| Transcript
for the Math
Jam "AoPS Classes Math Jam"
on Sep 30. |
| Math Jam hosted by rrusczyk
(Richard Rusczyk ). |
rrusczyk19:32:03
Hello, and welcome to an Art of Problem Solving Math Jam. Today we'll be discussing the Introduction to Number Theory, Introduction to Geometry, and Intermediate Algebra courses. We will go through a couple example problems from each class, and discuss both what these classes cover and how they work.
rrusczyk19:32:17
My name is Richard Rusczyk. I founded Art of Problem Solving and have written several Art of Problem Solving textbooks.
rrusczyk19:32:25
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.
rrusczyk19:32:37
The classroom is moderated: students can type into the classroom, but only the moderators can choose a comment to drop into the classroom. So, when you send a message, it will not appear immediately, and may not appear at all. This helps keep the class organized and on track. This also means that only well-written comments will be dropped into the classroom, so please take time writing responses that are complete and easy to read. Also, only moderators can enter into private chats with other people in the classroom.
rrusczyk19:33:17
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. I will let you know when to start asking questions about the classes.
rrusczyk19:33:34
The virtual classroom is LaTeX enabled. LaTeX allows users to make nice equations and other math expressions. If you would like to learn how to write in LaTeX, click on the tab on the left side panel of our site and there is a tutorial and reference guide there.
rrusczyk19:33:57
You do not need to learn LaTeX to use our classes or our classroom!
rrusczyk19:34:08
Using LaTeX in the virtual classroom is slightly different than using it on the message board or in a LaTeX editor. If anything you type up in a post uses LaTeX, then you must use a semicolon (;) to begin your post. For example, if you type
rrusczyk19:34:14
rrusczyk19:34:17
This message will look like this when posted in the classroom:
rrusczyk19:34:21
rrusczyk19:34:24
Just remember, if your post uses LaTeX, use the semicolon (;) to begin your post!
rrusczyk19:34:39
Again, you don't need to know or use LaTeX for our classes, so you can ignore that if you want!
rrusczyk19:35:03
One last thing: we recommend not to use a wireless connection while in the classroom. These have a tendency to cause disconnections. Please use a wired connection if possible.
rrusczyk19:35:14
In this Math Jam, I will briefly describe a course, then go through a few example problems. Then, I will hold a question-and-answer session about that class.
rrusczyk19:35:35
Are there any questions about this online classroom? (I'll take questions about the classes in a few minutes.)
rrusczyk19:36:06
All right, let's talk about . . .
rrusczyk19:36:07
Introduction to Number Theory
rrusczyk19:36:11
Many students who already know how to solve MATHCOUNTS level problems about divisibility, base numbers, and divisor counting might think they have little need for this class, but many of the students who have made the most of this class were participants at national MATHCOUNTS and the AIME and found many of the problems discussed in class very challenging.
rrusczyk19:36:49
The Introduction to Number Theory class covers divisibility problems, clever uses of prime factorization, base numbers, linear Diophantine equations, the Euclidean Algorithm, and covers the mechanics of modular arithmetic in thorough detail. There are also many other topics covered in this course that are rarely mentioned in math books at all, but these topics become increasingly important as you move to higher mathematics.
rrusczyk19:37:24
For example, the topics covered in this course are crucial to an understanding of such areas as cryptography and computer science.
rrusczyk19:37:33
The following mini-lesson is excerpted from one of the Number Theory classes.
rrusczyk19:37:59
COUNTING DIVISORS
rrusczyk19:38:11
Once we know how to find the prime factorization of numbers, we can begin to use this tool to solve other problems.
rrusczyk19:38:21
One such problem is answering the question, 'How many positive divisors does a particular integer have?' This kind of counting problem is common in number theory.
rrusczyk19:38:34
rrusczyk19:38:50
Using this, what can we say about any divisor of 200?
notehead19:39:19
multiple of either 2 or 5
gatorlover12419:39:19
its a multiple of 2 or 5
tominator657819:39:19
it is a multiple of either 2 or 5.
spiderboy9519:39:19
It must be a multiple of either 2, 5 or both.
rrusczyk19:39:42
Does a divisor of 200 have to be a multiple of 2 or 5?
binmu19:39:58
It is either divisible by 2, 5, or both. Unless it is 1
FantasyLover19:39:58
it coule be 1
Twix19:39:58
no 1 isnt
notehead19:39:58
no, could be 1
rrusczyk19:40:03
Indeed - it could be 1.
rrusczyk19:40:24
So, to be precise, what can we say about a divisor of 200 by looking at its prime factorization?
BOBCA19:40:55
every factor or 200 is a multiple of 2 or 5 except fofr 1?
Twix19:40:55
that is either divisible by 2, 5, or 1
binmu19:40:55
It is a multiple of 2, 5 or it can be 1
rrusczyk19:41:04
OK - is every multiple of 2 a divisor of 200?
BOBCA19:41:15
no
notehead19:41:15
no
Twix19:41:15
no
Y4KUZ419:41:15
No
spiderboy9519:41:15
no
kealand19:41:19
no, like 198
BOBCA19:41:20
4000000000 isnt ::
rrusczyk19:41:30
There are a couple examples that are not divisors of 200.
rrusczyk19:41:40
So, let's look at that prime factorization again:
rrusczyk19:41:44
rrusczyk19:41:53
What does this tell us about the divisors of 200?
tominator657819:42:40
to be precise, they have to be powers of 2 or 5 multiplied, like 2^2 and 5^2 , which equals 100.
shentang_219:42:41
the divisors can have a most 2^3 and 5^2
notehead19:42:41
can have up to 3 2's and 2 5's
rrusczyk19:43:25
From our prime factorization of 200, we see that a divisor of 200 can only have 2's and 5's in its prime factorization. If a number has any other prime in its prime factorization, it can't evenly divide 200.
rrusczyk19:43:45
Now, let's look at these claims about a maximum number of 2's or 5's in the prime factorization.
rrusczyk19:43:48
FantasyLover19:44:17
because 200 has only 3 2's
_gir_19:44:17
too many 2s
Kevin Zhang19:44:17
200 isn't divisible by 16
gatorlover12419:44:17
it exceeds 2^3...
