| Transcript
for the Math
Jam "AoPS Classes Math Jam"
on Feb 28. |
| Math Jam hosted by DPatrick
(Dave Patrick ). |
DPatrick19:28:52
Hello, and welcome to the Trigonometry/Complex Numbers and Special AIME Problem Seminar Classes Math Jam. Today we'll be discussing what these classes cover and how they work, and answering any questions you may have.
DPatrick19:29:41
My name is Dave Patrick, and I am an instructor and textbook author here at Art of Problem Solving.
DPatrick19:29:52
Before we get started I would like to take a moment to explain our Virtual Classroom to those who have not previously participated in a Math Jam or one of our online classes.
DPatrick19:30:08
The classroom is moderated: students can type into the classroom, but only the moderators can choose a comment to drop into the classroom.
DPatrick19:30:28
This helps keep the class organized and on track.
DPatrick19:30:37
Also, only moderators can enter into private chats with other people in the classroom.
DPatrick19:30:53
Note that it is not possible for the instructor to personally respond to every comment that you submit -- please do not take it personally if your comment is not posted or responded to! I will try to respond to all questions to the extent that I can.
DPatrick19:31:26
Today we're going to talk about our upcoming Trigonometry/Complex Numbers and Special AIME Problem Seminar Classes Math Jam.
DPatrick19:31:38
Intermediate Trigonometry/Complex Numbers
DPatrick19:31:56
The Intermediate Trigonometry/Complex Numbers course is for students who have mastered basic algebra and geometry, including: the Pythagorean Theorem, solving basic equations and systems of equations, graphing simple functions, the geometry of triangles, circles, and arcs, and the basic arithmetic of complex numbers. We do not assume any past experience working with trigonometry.
DPatrick19:32:15
The course begins on Thursday, March 13, lasts for 12 weeks, and meets for one 90-minute class each Thursday at 7:30 PM Eastern / 4:30 PM Pacific. If a student cannot attend a specific class, the student can access a full transcript of the class on our website within 24 hours after class end.
DPatrick19:32:32
Each class consists of the instructor leading the students through a series of increasingly challenging problems. An additional assistant instructor is present to help students who need extra attention.
DPatrick19:32:48
After each class, extra problems are given to the students on a course message board for full-class discussion. The course also includes 2 Challenge Sets, one for the first half of the course and one for the second half. The Challenge Sets consist of 20-30 problems for which students should write full solutions and submit them to us for evaluation. This evaluation will include commentary both on the accuracy of the solution, and on the student's ability to write mathematics effectively.
DPatrick19:33:04
Now Ian example of a trigonometry problem that we'll be doing in the course.
DPatrick19:33:11
oops...let me try that one again!
Quickster9419:33:17
who r the instructors for this class?
DPatrick19:33:23
The instructor for the course is Sean Markan. Sean participated in numerous math and science programs in high school, including the Math Olympiad Summer Program in 2001 and the US Physics Team in 2000 and 2002. He also won the Mandelbrot Competition in 2002. Sean graduated from MIT with a degree in Physics, and has previously taught Art of Problem Solving's Introduction to Algebra and MATHCOUNTS Problem Series courses.
DPatrick19:33:55
Now let me work through an example of a trigonometry problem that will be done in the course.
DPatrick19:34:17
The solution of this problem may use a couple of trigonometry terms or identities that you're not familiar with. Don't worry -- they will all be covered in the class!
DPatrick19:34:22
Here's the problem:
DPatrick19:34:26
DPatrick19:34:48
No calculators!!
DPatrick19:35:01
What should we do? Should we try to compute sin(20) and all the others?
xpmath19:35:13
express in terms of sines and cosines?
xpmath19:35:13
and double angle formula
DPatrick19:35:32
Let's put everything in terms of 10s rather than 10s and 20s. (In general, we always want to reduce the number of different angles we're dealing with if possible.)
DPatrick19:35:47
For that we can use the sine double angle formula.
DPatrick19:35:54
Again, don't worry if you don't know what any of this means!
DPatrick19:36:17
We'll be covering it all in the class -- the point of this problem is to give you a taste for the type of problems in the class and for how the class works.
