prove it during the test. It's made even worse by the obfuscation that $16 = 4^2$, so instead of trying things like 5x5 with 4 numbers, people would rather have tried 4x4 with 2 numbers. I really dislike the problem for this kind of test. It would make a good team round problem, though.
I would have liked the test a lot better if problem 10 were what is now problem 7, and an actual problem 7 were in the problem 7 slot, although I really don't like the current problem 7 as problem 10 either.
Now for team round. I liked the team round more than other rounds this year (although that might have been because we won), because I think there was actually a scaling of difficulty (and the ability to give partial credit helps immensely). However, some of the problems had minor issues.
Problem 1: This is pretty classic. I'm pretty sure that Dan is not wrong when he says that he has seen it before.
Problem 2: I feel like I have seen this problem before, although it may have been slightly different (and the key observation should be that every divisor of an odd number is odd).
Problem 4: I'm pretty sure this is way too classic (although I forgot to cover the case where the 2x2 system for x+y and xy is singular, oops!). Actually I'm wondering if it's even possible for A, B, C, and D to be rational except at x=0, y=0.
Problem 5: I think it was fine, except that ``decreasing'' is ambiguous because you write polynomials starting from the highest order term, so we had the (unanswerable) question of does
have decreasing coefficients or does 
? We did eventually settle on the one in the official solution, luckily.
Problem 6: Okay darn, I gave a pretty bad argument for the existence of an infinite ray being inside the set (A better argument is to just look at the furthest distance at each angle. It's clearly continuous and then it should have a maximum since
is compact, but that would mean it's bounded. Contradiction.). Mine can be made rigorous when you add in a weird continuity requirement and use the fact that is compact, but then you just get exactly the argument above. I actually like this problem, but I think that Jacob has mentioned that usually problems that have roots in college level math are rejected.
Problem 7: Maybe we're just bad at geometry, but it took Alex Zhu and I about 3 hours working together to solve this problem. Pretty sure this was harder than both 8 and 9 (and 10a, but having 10 be 10 is justified by 10b), but it was a good problem.
Problem 9: Maximum should run from i=1 to n, not i=0 to n-1, but I think that was pretty clear for most people. This problem was definitely easier than some of the ones that appear before it on the test. I'm not sure why it's a problem 9.
Problem 10: 10a is nice, but when Jacob says ``The idea for 10a works for 10b too after a few hours of work,'' it starts to look a bit unreasonable. I feel sad because I would have guessed
and now I'm wondering why I didn't write that down. Maybe we would have gotten a point!
Finally, guts.
I really liked most of the guts round (in fact, almost all of it). But there were a few issues:
Problem 12: No, it is not ``obvious'' that
does not need to be multiplied out. Replace the 9 by a 2010 and it would be. I don't see why that wasn't done.
Problem 17: Again, assuming people know (or can intuit) the equidistribution theorem (although in this case you don't actually need equidistribution) is a bit sketchy. However, I mind this a lot less in guts than in the other rounds.
Problem 32: I'm pretty sure our team had a fraction that we did not have time to turn into a decimal approximation. Without calculators, I find it a bit annoying that you would ask for a decimal to 5 places.
Problem 33: You have an exact form, so I'm not sure why the test is asking for the floor of
. I'd also like to point out that Vieta jumping tells you that 
immediately (and it's odd because this recurrence was used earlier in the round). I would have rather asked for the exact form, although perhaps it is impractical to grade? Regardless, I would avoid approximation problems that can be solved exactly.
As you can see, I have many fewer issues with the guts round than the other rounds. This is probably because I consider guts to have a vastly different style, so it is easier to write problems for it and also there are so many problems that it's almost impossible to get the issues like what happened on the calculus individual test.
I guess a large part of my complaint is that the calculus test had a huge wall at 29 points that really made it hard for people who took calculus to compete with the people who took the other tests. This definitely has happened in the past (such as with the even harder wall at 50 for geometry a few years ago), and I guess I'm just a bit bitter that it happened to my tests this year. I do think (looking at results again) it affected this year's competition a lot more than last year's. Last year calculus was the test that suffered from the most ties (which was probably from the test being a bit too straightforward), but it wasn't a four way tie for first.
