WOOT MATH JAM

Welcome to today's WOOT Math Jam!

Note: The two problems at the end of this Math Jam are not representative of the whole spectrum of WOOT problems. The first one is probably mid-level in difficulty and the second is easier than nearly all problems presented during WOOT classes.

MCrawford (19:30:30)
Before we get started I would like to take a brief moment to explain our virtual classroom to those who have not previously participated in a Math Jam or one of our online classes.

MCrawford (19:30:42)
As many of you know, this classroom is moderated, meaning that participants can type into the classroom, but only the moderators can choose a comment to drop into the classroom. This helps keep the session organized and on track. Also, only moderators can enter into private chats with other people in the classroom.

MCrawford (19:31:03)
Today we're here to answer questions that you may have about WOOT!

MCrawford (19:31:08)
WOOT stands for [b]W[/b]orldwide [b]O[/b]nline [b]O[/b]lympiad [b]T[/b]raining.

MCrawford (19:31:14)
Plus, it's fun to say... ""Woot!"" (At least, that's what Richard keeps insisting.)

MCrawford (19:31:21)
WOOT is a 7-month online training program for students who are preparing for their national math Olympiad or the International Mathematical Olympiad. WOOT 2006-07 will run from mid-September, 2006 to mid-April, 2007.

MCrawford (19:31:31)
WOOT is sponsored by D. E. Shaw group:

http://www.artofproblemsolving.com/Classes/Images/DESCO.gif

MCrawford (19:31:50)
If you can't click on that link, hold down the 'Ctrl' key and click.

MCrawford (19:32:01)
It means you have a popup blocker activated.

MCrawford (19:32:09)
Why WOOT?

MCrawford (19:32:14)
Most students have little or no Olympiad-level experience or instruction before they take a National Olympiad. WOOT remedies this problem by offering test-taking and solution-writing practice with feedback, classes with past Olympiad winners, and collaboration with the some of the top high school math students in the world.

MCrawford (19:32:27)
Specifically, WOOT will include the following activities:

MCrawford (19:32:32)
(1) 16 2-hour online lectures. Each class will be offered twice to alleviate time conflicts.

MCrawford (19:32:39)
(2) 8 handouts that will contain articles and practice problems, that will help students prepare for the lectures.

MCrawford (19:32:46)
(3) 3 Problem Sets, 2 Practice AIME-style tests, and 3 Practice Olympiad-style tests.

MCrawford (19:32:55)
(4) Detailed feedback on student work on each of the problem sets and practice Olympiads.

MCrawford (19:33:02)
(5) WOOT message board for discussing lectures, post-lecture problem lists, and past WOOT tests.

MCrawford (19:33:09)
(6) 24-hour online classroom access for WOOT participants to collaborate.

MCrawford (19:33:34)
In particular, the message board was extraordinarily active during the first year of WOOT.

MCrawford (19:33:42)
Numerous International Math Olympiad participants and USA Mathematical Olympiad winners (including the current Art of Problem Solving instructors) will be WOOT instructors.

MCrawford (19:33:57)
WOOT is ideal for students preparing for a National Olympiad such as the USA Mathematical Olympiad. Due to generous sponsorship from D. E. Shaw & Co., [b]2006 Math Olympiad Summer Program participants are invited to join WOOT for free[/b]. Therefore, all participants in WOOT will have the opportunity to train with the top high school students in the United States.

MCrawford (19:34:09)
The tuition for WOOT is $695, but if you register by July 15, the cost of the program is only $595.

MCrawford (19:34:14)
WOOT's page:
http://www.artofproblemsolving.com/Classes/AoPS_C_WOOT.php

MCrawford (19:34:25)
At this point I'll open the discussion for questions.

anitagade (19:32:41)
is woot open to everyone?

anitagade (19:34:00)
What grades is WOOT for?

MCrawford (19:34:39)
There is no age or grade limit for WOOT.