rrusczyk19:44:30
rrusczyk19:44:34
rrusczyk19:44:43
andrewnycpops19:45:08
that they must have up to 3 2's and up to 2 5's
cheeseater19:45:08
it must be an exponent of 2 from 0-3 multiplied by and exponent of 5 from 0-2
Y4KUZ419:45:08
when a is less than or equal to 3 and b is less than or equal to 2
FantasyLover19:45:08
a=0,1,2,3 b=0,1,2
cheeseater19:45:11
a = 0,1,2,3 b=0,1,2
spiderboy9519:45:14
a = 3,2,1,or 0
Twin Prime Conjecture19:45:14
a<4, b<3
rrusczyk19:45:17
rrusczyk19:45:35
Now that you know what the possible values for a and b are, how can you use this to count the total number of divisors of 200?
andrewnycpops19:46:10
4 x 3 = 12
FantasyLover19:46:10
a can be 4 numbers and b can be 3 numbers. 4*3=12. therefore 200 has 12 divisors.
Twix19:46:10
there would be 4 times three
tominator657819:46:10
multiply the possiblities of both numbers: 4x3=12
rrusczyk19:46:19
Consider this tree diagram:
rrusczyk19:46:26
cheeseater19:46:37
there are 4 possible values for a and 3 possible values for b therefore 4*3 = 12 positive integer divisors
rrusczyk19:46:39
Exactly.
rrusczyk19:46:45
rrusczyk19:46:57
Indeed we can see that this is true by listing the divisors:
1, 2, 4, 5, 8, 10, 20, 25, 40, 50, 100, and 200 are all the positive divisors of 200.
rrusczyk19:47:04
The reason we simply multiply the numbers of values for the exponents together is because we can select the values for each exponent independently from the values of the other exponent(s).
rrusczyk19:47:14
rrusczyk19:47:37
rrusczyk19:47:48
Any questions about that?
Mousepool19:48:00
(a plus 1)(b plus 1)
rrusczyk19:48:23
This is an interesting comment: Mouse suggests that we add 1 to each exponent, and multiply the results. Bold Claim! Let's take a closer look.
rrusczyk19:48:29
In general what can we say about the number of positive divisors of an integer n with a prime factorization
rrusczyk19:48:34
rrusczyk19:48:57
How can we find the number of divisors of this number?
FantasyLover19:50:14
Twin Prime Conjecture19:50:14
(e1+1)(e2+1)...
cheeseater19:50:14
(e1+1)(e2+1)...(em+1)
rrusczyk19:50:26
rrusczyk19:50:40
(You could also imagine a tree like the one we drew for 200)
rrusczyk19:50:47
rrusczyk19:50:56
rrusczyk19:51:11
To make this method more clear, we will now work through a couple exercises.
rrusczyk19:51:14
Exercise: How many positive divisors does 60 have?
rrusczyk19:51:21
Where do we start?
kealand19:51:36
prime factorization?
Y4KUZ419:51:36
prime factorization
Twix19:51:36
prime factorize it
gatorlover12419:51:36
prime factorization
rrusczyk19:51:42
What is the prime factorization of 60?
spiderboy9519:52:06
2^2 * 3 * 5
FantasyLover19:52:06
cheeseater19:52:06
2^2*3^1*5^1
Y4KUZ419:52:06
prime factorization is 2^2 x 3 x 5
tominator657819:52:06
2^2x3x5
Mousepool19:52:06
2^2 times 3 times 5
rrusczyk19:52:10
rrusczyk19:52:14
rrusczyk19:52:26
So, how many divisors does 60 have?
FantasyLover19:53:00
(2+1)(1+1)(1+1)=12
whowland19:53:00
12
binmu19:53:00
12 factors (3*2*2)
AARVIGI19:53:00
12
cheeseater19:53:00
(2+1)(1+1)(1+1)= 12
gatorlover12419:53:00
3 times 2 times 2
rrusczyk19:53:07
rrusczyk19:53:10
rrusczyk19:53:13
rrusczyk19:53:40
So, we have that little formula down.
rrusczyk19:53:50
But it's much more important to understand why the formula is true.
rrusczyk19:54:02
Then, you can do related problems that your formula might not solve so easily!
rrusczyk19:54:07
Like this one:
rrusczyk19:54:13
Find the number of positive integral divisors of 792 that are even.
rrusczyk19:54:17
This is a tougher problem than most similar problems at MATHCOUNTS. In fact, I took the last question on a sprint round and added 'that are even'. But I wanted you all to take a shot at a problem that requires a little more thought than plugging into a formula.
rrusczyk19:54:35
First, how many positive integral divisors does 792 have? (Just making sure you have our approach down.)
FantasyLover19:55:27
24
smoothiedood19:55:27
24
rrusczyk19:55:39
We start with the prime factorization:
shentang_219:55:41
2^3*3^2*11
rrusczyk19:55:49
rrusczyk19:55:59
rrusczyk19:56:08
OK, how do we count the number of even divisors?
tominator657819:56:27
you must have one two.
binmu19:56:27
each needs to have 2 as a factor
spiderboy9519:56:31
there has to be at least one two
shentang_219:56:33
all even divisors have to have a 2 as a divisor
rrusczyk19:56:38
How does this affect our count?
shentang_219:57:23
the exponent for 2 must be greater than 0
rrusczyk19:57:31
And how does that affect our count?
cheeseater19:57:39
we cannot have any values where the exponent of 2 is 0 so we must multiply (3+1-1)(2+1)(1+1) = 18
Eulers_Apprentice19:57:39
instead of 4*3*2, its 3*3*2
amacfie19:57:39
(3)(2+1)(1+1)
rrusczyk19:57:44
Exactly.
rrusczyk19:57:54
Here's what we did for any divisor:
rrusczyk19:58:00
rrusczyk19:58:06
tominator657819:58:10
3 x 3 x 2=18
rrusczyk19:58:19
rrusczyk19:58:38
Note: there are other ways to do this problem.