Overall, well done as always, but let's make next year's even better!
First up is Calculus. I think everyone should realize that a 4 way tie for first at 29 is a problem with the test. The problems that I liked on calculus were 1, 2, 3, and 8. The rest of them have some issues.
Problem 4: Everyone who thinks about this problem can probably get it, but I think it's not exactly kosher to assume that people that people know the equidistribution theorem.
Problem 5: Just differentiate 4 times...seriously? I mean there's the nicer approach where you can notice that you only get 4 copies of
when you differentiate the term 4 times, so you can directly pull out the coefficient by looking at just that term. By the time problem 5 rolls around I think you should be moving away from the stupidly straightforward problems.Problem 6: I didn't actually solve this problem, although I had enough intuition that I could have finished it rigorously somewhat quickly. I just said, ``Let's put the line through the inflection point'', which is exactly what you want to do as cubics are symmetric about the inflection point.
Problem 7: This problem shares the same issue as many of the problems on the test. The answer (set two equal and imaginary and the third one real) is guessable (although I don't think anyone did), but it's completely unreasonable to expect students to prove it in 50 minutes when there are 9 other problems to work on.
Problem 9: Nice solution, but do you really expect anyone to get it?
Problem 10: This one is definitely doable...but it basically has seeing it before as a prerequisite. I thought that was what we were trying to avoid after last year's #10. It's a nice technique, but I don't think anyone would be able to come up with it during the test.
Overall, calculus had relatively easy problems #1-#6, a doable #8, and impossible #7, #9, and #10. 29 was getting all the doable problems. It really doesn't help the test to put a bunch of impossible problems on. The difficulty just has such a huge jump between 6 and 7, with 8 in between somewhere. I'd not be surprised if there is not only a huge tie at 29, but also a huge tie at 23. Perfect scores aren't a problem; ties are.
Next up: Combinatorics. Most of this test was actually good. I only really have complaints about problems 7 and 10.
Problem 7: This problem is just so out of place at HMMT. Looking at the rest of the problems, there is absolutely no strenuous computation. This problem, in contrast, is a complete computation-fest, after a moderately silly manipulation with expected values.
Problem 10: Same issue as Calculus #7. It's somewhat possible (although I doubt anyone did) to guess the optimal configuration, but not reasonable to expect students to prove it during the test. It's made even worse by the obfuscation that $16 = 4^2$, so instead of trying things like 5x5 with 4 numbers, people would rather have tried 4x4 with 2 numbers. I really dislike the problem for this kind of test. It would make a good team round problem, though.
I would have liked the test a lot better if problem 10 were what is now problem 7, and an actual problem 7 were in the problem 7 slot, although I really don't like the current problem 7 as problem 10 either.
Now for team round. I liked the team round more than other rounds this year (although that might have been because we won), because I think there was actually a scaling of difficulty (and the ability to give partial credit helps immensely). However, some of the problems had minor issues.
Problem 1: This is pretty classic. I'm pretty sure that Dan is not wrong when he says that he has seen it before.
Problem 2: I feel like I have seen this problem before, although it may have been slightly different (and the key observation should be that every divisor of an odd number is odd).
Problem 4: I'm pretty sure this is way too classic (although I forgot to cover the case where the 2x2 system for x+y and xy is singular, oops!). Actually I'm wondering if it's even possible for A, B, C, and D to be rational except at x=0, y=0.
Problem 5: I think it was fine, except that ``decreasing'' is ambiguous because you write polynomials starting from the highest order term, so we had the (unanswerable) question of does
have decreasing coefficients or does ? We did eventually settle on the one in the official solution, luckily.Problem 6: Okay darn, I gave a pretty bad argument for the existence of an infinite ray being inside the set (A better argument is to just look at the furthest distance at each angle. It's clearly continuous and then it should have a maximum since
is compact, but that would mean it's bounded. Contradiction.). Mine can be made rigorous when you add in a weird continuity requirement and use the fact that is compact, but then you just get exactly the argument above. I actually like this problem, but I think that Jacob has mentioned that usually problems that have roots in college level math are rejected.Problem 7: Maybe we're just bad at geometry, but it took Alex Zhu and I about 3 hours working together to solve this problem. Pretty sure this was harder than both 8 and 9 (and 10a, but having 10 be 10 is justified by 10b), but it was a good problem.