MCrawford (19:35:02)
However, WOOT is for students mature enough to solve a few AIME problems and provide good explanations for their methods.

chiron7 (19:34:44)
what are the lectures about?

MCrawford (19:35:24)
Lectures include many topics. I'll try to name a few:

MCrawford (19:35:54)
Building Bridges is a lecture I will prepare for each year. The theme is combining different areas of mathematics in problem solving.

MCrawford (19:36:06)
Inequalities was a topic from last year. I'm sure there will be some of that again.

MCrawford (19:36:18)
Number theory was covered by 2 or 3 different instructors.

MCrawford (19:36:34)
Advanced methods in geometry (synthetic geometry).

MCrawford (19:36:40)
And many others.

MCrawford (19:37:13)
All the kinds of topics and methods that would appear on the AIME, USAMO, or IMO. (well, as many as we can cover in 16 lectures and a few papers.)

neelnal (19:35:48)
When will we get a schedule for all of the classes?

MCrawford (19:37:59)
I believe that a partial schedule will be posted in the next few days, though the schedule is being handled by somebody who is currently at the International Mathematical Olympiad, so I can't give an exact date.

sq53e (19:36:36)
How many different levels of instructions are there?

MCrawford (19:38:34)
That's a hard question to answer specifically, but I'll try to explain...

MCrawford (19:38:58)
Material will range widely from middle AIME level math all the way up to material at the hardest levels of the IMO and USAMO.

MCrawford (19:39:33)
But there will be many problems at each level to keep you working and thinking.

anitagade (19:36:54)
What will we be covering today?

MCrawford (19:39:48)
Today is just Q&A about the program itself.

MCrawford (19:40:07)
But I might throw out a problem or two at the end so that you can get an idea of what a lecture might be like.

Mr Hairy 2005 (19:36:55)
about how much of an improvement can you expect to see in terms of USAMO index/usamo score after WOOT?

MCrawford (19:40:25)
That's hard to say. It really depends on how much you work.

MCrawford (19:40:42)
But our goal is to provide you with a great deal of material to work on, and guidance along the way.

MCrawford (19:41:13)
Some students saw major improvements. There were several USAMO winners and IMO participants in WOOT last year, and many of the top AMC/AIME index scorers.

neelnal (19:39:50)
Will the classes end in time for the AIME and USAMO?

MCrawford (19:42:00)
WOOT will continue past the AIME, up until right before the USAMO (approximately).

iostream.h (19:42:01)
hi. i got here a few minutes late. is a transcript of this going to be posted somewhere?

MCrawford (19:42:10)
Yes, in the Math Jams area.

SplashD (19:42:05)
How much of the course does an average pariticipant understand or learn from?

MCrawford (19:42:24)
That depends on the student.

MCrawford (19:42:46)
Some students may learn well from 1/3 of the material, benefit from exposure to 1/3, and not be able to grasp the hardest 1/3.

MCrawford (19:43:09)
However, some participants (experienced students likely to do well on the USAMO already) might understand all of it -- or almost all of it.

MathDancer (19:42:33)
will we work any on proofs or is it mostly finding number answers as in the AIME?

MCrawford (19:43:29)
We will definitely work on proofs -- even for AIME level problems at times.

MCrawford (19:44:00)
Part of the lecture I'll be giving helps bridge the gap by unraveling mid-hard level AIME problems the way we would if we wanted to write a proof of the solution.

iostream.h (19:44:04)
how flexible are you with the exams. i'm often out of town on weekends, and i might not have sufficient time to do an exam within the schedule sometimes.

MCrawford (19:44:56)
Several days are given for each exam. You might make a post asking Naoki was the exact amount of time will be. I would ask him, but as I mentioned, he's out of the country at this time.

MCrawford (19:45:00)
But...

MCrawford (19:45:25)
Last year we made special consideration a few times for students involved in major projects or traveling during the time we distributed exams.

MCrawford (19:45:43)
In those cases, they were graded a little later.