rrusczyk19:58:41
Here's one:
Twix19:58:43
subtract odd divisors
rrusczyk19:59:03
Count the odd divisors (I'll let you think about how on your own), and then subtract from our total of 24.
rrusczyk19:59:20
As the course continues we begin to discuss more difficult questions such as 'How many of the divisors of 360 are even?' and 'How many of the divisors of 9800 are perfect squares?'
rrusczyk19:59:26
Here are a few more kinds or problems that we will be tackling in class:
rrusczyk19:59:53
rrusczyk20:00:04
rrusczyk20:00:09
rrusczyk20:00:14
rrusczyk20:00:18
rrusczyk20:00:28
rrusczyk20:00:36
(We won't be doing these tonight!!)
rrusczyk20:00:46
The Introduction to Number Theory class will be taught by Naoki Sato. Naoki joined AoPS in 2005 after a successful career in investment banking. He won first place in the 1993 Canadian Mathematical Olympiad, and represented Canada at the 1992 and 1993 International Mathematical Olympiads, winning a bronze and silver medal, respectively. He has also served as deputy leader for the Canadian IMO team in 1997, 2002, and 2006. A native of Toronto, Canada, Naoki earned a Bachelor's in mathematics from the University of Toronto, and a Master's in mathematics from Yale University.
rrusczyk20:01:08
One more word about this number theory class. If you do not know modular arithmetic well enough to use it on most any problem up to AMC-12 level, this is a class that you would benefit from taking.
rrusczyk20:01:11
The course will meet for 12 weeks on Mondays, starting October 6. Each class starts at 7:30 PM Eastern / 4:30 PM Pacific, and is 90 minutes long.
rrusczyk20:01:19
Are there any questions about the course and how it will work?
spiderboy9520:01:22
So Number Theory is all about how numbers are related?
rrusczyk20:01:34
Mainly, about how *whole numbers* are related.
triangle20:01:38
will the pace of the class be so fast in actual classrooms
rrusczyk20:02:28
It will be a touch slower than this -- and students will have supplementary reading from the textbook that they should have read before class (which will make the class seem like it's slowed down a bit more).
kdange20:02:32
how will our assignments be graded?
rrusczyk20:03:13
We don't give a formal grade usually (but can on request). Instead, we give something more valuable -- specific feedback on the accuracy and the quality of the work you do on the two Challenge Sets that are given during the course.
spiderboy9520:03:16
I am also a bit unsure of how assignments will work.
coolislander20:03:16
are we gonna have tests?
rrusczyk20:03:26
There are two types of "homework".
rrusczyk20:03:57
1) Message board problems. After each class, we post 10-12 problems on the class message board for you to discuss with your classmates.
rrusczyk20:04:18
Instructors will chime in on these as well, but you will not be formally evaluated on them.
rrusczyk20:05:12
2) Challenge Sets. There are two thorough Challenge Sets in the course. You have 6 weeks to do the problems and write full solutions -- you will receive thorough feedback on the accuracy of your solutions, and on your writing style (you don't get a letter grade unless a school needs one)
paganina20:05:14
and what dates does it start and end?
rrusczyk20:05:41
The course will meet for 12 weeks on Mondays, starting October 6. Each class starts at 7:30 PM Eastern / 4:30 PM Pacific, and is 90 minutes long.
musicda20:05:51
if our questions are not answered,how do we know what the answer is?
triangle20:05:54
if we do not understand certain things is their a way to get it clarified i.e can i ask for explainign once more
rrusczyk20:06:31
Absolutely -- during class, we have an assistant in the classroom, so you should speak up any time you don't understand something. Sometimes, the instructor will deal with it in the main room. Other times, the assistant will work with you 1-on-1.
rrusczyk20:06:48
After class, you can post questions on the course message board at any time, and an instructor will answer your questions.
Twix20:06:51
do we need to buy the books
rrusczyk20:06:56
The books are required for the course.
coolislander20:06:57
if we need help with anything is there any way we can contact an instructer?
rrusczyk20:07:06
The course message board, which you can access at any time.
coolislander20:07:15
why do you signify *whole numbers*?
rrusczyk20:07:32
I mean, number theory is mainly the study of the integers, as opposed to "all real numbers"
triangle20:07:35
does the text book come with solutions manual
rrusczyk20:07:40
Yes, with full solutions (not just answers)
smoothiedood20:07:43
do you teach any of the classes?
rrusczyk20:07:50
Yes, but not in the Number Theory course.
Twin Prime Conjecture20:07:52
Do you offer anything equivialent to epgy stanford number theory(152) ?
rrusczyk20:08:12
I don't know anything about that course, so I don't know. I would imagine our Intermediate class has some overlap.
mitlh20:08:14
what is modular arithmetic?
rrusczyk20:08:35
Take the course or read the book and see :) We certainly don't have enough time tonight to explain it :(
mitlh20:08:37
can you hold these classes on another day besides monday?
rrusczyk20:08:53
Not this fall -- when we offer it again next summer, it will probably be on a different day.
rorys20:08:56
will as many people be in it?
rrusczyk20:09:02
Not sure - a few dozen.
rrusczyk20:09:14
Are there any more questions about how the class will work?
triangle20:09:53
for ur geometry course do i need to take number theory course
rrusczyk20:10:07
No - you can take geometry without a background in Number Theory.
cheeseater20:10:09
will it be in a chat room like this?
rrusczyk20:10:11
Yes.
rrusczyk20:10:19
Here is a description of why we run our classes like this:
rrusczyk20:10:38
Specifically, it explains the choice to have no audio.
weirda120:10:42
I accidentally logged off when you were explaining how we turn in our assignments.
rrusczyk20:10:47
I didn't explain that :)
rrusczyk20:10:50
Here's the explanation:
rrusczyk20:11:09
For the Challenge Sets, you'll receive instructions for how to mail, fax, or email your solutions.
rrusczyk20:11:21
For the message board problems, you simply post your solutions on the message board.
smoothiedood20:11:33
What level of difficulty is the course (like AMC 8, 10, Mathcounts, etc.)
rrusczyk20:12:12
It covers number theory problems in the AMC 8 through AMC 12 (including AMC 10 and MATHCOUNTS) difficulty range. There's probably a bit more emphasis on AMC 8/10 and MATHCOUNTS level than advanced high school questions.
xuy_9820:12:15
basically i can read the transcript if i miss the class. mostly i will miss the class since i have sports on fridays
rrusczyk20:12:30
That's exactly right -- there's a full transcript made for every class that you can read whenever you want.