DPatrick19:36:30
So we know that sin(20)=2sin(10)cos(10).
chenhsi19:36:33
sin (2*10) = 2 * (sin 10) * (cos 10)
DPatrick19:36:38
right...I just beat you to it! :)
DPatrick19:36:53
DPatrick19:37:05
The ugliest thing left is the tan(10)+cot(10). What can we do about that?
ppzuan19:37:22
change tan and cot to sin and cos
chenhsi19:37:22
change to sin and cos
xpmath19:37:22
(tan10+cot10)=(sin10/cos10+cos10/sin10)
Quickster9419:37:22
tan and cot can be expressed as sin/cos or cos/sin
DPatrick19:37:38
Right, we're happier if we can write things in terms of sines and cosines.
DPatrick19:37:48
Then we get:
DPatrick19:37:51
charlestreykang19:38:00
and then distribute to simplify
DPatrick19:38:15
Right, we can plug this back into our original expression:
DPatrick19:38:24
xpmath19:38:45
now we cancel off and get 2*1=2
DPatrick19:38:53
Bingo!
DPatrick19:38:57
DPatrick19:39:36
This problem is on the easier end of the spectrum of problems covered in the class.
DPatrick19:39:46
And I want to stress again, don't worry if you don't have a trig background.
DPatrick19:39:53
The Intermediate Trigonometry/Complex Numbers course is for students who have mastered basic algebra and geometry, including: the Pythagorean Theorem, solving basic equations and systems of equations, graphing simple functions, the geometry of triangles, circles, and arcs, and the basic arithmetic of complex numbers. We do not assume any past experience working with trigonometry.
DPatrick19:40:22
The trig course will start from the very basic of trigonometry.
DPatrick19:40:41
Here are a few examples of more difficult trigonometry problems that we'll be covering in the class. (I'm not going to show the solutions...I just want to give you a taste for the problems.)
DPatrick19:40:48
DPatrick19:40:58
DPatrick19:41:08
DPatrick19:41:21
Trigonometry and complex numbers are not only very interesting topics in their own right, they also are extremely useful problem solving tools, and can be used to solve many problems which, on the surface, don't appear to involve trigonometry or complex numbers at all.
DPatrick19:41:33
The following two problems are examples of such problems. The first one is solved using trigonometry, the second one is solved using complex numbers. They are very, very hard to solve without these tools. We won't work through the solutions here, but we'll cover them in the course:
DPatrick19:41:40
DPatrick19:41:49
DPatrick19:42:05
In the course, we will cover the definitions and properties of the trigonometric functions, various trigonometric identities, the Law of Sines and Law of Cosines, the definition and properties of complex numbers and the complex plane, the exponential form of complex numbers, De Moivre's Theorem, the geometry of complex numbers, and roots of unity.
DPatrick19:42:34
And, as I mentioned before, the instructor for the course is Sean Markan. Sean participated in numerous math and science programs in high school, including the Math Olympiad Summer Program in 2001 and the US Physics Team in 2000 and 2002. He also won the Mandelbrot Competition in 2002. Sean graduated from MIT with a degree in Physics, and has previously taught Art of Problem Solving's Introduction to Algebra and MATHCOUNTS Problem Series courses.
DPatrick19:42:53
Are there any questions about the Trig/Complex Numbers course?
chenhsi19:43:06
it is 24 weeks right?
DPatrick19:43:09
No, just 12 weeks.
Quickster9419:43:14
who is the assis instructor?
DPatrick19:43:27
I actually don't know.
ppzuan19:43:31
what is the cost?
Unemployed19:43:31
whats the cost
DPatrick19:43:46
$195
Quickster9419:43:52
how will those challenge sets be submitted? They sound very interesting
DPatrick19:44:03
They'll be posted on the website, and you'll have 6 weeks to work on them.
DPatrick19:44:24
You can turn in your solutions by mail, email, or fax. (So you can handwrite them and mail them in if you like, that's probably most common.)
DPatrick19:44:38
Then we'll read them and give you detailed comments on your solutions.
DPatrick19:44:59
In other words, we won't give you a numeric score...we'll actually critique your solutions and let you know how you can improve.
ppzuan19:45:04
after class, can we access the material covered in the course?