MCrawford (19:46:09)
Are there any other questions about the WOOT program?

neelnal (19:46:13)
If we miss a class, is there a make up date? and will it be on the schedule?

MCrawford (19:46:28)
Each class is taught twice during a single week.

MCrawford (19:46:35)
Transcripts are made for both classes.

MCrawford (19:46:41)
But after that, we won't rerun them again.

SplashD (19:46:36)
Is there a recommended amount of background knowledge that you should have before you participate so you can grasp most of the material?

MCrawford (19:47:15)
What we usually tell students is that if you can get a 5 or higher on the AIME, then WOOT is designed to help you improve.

MCrawford (19:47:25)
Our goal is for half or more of the material to be good for each student.

MCrawford (19:47:35)
I think we do a pretty good job in that regard.

MCrawford (19:47:50)
Also, for younger students just getting into AIME problem solving who can just barely score that 5...

MCrawford (19:48:09)
Seeing a great deal of material that you might only understand in a year or two can still be very beneficial.

MCrawford (19:48:29)
You're introduced to new topics and have a chance to see a little about where you can take your own studies.

MCrawford (19:48:39)
Plus, you can save the transcripts and learn from them next year.

neelnal (19:46:47)
Do you know around what time they will be?

MCrawford (19:48:58)
http://www.artofproblemsolving.com/Classes/AoPS_C_WOOTSchedule.php

MCrawford (19:49:23)
Actually, I said earlier that we didn't have a schedule. I must have been wrong. That might have been posted while I was on vacation.

iostream.h (19:48:39)
what regular aops classes will woot conflict with?

MCrawford (19:50:00)
During the Monday times, possibly.

MCrawford (19:50:04)
Let me check.

MCrawford (19:50:48)
The MATHCOUNTS and AMC 12 problem series appear to be scheduled for Mondays, so there will be some conflict.

MCrawford (19:51:01)
It's up to each student (you) as to how you would prefer to handle that overlap.

SplashD (19:49:41)
do you change the material each year such as the problem sets and the lectures?

MCrawford (19:51:28)
Yes, though I might keep a few problems from my lectures last year if I think they're particularly good teaching problems.

MCrawford (19:51:43)
Though some lectures will be completely different. Perhaps most of them.

MCrawford (19:52:20)
But I would venture to guess that 95% of the class problems will be new and 100% of the test problems will be new.

iostream.h (19:52:08)
how long are the problem sets? are they midterm-length (around 25 hard problems)? shorter/longer?

MCrawford (19:53:21)
I don't know the length of the problem sets. But that's a pretty good guess as to their length.

neelnal (19:52:36)
Where can we find schedule of the other AOPS classes? I can't find the page....

MCrawford (19:53:36)
http://www.artofproblemsolving.com/Classes/AoPS_C_Enroll.php

MCrawford (19:53:45)
Those are classes we'll be offering soon,.

MCrawford (19:53:53)
http://www.artofproblemsolving.com/Classes/AoPS_C_ClassesS.php

MCrawford (19:53:58)
Those are classes we offer in general.

nebula42 (19:53:33)
Is the actual WOOT classroom the same as this one (the one used for Math Jams)?

MCrawford (19:54:30)
It's functionally the same, but seperated from the Math Jam classroom.

MCrawford (19:55:06)
But, when WOOT classes are not in session, the WOOT classroom serves as an open chat room for WOOT students.

neelnal (19:54:58)
That's good. I like this format. It is organized.

SplashD (19:55:03)
Would WOOT eliminate the necessity to take intermiate classes and or the AIME Problem Series?

MCrawford (19:55:24)
That depends on the student.

MCrawford (19:55:44)
The AIME Problem Series can be valuable practice at a specific level.

MCrawford (19:55:56)
Our Intermediate classes follow more of a complete and set curricular path.

MCrawford (19:56:03)
So there are distinct differences.