Mousepool20:12:45
in the class, are the students not allowed to talk with each other like in the math jam
rrusczyk20:13:23
During the class, that is correct. Usually in each course, we allow open chat before and after class. Also, there are times during each day when the classroom is open for students to gather and collaborate on assignments and chat.
rrusczyk20:13:41
Let's do some geometry, and then I'll take more questions.
rrusczyk20:13:44
Introduction to Geometry
rrusczyk20:13:50
In the Introduction to Geometry class we cover all the fundamentals of geometry. We will start with a few days covering the basic tools such as triangle congruence, similarity, power of a point, relationships between angles and circles, etc., then dive into using those tools and more to solve increasingly difficult problems.
rrusczyk20:14:00
Most of the problems in the course will be at the MATHCOUNTS and AMC-10 level of difficulty, but we will be throwing in a few harder problems occasionally to show how to use very basic ideas to solve very challenging problems.
rrusczyk20:14:12
Let's see a couple problems.
rrusczyk20:14:22
In the diagram shown, DEOC is a square. The radius of circle O is 6 in. What is the number of inches in AC? Express your answer in simplest radical form.
rrusczyk20:14:44
rrusczyk20:14:52
In order to find AC, what will we need?
_gir_20:15:12
CO
coolislander20:15:12
length of CO
amacfie20:15:12
CO
rrusczyk20:15:26
A note: please use capital letters when referring to points.
rrusczyk20:15:29
We need AO, which we already have, and OC. How can we find OC?
rrusczyk20:15:35
kealand20:16:20
we can use the line DO, which is the radius of the circle to get CO
maxmk20:16:20
Because DEOC is a square
BOBCA20:16:20
and since OCDE is a sqaure, OED is a icoceles triangle
Y4KUZ420:16:20
45-45-90 triagle
rrusczyk20:16:48
OC is a side of square OCDE. OD is a diagonal of this square. OD is also a radius of the circle, so OD = AO = 6.
rrusczyk20:16:54
So, what is OC?
Y4KUZ420:17:16
and OC would be 3 rad2
BOBCA20:17:16
or OC is 3 sqrt 2
kealand20:17:16
3 sqrt2
Twin Prime Conjecture20:17:16
6/sqrt2
gatorlover12420:17:18
3*square root of 2
shentang_220:17:18
3sqrt2
rrusczyk20:17:24
rrusczyk20:17:26
Because the diagonal has length 6 and OC is a side, we have:
rrusczyk20:17:29
rrusczyk20:17:38
So, what is AC?
gatorlover12420:18:16
3*square root of 6!
tominator657820:18:16
3 sqrt 6
kealand20:18:16
3 sqrt 6
cheeseater20:18:16
3radical(6)
Y4KUZ420:18:16
3 rad 6
rrusczyk20:18:20
We have a right triangle, so we use the Pythagorean Theorem:
rrusczyk20:18:26
rrusczyk20:18:34
rrusczyk20:18:51
Let's try one more.
rrusczyk20:18:56
Sector OAB is a quarter of a circle of radius 3 cm. A circle is drawn inside this sector, tangent at three points as shown. What is the number of centimeters in the radius of the inscribed circle? Express your answer in simplest radical form.
rrusczyk20:19:02
rrusczyk20:19:16
We need the radius of the small circle, so what should we do with our diagram?
Y4KUZ420:19:29
draw it
rrusczyk20:19:33
Let's draw the radius. Where should we draw it?
Y4KUZ420:19:59
tangent to the sides
gatorlover12420:19:59
from the tangent points
Twin Prime Conjecture20:19:59
to the tangent point
Y4KUZ420:20:11
well touching the sides wheere the small circle meets the sector
rrusczyk20:20:12
Draw a radius to each point of tangency and label each with length r. We like to draw radii to points of tangency because we get right angles.
rrusczyk20:20:19
rrusczyk20:20:26
Now what?
rrusczyk20:21:02
Notice anything interesting?
gatorlover12420:21:14
the length of the square = the radius...
_gir_20:21:16
square
rrusczyk20:21:21
We notice that HOGP is a square. How will the square help us?
rrusczyk20:21:28
kealand20:21:35
find PO (assuming there was a line)
gatorlover12420:21:48
45-45-90 ratio
_gir_20:21:48
PO
rrusczyk20:21:49
Interesting. In terms of r, what is PO?
Y4KUZ420:22:07
r rad 2
AARVIGI20:22:07
r radical 2
maxmk20:22:07
sqrt(2)r
rrusczyk20:22:18
Because the side length of the square is r, the diagonal is r*sqrt2.
rrusczyk20:22:23
thing120:22:27
are F P and O colinear?
rrusczyk20:22:40
Very interesting question. Are they?
Y4KUZ420:23:13
yes
binmu20:23:13
yes
maxmk20:23:13
yes
Twix20:23:13
yes
gatorlover12420:23:13
YES!
kealand20:23:26
they should be since the quarter circle is symmetrical
rrusczyk20:23:32
Yes--by symmetry, both P and F are equidistant from AO and BO. So, O, P, and F are collinear, which means they all are on the same line.
rrusczyk20:23:37
rrusczyk20:23:41
How does this help?
Twix20:24:20
and add PO to r and let it equal 3
Twix20:24:20
so you can add FP and PO and let it equal 3
mathbear_220:24:20
r+r sqr2 =3
BOBCA20:25:03
r+r root 2 = 3
rrusczyk20:25:06
We set up an equation and solve it. We know that OF is a radius, so OF = 3. However, we can write OF = OP + FP = r*sqrt(2) + r.
rrusczyk20:25:14
Now we have an equation and we can solve for r. What do we get?
