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?

• 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?
  

I'm not sure of the answer to this question, so I'm interested to see what the readers of my blog think.

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.

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 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!

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

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!

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:

Motivation

On Friday, the 22nd of January, 2010, I was in my AP Government class for the last day of a model congress. The model congress was a great idea and one that worked very well for the first day. By the end of the simulation, however, the students decided they would rather screw with the system than actually try to learn. The result? Bills were passed that included provisions such as every April 23 is now National Korean Appreciation Day, also known as Kimchi Day. Those two weeks of class were not a model of congress. They were a mockery. At least I hope they were, or else I just lost all my faith in congress.

Very often, people bring up the issue of America's lackluster education performance. Efforts to alleviate the problem have been instituted, but they don't address the real cause of the problem. Our students simply don't want to learn. Throw as much money as you'd like at the education system, but if you don't change the attitude toward education, nothing will get done.

When a student goes through school, he or she is constantly bombarded with two conflicting messages: on the one hand, they have the American dream in one form or another, and at some level they understand that education is necessary to achieve this, but on the other hand, doing well in school is simply uncool. So students are essentially presented with a choice: work for a better future or shun school to become popular. It's obvious which one is chosen more, and not unreasonably.

When the issue goes to the government, they don't see this choice. All they see is under-performing students making their county/state/country look bad. They can make a law that requires improvement on standardized tests, but the people in charge of education locally, be it the school level, the county level, or the state level, won't make education better to meet the standards. Instead, they'll lower the standards so that more people can pass without increasing costs at all. Again, this is a rational decision, especially when the result of not meeting standards is punishment rather than help. Instead of improving, they cheat the system.

The same problem is present on a much smaller scale. I know many people who can say that they have what are colloquially known as "Asian Parents". As many of you know, Asian cultures value education highly. Unfortunately, the way that some parents carry this out is by punishing their child for every bad grade that they get. And a bad grade means a B+ (or an A- now, I guess). What do the kids do? They cheat the system. I don't mean that they necessarily cheat, but they get good grades without learning.

I'll do another blog post at some point on what learning is, exactly, but for this entry I want to talk about why students don't want to learn. The answer comes from History of Science. What Mr. Kelly said, and I think that this is very true, is that there are two types of motivation: intrinsic and extrinsic.

There are many reasons why students study for school: they need good grades to get in to college, they need good grades so that they don't get grounded, they need good grades so they don't get hit on the head, et cetera. Of all of those reasons, none of them are intrinsic. And that's the problem.

Think about the things that you do for fun. Do you do them to get in to college? Do you do them because your parents will hit you on the head if you don't? Do you have any reason to suspect that it will help you in the future? It's possible for school to be that way too, but a lot of things need to change.

First, people need to stop taking pride in their senioritis. Yes, your grades mostly don't matter for college now. Yes, you can slack off a little bit. But it's not something to be proud of.

Second, people need to realize that graduation is not the end of life. I'm sick of people telling me that they aren't going to work at math team because "It's too late" or that they don't want to do anything academic because they're "already in college" or have "already submitted [their] application." You aren't stepping off the face of the earth after you get your diploma, so stop acting like it.

Third, people need to stop telling other people to stop trying. Just because you want to be a slacker in your eighth semester of high school doesn't mean everyone does. Unfortunately, most people do. So this change needs to happen in teachers too. Half of the reason that geosystems is a horrible class is that the students don't care. The other half is that the teacher knows the students don't care and facilitates their not caring.

People wonder why our math team isn't doing well. I can answer that, but I'm also powerless to fix it without your help. You also won't like my answer. We aren't doing well because none of you want to do well. The only reason why you're going to competitions is for the prizes.

Why do I say this? Because none of the seniors are coming to math team anymore. They have no intrinsic motivation to do math or math team. They did it because it looks good on college applications. They ran for an officer position not because they wanted to help the team, but because they wanted to buff up their resume. But none of them actually say that straight up. Instead, when I ask a senior why they weren't at math team, they make an excuse.

At PUMaC, I said we had a chance to win HMMT if the team worked. From what I've seen, the team hasn't worked. I will be at HMMT and put in my part, and we will still lose. Chances are I won't be at ARML again. If the math team keeps going how it's been going, then this year will be our worst year yet.

Come on, prove me wrong.

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:

• 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.
  

Without further ado, I declare the beta for POTW open! The actual POTW will hopefully start shortly after break (maybe with 1-2 weeks in between). Note: This beta is completely voluntary and will not count for anything.

Ohaithar