MCrawford (19:56:35)
Personally, I wouldn't suggest skipping over Intermediate subject classes if you're a student wanting the most complete education in regards to math and problem solving.

MCrawford (19:57:12)
WOOT is a different program than our other classes in that the curriculum is not linear or as focused, but there is a great deal more attention to rigorous proofs.

MCrawford (19:57:59)
Are there any other questions about WOOT?

samath (19:58:12)
Can you describe the type of feedback we will get back from the instructors on our work. Do we get feedback on the practice tests and the problem sets?

MCrawford (19:59:13)
Feedback on the Olympiad-style tests will include comments on your rigor, writing style (clarity), methods (what approach might be better), and other helpful commentary.

MCrawford (19:59:39)
For those who have turned in an exam in any of our other classes, it will be similar, except that with fewer problems, there will be more commentary per problem.

MCrawford (20:00:09)
I don't believe there will be feedback on the AIME-style tests however.

MCrawford (20:00:16)
Though I'm not actually sure.

iostream.h (19:59:27)
what is the dropout policy for woot? like, if i find out that i'm in over my head after 2 classes, do i get a partial/full refund when i drop out?

MCrawford (20:00:50)
Let me try to look that up.

MCrawford (20:00:57)
Richard set that policy while I was on vacation...

neelnal (20:01:02)
How do we hand in tests? Do we have to type up our answers?

MCrawford (20:01:24)
I'll come back to the previous question.

MCrawford (20:01:27)
You fax in your tests.

MCrawford (20:01:40)
Most students write in dark pencil or in pen.

MCrawford (20:01:59)
You should take the Olympiad-style tests like you would a USAMO, which is to say in writing.

MCrawford (20:02:01)
Writing style matters!

MCrawford (20:02:10)
But, on the problem sets, you can TeX up your work if you like.

MCrawford (20:02:21)
Many students do this for our subject class exams.

samath (20:01:53)
Would you say the level and range of difficulty of the Olympiad practice test and problems be comparable to those on USAMO?

MCrawford (20:02:33)
Yes, but there will be options...

MCrawford (20:02:40)
There will be more than 3 problems...

MCrawford (20:02:56)
The easiest three will be like mid-AIME to easy-USAMO level problems.

MCrawford (20:03:19)
The hardest three will be harder than a typical day at the USAMO/IMO.

MCrawford (20:04:00)
I can't find the drop policy, but I can say this...

MCrawford (20:04:15)
I think it's two lectures and one article.

MCrawford (20:04:30)
After that, students should have a good idea as to whether or not the program is for them.

nebula42 (20:04:09)
What if we want to TeX our solutions but there are geometry problems where we cannot draw the diagram for the problem? Also, is there a way to submit solutions other than by fax?

MCrawford (20:05:07)
Problems should be faxed because they must be entered quickly into a database and graded at a different location using that database.

MCrawford (20:05:31)
If you can't find a way to draw a diagram (metapost? geometer sketchpad?) for an assignment, then just draw it by hand.

MCrawford (20:05:35)
In fact...

MCrawford (20:05:54)
Learning to construct your diagrams alone is one of the most useful and important lessons in geometric problem solving.

MathDancer (20:06:23)
about how large are the WOOT classes?

MCrawford (20:07:12)
Last year there were just over 100 total students in WOOT and when I taught there were usually 30-45 in each class (where each was taught twice).

MCrawford (20:07:24)
I don't think classes got any larger...

MCrawford (20:07:37)
But a few of the classes were smaller (when the material was the most difficult).

MCrawford (20:08:16)
Are there any questions, or would you like to see one of the problems I covered in class last year?

neelnal (20:08:15)
Were the make-up classes just as large as the regular sessions?

MCrawford (20:08:30)
They aren't make-up classes. Just different times.

MCrawford (20:08:35)
So they were usually about the same size.

MCrawford (20:09:11)
Last year I gave a class called Building Bridges...

MCrawford (20:09:18)
There were two parts to the lecture...