Twin Prime Conjecture20:26:31
r=3sqrt2-3
cheeseater20:26:31
r = 3/(rad(2)+1)
maxmk20:26:37
3/(sqrt2+1)
rrusczyk20:26:40
Let's see who is right.
rrusczyk20:26:47
Here's how we can solve the equation:
rrusczyk20:27:27
rrusczyk20:27:34
rrusczyk20:27:45
rrusczyk20:28:08
gatorlover12420:28:26
no
musicda20:28:26
no
AARVIGI20:28:33
no. they rationalized the denominator
_gir_20:28:33
no
gatorlover12420:28:35
rationalize the denominator!
mathbear_220:28:35
no. It is simplified
rrusczyk20:28:52
We multiply both the top and bottom of our value of r by 1-sqrt2 to rationalize the denominator.
rrusczyk20:28:56
rrusczyk20:28:59
Notice that we don't just sit and stare at the problem and wait for it to solve itself. We have to add lines and variables so we can build equations.
rrusczyk20:29:05
Now, here's a very interesting question:
thing120:29:07
is it possible to prove that F,P,O must be colinear?
rrusczyk20:29:17
rrusczyk20:29:28
We kind of ducked this question by arguing symmetry.
rrusczyk20:29:37
The more satisfying proof is somewhat subtle.
rrusczyk20:30:08
First, it should be clear that P is the same distance from AO and BO (the distance from P to each is the radius of the little circle).
rrusczyk20:30:17
So, P is on the angle bisector of <AOB.
rrusczyk20:30:42
What's not as clear is why F must be equidistant from AO and BO.
kealand20:31:06
F is the point of symmetry, the arcs are equal
rrusczyk20:31:08
Why?
rrusczyk20:31:19
(Why are the arcs equal?)
rrusczyk20:32:48
I see a lot of people arguing "symmetry" or "because F is the tangent point"
rrusczyk20:32:53
Can anyone be more specific?
rrusczyk20:33:04
rrusczyk20:33:44
I see some of you arguing that the arcs AF and BF are equal because FO bisects angle AOB.
rrusczyk20:33:52
Note that this is circular logic!
rrusczyk20:34:17
Here's a question. Imagine that F were not on the angle bisector of OP.
rrusczyk20:34:26
Then what could we conclude?
rrusczyk20:36:09
Again, several of you are telling me that <AOF is 1/8 of the circle -- you are assuming what we are asked to prove. You can't do that!
cheeseater20:36:21
Then circle P would not have 3 tangent points
rrusczyk20:36:37
If F were not on the angle bisector of OP, then consider the reflection of F over OP. Let's call that point Y. This point would then be on both the little circle and the arc. So, the two intersect at two points, which is impossible becasue they are tangent.
rrusczyk20:36:46
Here's another way to look at it:
thing120:36:48
extend OP to F' such that PF'=PF then uses triangle inequality to argue that F' is outside the circle
rrusczyk20:37:15
That last proof was a bit of a digression.
rrusczyk20:37:28
There are a few proofs in the class that are that subtle.
rrusczyk20:37:56
The first of those two problems is on the easy end of problems we will discuss. The second is a bit easier than average (except for the proof at the end, which I just added on because thing1 asked such an interesting question). All the geometric tools we use to solve problems, such as all the special relationships we used to solve these two problems today, will be taught in the class. We don't expect students to have any background knowledge in geometry.
rrusczyk20:38:05
You can find more questions like those we cover in the course by checking out the Post Test for the course here:
rrusczyk20:38:13
The course will meet for 24 weeks on Fridays, starting October 10, at 7:30 PM Eastern / 4:30 PM Pacific. Each class is 90 minutes.
rrusczyk20:38:23
The course will be taught by Joshua Zucker. Joshua joined AoPS as a part-time instructor in 2007. He discovered his love for number theory at Dr. Arnold Ross's summer program at Ohio State University a bit over 20 years ago. Joshua has been a Math Olympiad Summer Program invitee, a member of the first US Physics Olympiad team, and a top-10 scorer on the Putnam. He holds a BS in physics and an MS in mathematics from Stanford, as well as an MS in astrophysics from UC Berkeley. He has taught at levels ranging from summer camps for gifted elementary school students through remedial arithmetic at community college. He was a middle and high school teacher in Palo Alto, CA, for ten years was a problem writer for MATHCOUNTS, and is the proud father of three children
rrusczyk20:38:53
This course will use a textbook in conjunction with the course: our own Introduction to Geometry book. The material covered in the textbook is roughly equivalent to the material covered in the course. You can see the table of contents and some excerpts from the book here:
rrusczyk20:39:13
The book is required for the course. Students will be able to read additional material that complements the lectures, and will have access to a large number of practice problems at varying levels of difficulty. We recommend that students read the corresponding chapter(s) in the book before each lecture, and attempt some of that chapter's Review and Challenge Problems after each lecture.
rrusczyk20:39:23
The homework for the class consists of weekly problems that will be posted to the class message board -- for these problems, you do not turn your solutions in, however you may post them to the message board if you like. The class also has 4 longer problem sets for which you should write up your full solutions and submit them. These solutions will be read, and you will receive detailed feedback.
rrusczyk20:39:34
Are there any questions about this class or the textbook?
spiderboy9520:40:02
How much homework will we have?
rrusczyk20:40:04
You should expect to spend 5-7 hours a week on the course. What you get out of the course will depend a great deal on what you put into it.
musicda20:40:06
do you give us the challenge sets, or do we have them in the textbooks?
rrusczyk20:40:21
We post them on the class home page (which is also where the class transcripts are).
binmu20:40:23
what pace is it like
rrusczyk20:40:52
It is a *much* faster-paced class than what you have in your usual school classes. It is more like tonight (maybe a tiny bit slower than tonight).
Mousepool20:40:57
are there message board problems for this class as well
rrusczyk20:40:59
Yes.
coolts20:41:11
will there be any large exams? (tests? quizzes?)
rrusczyk20:41:18
There are no timed tests or quizzes.
rrusczyk20:41:27
Just the Challenge Sets and Message Board problems.