DPatrick19:45:19
Yes: each class session is transcripted, and you can access the transcripts on our website at any time.
DPatrick19:45:27
Also there will be a private message board just for the students in the class.
yenleo19:45:52
when will it start
DPatrick19:46:11
It starts on March 13 (2 weeks from today), and runs until May 29.
DPatrick19:46:29
Any other questions about the Trig/Complex Numbers course?
ppzuan19:46:46
how long will each class be?
DPatrick19:46:51
90 minutes.
FloodFilter:19:46:59
where are the transcripts located
DPatrick19:47:17
They'll be posted on our website.
DPatrick19:47:51
I'll take more questions at the end, but now let's move on to the Special AIME Problem Seminar.
DPatrick19:48:13
The Special AIME Problem Seminar is on Saturday, March 8, and Sunday, March 9 from 3:30 - 6:30 PM ET (12:30 - 3:30 PM PT) each day. During this class we will discuss both general test-taking strategies and specific mathematical tactics for the AIME. We will discuss several problems both from past AIMEs and from other challenging competitions that exemplify some of the most common problem-solving tactics on the AIME.
DPatrick19:48:35
The Saturday session will cover problems in Counting & Probability and Algebra, and will be taught by David Patrick (that's me). I was a 2-time Math Olympiad Summer Program invitee and a winner of the USA Math Olympiad back in high school. I have a Ph.D. in Mathematics from MIT and I'm the author of two of Art of Problem Solving's textbooks: Introduction to Counting & Probability and Intermediate Counting & Probability.
DPatrick19:48:48
The Sunday session will cover problems in Number Theory and Geometry, and will be taught by Valentin Vornicu. Valentin was a 2-time participant at the International Math Olympiad, and currently serves as a coordinator for the IMO. Valentin has a Master's degree in Mathematics from the University of Bucharest and founded the MathLinks website in 2002 (which merged with Art of Problem Solving in 2004).
chenhsi19:48:58
it says on the website that the seminar is 5 hours
chenhsi19:48:58
but you said that is is 3 hours per day
DPatrick19:49:08
Indeed...good attention to detail :)
DPatrick19:49:16
The class will have a 30-minute break in the middle of each 3-hour section. During the break time we won't do any problems, but the instructor will be available to discuss general test-taking strategies.
DPatrick19:49:38
(3 hours is a long time to sit in front of a computer without a break!)
DPatrick19:50:09
Let me discuss a couple of problems that are typical of those that we'll be covering in the seminar.
DPatrick19:50:14
xpmath19:50:46
6=7-1,8=7+1, Binomial Theorem?
ppzuan19:50:46
6 = 7-1 and 8 = 7+1
unimpossible19:50:49
6=7-1 and 8=7+1
DPatrick19:51:00
DPatrick19:51:12
ppzuan19:51:24
expand it
DPatrick19:51:36
Right, we can write out the expansions of (7-1)^83 and (7+1)^83:
DPatrick19:51:41
DPatrick19:51:47
DPatrick19:52:00
OK, that's icky. But what's pretty nice about it?
xpmath19:52:04
most of the terms are divisible by 49
unimpossible19:52:13
just look at the last 2 terms
DPatrick19:52:32
That makes us happy...most of these terms have factors of 7^2, so they're evenly divisible by 49.
DPatrick19:52:42
So for the purposes of solving the problem, we can ignore them!
DPatrick19:53:21
So all we have left is (83*7 - 1) from the first term, and (83*7 + 1) from the second term.
BOGTRO19:53:33
and we are adding them
DPatrick19:53:46
Right: we add them, the "1"s cancel, and all we have is 83 * 14.
DPatrick19:53:57
That's 1162.
BOGTRO19:54:04
so 35
DPatrick19:54:19
Yes: 1162/49 leaves a remainder of 35, which is the answer.
DPatrick19:54:36
Let's do one more, then I'll take questions.
xpmath19:54:49
that didn't seem too hard for an AIME problem
DPatrick19:54:55
It was a #6 I believe.
DPatrick19:55:00
The next one was a #9.