MCrawford (20:09:39)
There was learning to use different areas of math in combination (as is often the theme on AIME or ARML problems).

MCrawford (20:09:46)
Then I got into more proof-like problems.

MCrawford (20:09:55)
So the class got harder and harder as it progressed.

MCrawford (20:10:19)
Here is one of the easier problems from the second half of that class (though slightly harder than the average problem in that class).

MCrawford (20:10:25)


MCrawford (20:10:28)
http://www.artofproblemsolving.com/Classroom/cbe6/images/lx-70493774.gif

MCrawford (20:10:37)


MCrawford (20:10:53)
We can view this factorization in a number of ways. Those of you who know it already might see that a difference of squares cuts out any terms between the leading term and the constant. However, it might be misleading to focus on that fact.

MCrawford (20:11:05)
Is there some way we can approach this factorization problem such that we know for certain that we have all possible answers?

SplashD (20:11:20)
work backwards?

MCrawford (20:11:29)
In what sense?

SplashD (20:11:46)
assign variables for the coefficients?

MCrawford (20:12:01)
That's going to get messy.

iostream.h (20:11:55)
shouldn't all (x^2 + nx + n)(x^2 - nx + n) generate such a polynomial?

MCrawford (20:12:23)
You tell me. Does it? How would we know if we've covered all possible cases?

MCrawford (20:12:54)
What in algebra to we often relate to factoring? What does factoring help us [b]do[/b]?

samath (20:12:52)
That's only true if 2n=n^2

iostream.h (20:13:00)
get roots

MCrawford (20:13:11)
Ah, good observation.

nebula42 (20:13:08)
solve for roots

MCrawford (20:13:23)
How do examine the roots of the given polynomial?

SplashD (20:13:40)
complex

MCrawford (20:13:59)
Let's start as simple as possible.

nebula42 (20:14:00)
So finding the roots of $x^4+n$ will help us factor it

MCrawford (20:14:19)
There are no such things as roots of an expression, only of an equation.

MCrawford (20:14:29)


MCrawford (20:14:34)
Now what?

Xantos C. Guin (20:15:04)
Subtract n from both sides?

MCrawford (20:15:17)


MCrawford (20:15:26)
What is it that we can do now that makes taking the fourth root relatively easy?

MCrawford (20:15:48)
SplashD had a general answer. What can we do specifically?

Piggypi314 (20:15:56)
-1 = cis (pi)

iostream.h (20:16:00)
x^2 = in, then use demoivre?

MCrawford (20:16:09)


MCrawford (20:16:31)
We could have written the variable k into the first line above, but it's only in the second line when it comes into play.

MCrawford (20:16:38)
We have now identified the roots of the polynomial.

Now what?

MCrawford (20:17:06)
What do the roots give us that we like?

iostream.h (20:17:16)
so we have x^4 + n = product of all (x - (ugly terms))

MCrawford (20:17:23)
Good.

MCrawford (20:17:35)
What must we do to make what's ugly into a swan?

MCrawford (20:17:54)


samath (20:18:18)
Use (a+b)(a-b)=a^2-b^2

MCrawford (20:18:30)
How do these help us identify trinomial factors?

MCrawford (20:18:44)
What is samath hinting at?

iostream.h (20:19:01)
the im part of the exponential terms will vanish

MCrawford (20:19:12)


MCrawford (20:19:25)
We now have a factorization into trinomials with real coefficients, but our goal is to find n such that the coefficients are integers.

What is left to be done?

iostream.h (20:19:35)
just find all n such that the coefficients are integral

MCrawford (20:19:56)
So...?

Piggypi314 (20:20:09)
n = 4k^4

MCrawford (20:20:17)
Why?

iostream.h (20:20:22)
so it has to have a 2^2 term to get rid of the 4 under the fourth roots, and another 2^2 term to get rid of the denominator in the linear term

nebula42 (20:20:34)
4n must be a perfect fourth power, and n must be a perfect square

Piggypi314 (20:20:35)
because n is a perfect square and 4n has to be a fourth power

MCrawford (20:21:03)
While those are correct answers, let's remember to be straight-forward in our reasoning.