Fly_DIA20:41:29
How strict are the deadlines?
rrusczyk20:41:47
For the Challenge Sets, pretty strict. A couple days here and there may not be a big deal, but weeks are.
spiderboy9520:41:51
How about exams? Is there a "final", or like a midterm?
rrusczyk20:42:07
Only the Challenge Sets and the Message Board problems. No timed finals.
Twin Prime Conjecture20:42:09
Is there a grade for this course?
rrusczyk20:42:21
If you need one, then we can assign one, but you should tell us at the start of the course.
rrusczyk20:42:37
(Email classes@artofproblemsolving.com to let us know)
binmu20:42:39
How will we know if we got problems right
rrusczyk20:42:45
You'll be given solutions for everything.
rrusczyk20:43:00
And we will give you thorough feedback on the Challenge Sets.
coolislander20:43:04
Is their a placement test or anything to see whta level we're at?
rrusczyk20:43:27
Yes, there are pre-tests and post-tests for each class here:
binmu20:43:30
Will we go over the problems in class?
rrusczyk20:43:50
Not for the Challenge Sets, since everyone will have the solutions. If you have questions about them, you can ask them on the class message board.
Mousepool20:43:52
will students be able to draw up geometric drawings like you did today
rrusczyk20:44:16
For their homeworks, yes. There's not a whiteboard to draw on in the classroom (though we hope to implement one in the next year).
spiderboy9520:44:29
Just one question (and I realize I am going a little off-topic here) but is there an age range for this class? Like, what grade are most of the other students in?
rrusczyk20:44:38
Most of the students in the geometry class are grades 7-10.
rrusczyk20:44:58
Any more questions about the geometry class?
rrusczyk20:45:17
Let's do some . . .
rrusczyk20:45:18
Intermediate Algebra
rrusczyk20:45:21
Our Intermediate Algebra class contains much of the algebra of a typical Algebra II class, all of the non-trig, non-matrices algebra of a typical precalculus class, plus a number of advanced topics that are excluded from the standard curriculum.
rrusczyk20:45:31
The course starts with a review of linear and quadratic equations, functions, and complex numbers, then goes on to cover conics, polynomials, advanced factoring techniques, classical inequalities, techniques for solving hard systems of equations, symmetric polynomial sums, sequences and series, identities and induction, greatest and least integer functions, advanced methods for dealing with logarithms, functional equations, and much more.
rrusczyk20:46:06
The textbook for the course is our new Intermediate Algebra text, by Richard Rusczyk and Mathew Crawford. The text is required for the course.
rrusczyk20:46:26
I'll now proceed with a couple more challenging problems from the course. If you find these problems very, very easy, then you might be too advanced for the course. If you find them challenging but not completely impossible, then the class is probably a good fit for you.
rrusczyk20:46:32
rrusczyk20:47:08
(Note: if you were here for the Intro class discussions, these problems might be too hard for you. Don't panic -- your time will come.)
rrusczyk20:47:13
Where might we start?
Twin Prime Conjecture20:47:15
add these 2 equations
rrusczyk20:47:21
rrusczyk20:47:27
How does this help?
rrusczyk20:48:00
What can we do with this equation that might be useful?
gatorlover12420:48:12
you can factor the x^3 + y^3
AARVIGI20:48:12
factor
Twin Prime Conjecture20:48:22
factor out x+y
rrusczyk20:48:27
rrusczyk20:48:39
OK. Not yet clear how that will help, but it's something.
rrusczyk20:48:49
Let's look back at the problem:
rrusczyk20:48:50
gatorlover12420:48:53
and the second part(x^2 - xy + y^2) is useful
rrusczyk20:49:16
Indeed, we have information now about squares of x and y. But we're still not done. What else might we have done with the original equations?
Twin Prime Conjecture20:49:28
then subtract these equations
rrusczyk20:49:35
rrusczyk20:49:39
Why is this helpful?
Twin Prime Conjecture20:50:00
factor
cheeseater20:50:00
factor out the x-y
rrusczyk20:50:09
Factoring worked once, let's try it again.
rrusczyk20:50:13
rrusczyk20:50:18
gatorlover12420:50:20
and the second part of that equation is x^2 + xy + y^2
rrusczyk20:50:27
Aha!
rrusczyk20:50:28
rrusczyk20:50:34
These are the two equations we just found.
rrusczyk20:50:39
What can we do with these?
Twin Prime Conjecture20:50:56
subtract
AALAP20:50:56
add them
rrusczyk20:51:01
Which should we do?
Mousepool20:51:05
subtract
mathbear_220:51:05
Add
rrusczyk20:51:17
Which one?
binmu20:51:31
add them
amacfie20:51:31
subtrace
binmu20:51:31
add
Mousepool20:51:31
subtract
rrusczyk20:51:42
This is fun. Still some difference of opinion.
rrusczyk20:51:44
So . . .
rrusczyk20:51:47
Let's do both!
rrusczyk20:51:54
mathbear_220:51:56
Subtraction gets rid of the x^2 and y^2
rrusczyk20:52:04
rrusczyk20:52:42
Time to go back and look at what we wanted.
rrusczyk20:52:46
Here's the problem again:
rrusczyk20:52:50
rrusczyk20:53:17
rrusczyk20:53:36
(Note that we don't have to actually find x and y. We need (x^2 - y^2)^2)
gatorlover12420:53:41
yes
AALAP20:53:41
yes
Mousepool20:53:41
yes
rrusczyk20:53:46
gatorlover12420:54:26
x^4 -2x^2y^2 + y^4
Mousepool20:54:31
x^4 - 2x^2 y^2 plus y^4
rrusczyk20:54:33
rrusczyk20:54:53
Hmm. Not quite there. How can we write that right side in terms of x^2 + y^2 and xy (since we know what these are).