DPatrick19:55:06
xpmath19:55:31
the denominator (not reduced) is 1024
DPatrick19:55:59
Right, that's a good place to start. There are 2^10 = 1024 possible outcomes for the series of ten tosses. So that'll be our denominator, before taking "lowest terms" of course (otherwise the final answer would be too big!).
DPatrick19:56:06
How can we count the number of tosses that include no consecutive heads?
unimpossible19:56:33
casework: 5 heads, 4 heads, 3 heads, 2 heads, 1 head, 0 heads
DPatrick19:56:59
That's certainly a very valid solution method. If there are no consecutive heads, then we know that there are at most 5 heads, and we can count the possibilities using casework.
DPatrick19:57:14
And that's a perfectly fine solution.
DPatrick19:57:25
I'd like to discuss another approach though.
DPatrick19:57:57
One thing you might do, while exploring the problem, is to look at smaller cases.
DPatrick19:58:23
For example: one toss outcomes without HH:
H
T
DPatrick19:58:24
That's 23
DPatrick19:58:26
oops
DPatrick19:58:27
That's 2
DPatrick19:58:34
(my finger slipped there :))
DPatrick19:58:38
Two toss outcomes without HH:
HT
TH
TT
DPatrick19:58:41
That's 3
DPatrick19:58:47
Three toss outcomes without HH:
HTH
HTT
THT
TTH
TTT
DPatrick19:58:49
That's 5
DPatrick19:58:56
2,3,5,... might look a little familiar...
xpmath19:59:10
Fibonacci? (I can't spell)
BOGTRO19:59:10
Fibbonacci?
arkantosstevius19:59:10
fibonacci?
DPatrick19:59:22
And if we were still suspicious, we could do 4 tosses:
DPatrick19:59:28
Four toss outcomes without HH:
HTHT
HTTH
HTTT
THTH
THTT
TTHT
TTTH
TTTT
DPatrick19:59:30
There's 8.
DPatrick19:59:33
So now it's 2,3,5,8...
DPatrick19:59:51
So now it really looks like Fibonacci.
unimpossible19:59:55
oh its a recursion because it depends on the previous results
DPatrick20:00:17
Aha...can we set it up as a recursion so that we get the Fibonacci numbers?
DPatrick20:01:02
Can we say how "n flips without HH" equals "(n-1) flips without HH" plus "(n-2) flips without HH"?
xpmath20:01:38
after taking one away, we have a previous set?
unimpossible20:01:51
it can only be heads if the previous was not a head, so if a flip is a head, the one immediately after must be tails. then we have a conversion with Fn=F(n-1)+F(n-2)
DPatrick20:01:54
Right.
DPatrick20:02:11
If the first flip is H, then the second flip must be T, and the remaining (n-2) flips are a sequence without HH.
DPatrick20:02:27
If the first flip is T, then there's no such restriction, and the remaining (n-1) flips are a sequence without HH.
DPatrick20:03:02
charlestreykang20:03:12
then are we doing the fibonacci 9 times?
DPatrick20:03:28
Exactly. We started with a_1 = 2 and a_2 = 3, so we just extend this sequence until we get to a_10.
BOGTRO20:03:36
2,3,5,8,13,21,34,55,89,*144*
144/1024=72/512=36/256=18/128=9/64. so 73?
chenhsi20:03:39
144
DPatrick20:03:50
That's it. The 10th term (where the 1st term is 2 and the 2nd term is 3) is 144.
DPatrick20:03:55
We can now say that i/j = 144/1024 = 9/64, so i + j = 73.
DPatrick20:04:23
This is one of those problems that shows the usefulness of testing small cases and then observing the way that we generate those cases.
DPatrick20:05:02
By the way, the casework solution at the beginning would have worked out to another formulation of the Fibonacci numbers...you can work it out yourself for fun and see!
DPatrick20:05:13
Are there any questions about the Special AIME Problem Seminar?
arkantosstevius20:05:25
Is the special AIME seminar the same as the AIME course offered in the summer?
DPatrick20:05:35
It has much of the same problems.
DPatrick20:05:46
It is different from the class that just ended that ran during the school year, though.
arkantosstevius20:05:54
but the summer course covers more problems right?
DPatrick20:06:01
Right, the summer course is a full 12-week course.