MCrawford (20:21:09)
That's the stuff of good proofs:

MCrawford (20:21:11)


MCrawford (20:21:29)


MCrawford (20:21:48)


MCrawford (20:22:00)
This problems explores both the use of roots in developing factorizations as well as integrating our understanding of complex numbers into our understanding of polynomial algebra, something that should feel more and more natural as you explore further into Olympiad level algebra. We also used a touch of number theory to hone in on our generalized solution.

iostream.h (20:22:13)
that was cool.

MCrawford (20:22:26)
I think so too.

MCrawford (20:22:38)
I'll try to pick out more cool problems to replace that one in class.

Mr Hairy 2005 (20:22:35)
Are we gonna move at this pace through most of the lectures?

MCrawford (20:23:01)
Different lectures will be taught differently depending on the content and style of the instructor.

MCrawford (20:23:23)
I tend to move quickly through material and problems and let students go back and review transcripts to get the finer points down.

MCrawford (20:23:28)
That allows me to cover more material.

MCrawford (20:23:42)
Some classes give more time to ponder the problems, which is valuable in a different way.

SplashD (20:22:36)
What level would this problem be considered?

MCrawford (20:23:59)
I think that problem would be on part with an easier USAMO problem.

MCrawford (20:24:13)
Of course, you'd have a lot of time to work on it if you saw it on the USAMO.

nebula42 (20:22:59)
Yes, that was a very quick pace.

MCrawford (20:24:49)
Indeed. I challenge the class to provide all the pieces together, which is different from individual exploration, but as I said, it allows greater coverage.

sq53e (20:24:30)
Can we have another problem?

MCrawford (20:25:05)
Here's one more I taught in that same class last year. It's one of the easier problems.

MCrawford (20:25:08)


MCrawford (20:25:24)
I'm not going to go through this one now.

MCrawford (20:25:36)
But if you work it, you'll know quickly that it's a much easier problem.

MCrawford (20:25:48)
Are there any other questions about WOOT itself before I go?

MCrawford (20:26:29)
Here's a hint for that problem for those who are wondering:

MCrawford (20:26:34)


iostream.h (20:26:20)
nope, this seems awesome. thanks for hosting the jam

Xantos C. Guin (20:26:25)
Are students allowed to attend both dates for a lecture?

MCrawford (20:27:00)
We don't encourage it, but if you sit silently during the second lecture, it won't be a problem.

MCrawford (20:27:18)
We don't want students hinting at the answers they already know.

Piggypi314 (20:27:10)
hey, look, a plus sign

nebula42 (20:27:18)
Thank you for the jam!

MCrawford (20:27:30)
Yes, plus sign. That's what I was getting at...

MCrawford (20:27:49)
Thank you everyone for coming to the WOOT Math Jam.

MCrawford (20:27:50)
I hope you enjoyed it.

MathDancer (20:27:41)
thank for the jam - looking forward to the class!

Mr Hairy 2005 (20:27:42)
Thanks Mr. Crawford :-)

neelnal (20:27:43)
LOL hahaha. thanks for the jam!

iostream.h (20:27:50)
oh, i thought that was a representation of the area of the solution set :/

iostream.h (20:27:52)
thanks!

samath (20:27:58)
thank you!

Xantos C. Guin (20:28:05)
Thanks for hosting it.

sq53e (20:28:08)
Thanks!

Piggypi314 (20:28:12)
It is pitch black. You are likely to be eaten by a grue.

MCrawford (20:28:21)
pfffft!

MCrawford (20:28:36)
I didn't know your generation knew of grues.

neelnal (20:28:22)
This was fun.

iostream.h (20:28:26)
rofl aops is not zoids

iostream.h (20:28:32)
it's much much scarier

MCrawford (20:29:11)
Good night all!