Twin Prime Conjecture20:56:08
x^2+y^2=15
rrusczyk20:56:17
What can we do with this to get x^4 and y^4?
cheeseater20:56:28
add 4x^2y^2
rrusczyk20:56:43
This is an interesting idea (this refers to our earlier equation):
rrusczyk20:56:45
rrusczyk20:56:56
When we add 4x^2y^2 to this, we get
rrusczyk20:57:13
rrusczyk20:57:18
How does this help?
mathbear_220:57:41
(x^2+y^2)^2
rrusczyk20:57:46
The right side is just (x^2 + y^2)^2!
cheeseater20:57:50
it equals (x^2+y^2)^2
rrusczyk20:58:06
We have
rrusczyk20:58:07
rrusczyk20:58:24
Aha! Now, can we finish? Can we find (x^2 - y^2)^2
cheeseater20:59:08
15^2 = 225
rrusczyk20:59:25
We know (x^2 + y^2) = 15, so (x^2 + y^2)^2 = 225.
rrusczyk20:59:29
Now, we have
rrusczyk20:59:40
rrusczyk20:59:45
Keep going!
gatorlover12421:00:25
4x^2y^2 is 2xy squared
rrusczyk21:00:29
We know that xy = -4, so 4x^2y^2 = 4(-4)^2 = 64.
rrusczyk21:00:35
So, what is (x^2 - y^2)^2?
AALAP21:00:50
161
Twin Prime Conjecture21:00:50
161=225-64
rrusczyk21:00:53
rrusczyk21:01:00
The key to this problem was simply working with the information that we were given long enough to see the factorizations that we know how to perform. A couple of times we cleverly added and subtracted equations to create new equations that we could manipulate in ways we know how.
rrusczyk21:01:30
We'll look at one more problem.
rrusczyk21:01:33
rrusczyk21:01:42
This problem comes from the AIME.
rrusczyk21:01:53
That's a link to the problem that you can refer back to.
rrusczyk21:01:57
We have a very wordy problem. We'll have to convert those words to equations.
rrusczyk21:02:11
The sentence that we have to focus on is, "This function has the property that the image of each point in the complex plane is equidistant from that point and the origin." We have to turn this into an equation somehow. We'll do it step-by-step.
rrusczyk21:02:22
In terms of f and z, what is "the image of the point z"?
rrusczyk21:03:18
What mathematical expression means "the image of the point z" when we apply the function f?
rrusczyk21:03:50
The image is what you get when you apply the function.
rrusczyk21:04:07
So, all this phrase means is f(z). Nothing fancy here.
rrusczyk21:04:15
The image of the point z under the function f is simply f(z). So, we can replace "the image of each point" with f(z) and "the point" with z. This leaves us the sentence, "This function has the property that f(z) is equidistant from z and the origin."
rrusczyk21:04:31
All we are doing here is converting the words in the problem to algebraic expressions.
rrusczyk21:04:52
Now, what is an expression for the distance between f(z) and z in the complex plane?
rrusczyk21:06:00
Let's take a step back.
rrusczyk21:06:18
Suppose z is a complex number. What is the distance between z and the origin on the complex plane?
Twin Prime Conjecture21:07:06
the modulus
rrusczyk21:07:08
The distance between z and the origin is just the magnitude of z, which we write as |z|.
rrusczyk21:07:22
We can also write this as |z - 0|.
rrusczyk21:07:46
Similarly, if w and z are two different complex numbers, then the distance between them on the complex plane is |w-z|.
rrusczyk21:08:05
So, I ask again, what is an expression for the distance between f(z) and z in the complex plane?
Twin Prime Conjecture21:08:37
|f(z)-z|
rrusczyk21:08:48
The distance between f(z) and z is |f(z) - z|.
rrusczyk21:08:51
And the distance between f(z) and the origin?
Twin Prime Conjecture21:09:18
|f(z)|
Mousepool21:09:18
l f(z) - 0 l
cheeseater21:09:23
|f(z)-0|
rrusczyk21:09:27
This is simply |f(z)|. So, what is our equation?
rrusczyk21:09:52
Here was the problem again:
rrusczyk21:09:57
rrusczyk21:10:40
With all that we've just learned, what equation can we write with the information "the image of each point in the complex plane is equidistant from that point and the origin?"
rrusczyk21:12:01
"the image of each point is the same distance from that point as it is from the origin"
rrusczyk21:12:11
We just found expressions for both of these distances:
mathbear_221:12:13
I f(z) I =I f(z)-z I
rrusczyk21:12:20
rrusczyk21:12:36
All we did here was convert the words in the problem to an equation. There was nothing magic here.
rrusczyk21:12:51
And yet it is probably the hardest and most important part of many AIME and Olympiad problems.
rrusczyk21:12:59
(And is probably the hardest part of this problem).
rrusczyk21:13:08
Now that we have this equation, what can we do with it?
rrusczyk21:13:41
rrusczyk21:13:49
Anyone see a way to simplify this?
mitlh21:14:46
z(a+bi-1)=z(a+bi)
rrusczyk21:14:52
We can factor the left side:
rrusczyk21:15:09
rrusczyk21:15:19
What can we do with this equation to cancel it?
rrusczyk21:15:37
*simplify it, I mean (a little hint slipped out there :) )
mitlh21:15:42
cancel out the z
rrusczyk21:15:45
rrusczyk21:16:04
Now what?
rrusczyk21:16:31
Can we get rid of the absolute value signs now? How can we write |c+di| for any complex number?
rrusczyk21:17:21
rrusczyk21:17:45
It's basically the distance from c+di to the origin on the complex plane, just like |x| is the distance from x to 0 on the number line.
rrusczyk21:17:59
So, we apply this definition to the equation we had above.
rrusczyk21:18:04
rrusczyk21:18:18
However the problem asked us to find the value of b^2. Is there any piece of information in the problem that we haven't used yet? (Whenever you're stuck on a long, complicated problem, this is a good question to ask yourself: What haven't I used yet?)