DPatrick20:06:26
The weekend seminar is just a "crash course" of AIME-prep held right before the contest itself (the contest is on March 18).
arkantosstevius20:06:32
i wish to take the summer AIME course too, does that mean the seminar wouldn't be a wise choice?
DPatrick20:07:05
That's correct: if you're planning to take the summer course, then there's not really a good reason to take the weekend seminar (unless you really want to cram for this year's AIME for some reason).
xpmath20:07:12
hmmm...I have states on the 8th, so should I still do it?
DPatrick20:07:34
The two days are independent, so you can attend one and read the transcript of the other one. (The seminar will be transcripted, just like our regular courses.)
chenhsi20:07:44
about what are the range of the difficulty of the questions?
DPatrick20:08:13
All across the spectrum...each "unit" (for example, "Algebra") will start with easier problems and work up to harder ones.
DPatrick20:08:33
They'll generally start with a problem in the 3-5 range and by the end of the unit we'll be doing problems in the 11-15 range.
DPatrick20:09:06
The goal is not so much the specific problems but to give you an exposure to the variety of strategies that you can use to solve AIME problems in the different subjects.
DPatrick20:09:31
As I mentioned before, each unit will start with 5-10 minutes of general discussion of the different strategies applicable to that topic.
Quickster9420:09:41
the AIME is the 18th this yr rite>?
DPatrick20:09:57
The "primary" date is March 18. The alternate date is April 2.
DPatrick20:10:16
And if you didn't know, the AMC 10A/12A cutoff scores have kindly been posted by the AMC Director on our AMC forum.
etan20:10:56
where is that?
DPatrick20:11:30
Click "Community" on the website, then scroll down to the "American Mathematics Competitions" section (under "Contests & Programs").
DPatrick20:11:42
Actually, I'll just post the message here:
DPatrick20:11:53
The AIME invitation score for the 2008 AMC 10 A is 117.0
The AIME invitation score for the 2008 AMC 12 A is 97.5
DPatrick20:12:22
The 10B/12B cutoffs will not be available until next week at the earliest, and possibly not until the week after that.
DPatrick20:12:35
Any other questions about the upcoming courses?
ppzuan20:12:56
what is the cost for AIME seminar?
DPatrick20:13:06
$65
DPatrick20:14:20
Let me also add the disclaimer that we have on our website:
DPatrick20:14:26
This class is appropriate for students who are hoping to pass the AIME. If a student already consistently scores above 10 on the AIME, this class is probably not necessary, and if a student is unlikely to answer more than 1 or 2 questions correctly, then that students should start with some of our Introduction series of classes.
DPatrick20:14:41
In this context, "pass" means "qualify for the USAMO".
arkantosstevius20:14:51
which years' AIME problems will be covered during seminar? will the years be announced beforehand?
DPatrick20:15:09
There will be several different problems from many different years, and a few non-AIME problems that are similar in style to AIME problems.
DPatrick20:15:47
Are there any other questions?
arkantosstevius20:15:59
what's the AIME cutoff for the USAMO qualification?
DPatrick20:16:24
It varies from year to year, and for students beyond grade 10 the AMC score is also a factor. You can check the AMC website for past years' statistics.
DPatrick20:16:49
Usually it's around 7 or 8 for students in grade 10 and below. But it varies.
arkantosstevius20:16:56
will the years of the AIME questions covered be announced beforehand?
DPatrick20:16:59
No.
DPatrick20:17:14
They will cover all years. We've got around 25 years of AIMEs to work with.
etan20:18:28
are there any other classes to cover in this Math Jam?
DPatrick20:18:46
No, our other Spring classes were covered in the Math Jam last week; the transcript is available on our web site.
DPatrick20:19:02
(Click on the "Math Jams" tab under "Community", and then click "Transcripts" to access.)
DPatrick20:19:39
We'll have another full schedule of classes starting in June, and we'll have Math Jams in late May / early June to discuss them.
DPatrick20:20:10
OK...thanks for coming tonight, and if you have further questions you can post them on the message board (in the "Classes Information" forum), or send email to classes@artofproblemsolving.com
DPatrick20:20:13
Have a good evening!