Problem 9: Maximum should run from i=1 to n, not i=0 to n-1, but I think that was pretty clear for most people. This problem was definitely easier than some of the ones that appear before it on the test. I'm not sure why it's a problem 9.
Problem 10: 10a is nice, but when Jacob says ``The idea for 10a works for 10b too after a few hours of work,'' it starts to look a bit unreasonable. I feel sad because I would have guessed
and now I'm wondering why I didn't write that down. Maybe we would have gotten a point!Finally, guts.
I really liked most of the guts round (in fact, almost all of it). But there were a few issues:
Problem 12: No, it is not ``obvious'' that
does not need to be multiplied out. Replace the 9 by a 2010 and it would be. I don't see why that wasn't done.Problem 17: Again, assuming people know (or can intuit) the equidistribution theorem (although in this case you don't actually need equidistribution) is a bit sketchy. However, I mind this a lot less in guts than in the other rounds.
Problem 32: I'm pretty sure our team had a fraction that we did not have time to turn into a decimal approximation. Without calculators, I find it a bit annoying that you would ask for a decimal to 5 places.
Problem 33: You have an exact form, so I'm not sure why the test is asking for the floor of
. I'd also like to point out that Vieta jumping tells you that immediately (and it's odd because this recurrence was used earlier in the round). I would have rather asked for the exact form, although perhaps it is impractical to grade? Regardless, I would avoid approximation problems that can be solved exactly.As you can see, I have many fewer issues with the guts round than the other rounds. This is probably because I consider guts to have a vastly different style, so it is easier to write problems for it and also there are so many problems that it's almost impossible to get the issues like what happened on the calculus individual test.
I guess a large part of my complaint is that the calculus test had a huge wall at 29 points that really made it hard for people who took calculus to compete with the people who took the other tests. This definitely has happened in the past (such as with the even harder wall at 50 for geometry a few years ago), and I guess I'm just a bit bitter that it happened to my tests this year. I do think (looking at results again) it affected this year's competition a lot more than last year's. Last year calculus was the test that suffered from the most ties (which was probably from the test being a bit too straightforward), but it wasn't a four way tie for first.
Overall, well done as always, but let's make next year's even better!
Wednesday, February 10, 2010
Please Never Use This Problem On A Contest
2010 AMC 12A #24
There are so many things wrong with this problem that it made me make a blog post about it. The problem, of course, is that it relies on several conventions that are taught in math classes, but are not the conventions when you actually do math (or at least they aren't the conventions in every field of math).
First of all, I strongly object to the use of the word ``domain'' in this context. The domain of a function is absolutely not dependent on the definition of the function. A function is defined with a specified domain and codomain, of which this problem specifies neither. Instead, it tries to implicitly define the domain from the properties of the function. This is commonly used in math classes. I know I learned in some math class ``how to find the domain of a function'' such as
, but I have never seen this outside of math class and a few competitions (and all of the competitions that I've seen it on, including the AMC, are very clearly tailored for average math class students, or at least students who don't have math education beyond that which you get in the classroom). Nevertheless, while I object to the use of the word ``domain'', it was clear what the AMC meant, so that would be admissible.
However, the real problem comes in the use of
, which is clearly and unambiguously defined as 
. Furthermore, 
has a well-defined value. The problem is that 
has different meanings in different fields, and there is no way to know which one the AMC wants, except for the fact that people who have not learned math outside of the classroom can only be expected to know one of them.
The AMC never specfiied a codomain. And actually, since the AMC assumes the knowledge of complex numbers, this is a huge problem.
is, for a vast number of fields, given the value of 
, even though any one of 
would work just as well. However, the point is that it is defined.