AARVIGI21:18:29
the question gave the value of (a + bi)
rrusczyk21:18:38
rrusczyk21:18:51
How can we use this along with a=1/2 to finish the problem?
rrusczyk21:19:36
gatorlover12421:19:48
so plug it in
rrusczyk21:20:03
Exactly. We plug it in and crank it out and get:
mathbear_221:20:04
b^2 =255/4 =m/n
rrusczyk21:20:09
rrusczyk21:20:40
Now, in the regular Intermediate Algebra class, you would have spent much of a class developing the ideas of |a+bi| and the complex plane before seeing this problem.
rrusczyk21:21:01
I can tell that very few of you in the class tonight have seen these ideas before (which is fine! Have to learn some time).
rrusczyk21:21:14
So, in the usual Intermediate Algebra class, this problem would not have seemed so impossible.
rrusczyk21:21:32
I mainly include it here to give you an idea where you'll be if you take the course and pursue it seriously.
rrusczyk21:21:36
The Intermediate Algebra class also involves a variety of other algebraic topics including methods of substitution, functions, polynomials, sequences and series (including the use of difference equations), binomial expansion, logarithms, advanced systems of equations, and greatest/least integer functions. Here are a few harder problems we will tackle in the course:
rrusczyk21:21:41
rrusczyk21:21:45
rrusczyk21:21:51
rrusczyk21:21:58
rrusczyk21:22:04
rrusczyk21:22:10
The course will meet for 24 weeks on Thursdays, starting October 9. Each class starts at 7:30 PM Eastern / 4:30 PM Pacific, and is 90 minutes long.
rrusczyk21:22:16
The course is taught by Valentin Vornicu. Valentin joined AoPS in 2004. He founded the MathLinks site for Olympiad students in 2002. Valentin was a Gold Medal and Special Award winner at the Junior Balkan Math Olympiad in 1998 and a Silver Medal winner at the Balkan Math Olympiad in 2001. He also took 2nd Prize at the IMC in 2003 and 2004. He won first or second prize in the Romanian National Olympiad 1997-2002, and was a member of the Romanian International Mathematical Olympiad (IMO) team in 2001 and 2002, winning a bronze medal in 2002.
yaor89021:22:19
Is intermediate Alg similar to Alg2?
rrusczyk21:23:07
It has all the algebraic non-trig, non-vector material of Algebra 2 and Precalculus, plus a lot harder and more advanced material (for example, you won't see that problem we just did in any typical Algebra 2 class or Precalculus class).
thing121:23:09
is it still worthwhile to take the class if we have read the textbook? Is there material in the course thats not covered in the textbook?
rrusczyk21:23:38
It depends on how thoroughly you have gone through the book. If you're at the point where you can do many of the Challenge Problems, then you surely don't need the class.
thing121:23:40
at what level does the course begin at easy amc hard amc easy aime etc
rrusczyk21:24:14
There are problems that you might call easyish AMC at the start of each class, just to illustrate the basic concept. The class tops out at beginning Olympiad level.
Mousepool21:24:17
what courses do you teach
rrusczyk21:24:24
This fall, I'll be teaching Introduction to Algebra.
mathbear_221:24:27
how much time is required to spend on this course and do the challenge sets?
rrusczyk21:25:22
I recommend that to get the most out of the class you should expect to spend at least 5-7 hours a week on the course. The students who get the most out of the course tend to spend a little bit more. They're also the ones who do the assigned reading before class, work on lots of extra problems, etc.
Twin Prime Conjecture21:25:24
do you do hard AIME problems in this course?
rrusczyk21:25:46
Yes. The second problem we did tonight is a mid-level AIME problem, for example (I think maybe a number 8 or 9).
rrusczyk21:25:57
There are plenty of these sprinkled throughout the class and book.
rrusczyk21:26:00
Any more questions?
AARVIGI21:26:17
Why is the 'Do You Need This?' for Intro Algebra more difficult than the 'Are You Ready?' for the Intermediate Algebra?
rrusczyk21:27:06
Because you go a little bit farther in the Intro Algebra class than you absolutely need for Intermediate Algebra. Also, because students usually overestimate how much they know based on having already taken certain classes in school (which don't correlate very well to ours).
musicda21:27:10
Are there transcripts of the MathJams
rrusczyk21:27:20
Yes; they are in the Math Jams section of the website.
Mousepool21:27:37
on the course: Mastering Mathcounts/ AMC 8, around what level of Mathcounts would they be? (Chapter, State, National)
rrusczyk21:27:48
Mostly state and national.
rrusczyk21:28:44
Any more questions?
Twin Prime Conjecture21:29:15
Do we get the materials needed for USAMO in this course?
rrusczyk21:29:34
Not for every algebra problem on the USAMO, but you'll certainly get some tools that will help with some USAMO problems.
Mousepool21:29:36
is there an intro to probability course
rrusczyk21:29:46
Yes, but not this fall. It will next be offered in spring, 2009
thing121:30:13
Will we be studying advanced techniques for solving systems of equations ie parameters
rrusczyk21:30:40
There are some advanced techniques for systems of equations in Interm Algebra, and others in the Intermediate Trig/Complex Numbers class.
rrusczyk21:31:30
Any more questions before we call it a night? If you think of questions later, you can email them to classes@artofproblemsolving.com
musicda21:32:16
Does Introduction to Alg and Intermediate algebra replace algebra 1 & 2 or complement them?
rrusczyk21:33:08
It's much harder than the typical school sequence Algebra 1 and Algebra 2. A student who has taken our Intro and Intermediate Algebra (and done the reading and work, etc), should not need to take Algebra 1 or Algebra 2 (and will find most of precalculus in a regular class pretty simple, too)
kelleyzhao21:33:34
AMC 10 NEED THIS INTER ALG?
rrusczyk21:34:10
There might be one question at most on the AMC 10 that requires information we teach in Intermediate Algebra. Introduction to Algebra should be enough for the AMC 10.
Y4KUZ421:34:14
Would taking alg 2 and intermediate alg at the same time conflict or benefit from each other?
rrusczyk21:34:45
It wouldn't 'conflict'. To the extent you work hard in our class, it will make your algebra 2 class much easier.
Twin Prime Conjecture21:34:47
Does this course and intermediate trig/complex number course overlap?
rrusczyk21:35:01
Only about 1/2 day of the introductory complex number material, at most.
rrusczyk21:35:28
Thank you all for coming. That's it for tonight. If you think of classes later, you can write us at classes@artofproblemsolving.com.