If I were to ask someone what is the domain of
, I would almost certainly get the answer 
. But then, what if I say, ``Oh but 
is defined as 
!''? Then the person I'm talking to will, in many cases, revise their answer to all of 
. The exact same problem exists with . Is the domain 
or 
? That question comes directly from the question as to whether the codomain is or 
.
So please, if you want to use this problem on a contest, word it like this:
There are so many things wrong with this problem that it made me make a blog post about it. The problem, of course, is that it relies on several conventions that are taught in math classes, but are not the conventions when you actually do math (or at least they aren't the conventions in every field of math).
First of all, I strongly object to the use of the word ``domain'' in this context. The domain of a function is absolutely not dependent on the definition of the function. A function is defined with a specified domain and codomain, of which this problem specifies neither. Instead, it tries to implicitly define the domain from the properties of the function. This is commonly used in math classes. I know I learned in some math class ``how to find the domain of a function'' such as
, but I have never seen this outside of math class and a few competitions (and all of the competitions that I've seen it on, including the AMC, are very clearly tailored for average math class students, or at least students who don't have math education beyond that which you get in the classroom). Nevertheless, while I object to the use of the word ``domain'', it was clear what the AMC meant, so that would be admissible.However, the real problem comes in the use of
, which is clearly and unambiguously defined as . Furthermore, has a well-defined value. The problem is that has different meanings in different fields, and there is no way to know which one the AMC wants, except for the fact that people who have not learned math outside of the classroom can only be expected to know one of them.The AMC never specfiied a codomain. And actually, since the AMC assumes the knowledge of complex numbers, this is a huge problem.
is, for a vast number of fields, given the value of , even though any one of would work just as well. However, the point is that it is defined.If I were to ask someone what is the domain of
, I would almost certainly get the answer . But then, what if I say, ``Oh but is defined as !''? Then the person I'm talking to will, in many cases, revise their answer to all of . The exact same problem exists with . Is the domain or ? That question comes directly from the question as to whether the codomain is or .So please, if you want to use this problem on a contest, word it like this:
Monday, December 28, 2009
POTW Beta
As I hinted toward recently, I was thinking about running problems of the week. Well now I have created a beta test of the system to run over winter break. I am calling on you to help me test in the time between all your fun winter break activities. If you are a current member of VMT, go to http://activities.tjhsst.edu/vmt/pages/training/index.php after logging in to the wiki and you should be able to access the problems by clicking on the link that says POTW Beta. If you are not a member of the TJ math team, that link should direct you to a login/registration page where you can register for an account and log in as a guest. This beta will be open until the real POTW starts.
A note about the interface: All of the answers for this beta are positive integers. Therefore, there is no need to use the preview button. That is there so that if a problem has a more complicated answer, you can check to make sure that the system is parsing it correctly, and that the fact that your answer is incorrect is a result of your answer being wrong, rather than that it is formatted incorrectly.
I ask the following from you if you decide to participate in the beta:
A note about the interface: All of the answers for this beta are positive integers. Therefore, there is no need to use the preview button. That is there so that if a problem has a more complicated answer, you can check to make sure that the system is parsing it correctly, and that the fact that your answer is incorrect is a result of your answer being wrong, rather than that it is formatted incorrectly.
I ask the following from you if you decide to participate in the beta:
- Use your real name and grade and keep IDs appropriate. I will delete accounts that violate either one of these conditions.
- Report any bugs with the system to me! These include broken links, bad formatting, anything that you think could be improved. This also includes feature requests!
- Do not look up the problems or cheat on them in any other way. Because this is just a beta, I did not use original problems. However, these are still good practice problems and I think that much of the TJ math team can benefit from actually doing them.
- Tell your friends! I want to get as many people as possible into the POTW system and to do that I need your help. However, make sure that they don't leave between now and the start of the real POTW.
Sunday, November 22, 2009
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:
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.
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
satisfying the following properties:- If
, then
- If
then
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
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