Transcript for the Math Jam "AoPS Intro Geometry and AMC 10" on Sep 27.

Math Jam hosted by rrusczyk (Richard Rusczyk ).

DPatrick (19:28:48)
Hello, and welcome to the first AoPS Fall 2006 Classes Math Jam!

DPatrick (19:28:55)
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.

DPatrick (19:29:05)
The classroom is moderated: students can type into the classroom, but only the moderators can choose a comment to drop into the classroom. 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.

DPatrick (19:29:22)
In addition, the virtual classroom is TeX enabled. TeX allows users to make nice equations and other math expressions. If you would like to learn how to write in TeX/LaTeX, click on the tab on the left side panel of our site and there is a tutorial and reference guide there.

DPatrick (19:29:38)
Using TeX in the virtual classroom is slightly different than using it on the message board or in a TeX editor. If anything you type up in a post that uses TeX then you must use a semicolon (;) to begin your post. For example, if you type

DPatrick (19:29:44)


DPatrick (19:29:51)
This message will look like this when posted in the classroom:

DPatrick (19:29:55)


DPatrick (19:30:15)
Just remember, if your post uses TeX, use the semicolon (;) to begin your post!

DPatrick (19:30:26)
Today we will be discussing our upcoming fall classes:
- AMC 10 Problem Series (starts October 13)
- Introduction to Geometry (starts October 5)

DPatrick (19:30:37)
AMC 10

DPatrick (19:30:44)
The AMC 10 class starts on October 13, and meets every Friday from 7:30-9:00 PM Eastern, and lasts 12 weeks. Naoki Sato will be the instructor.

DPatrick (19:30:57)
This class is a Problem Series class, meaning that the major focus of the class will be working through various AMC 10 or AMC 12 problems. Although there will be weekly problem sets for each class, students do not submit their homeworks to be graded, and there is no individual instructor feedback. (As a result, these classes are somewhat less-expensive that our regular subject classes.)

DPatrick (19:31:10)
The AMC 10 Problem Series is a 12-week course designed to cover a large portion of the curriculum tested on the AMC 10 exam.

DPatrick (19:31:20)
The following are excerpts of a couple of the areas of problem solving covered in the AMC 10 Problem Series.

DPatrick (19:31:31)
ARITHMETIC SEQUENCES

We have all seen many arithmetic sequences, but I would like you to pay close attention to the ways in which we manipulate facts according to our understanding of arithmetic sequences in the following problems. In particular, arithmetic sequences involve common differences that are constant. Constants are our friends and we should remember how useful they can be.

DPatrick (19:31:39)


DPatrick (19:32:02)
How can we use the given information to find the number of sides of the convex polygon?

DPatrick (19:33:09)
Can we list the angles in this polygon?

lyra (19:33:15)
yes

DPatrick (19:33:27)
What are they?

DPatrick (19:34:35)
For example, we know that the largest is 160 degrees. What's the next largest?

lyra (19:34:42)
155

galbatorix (19:34:43)
155

HoratioK (19:34:46)
155 degrees

DPatrick (19:34:57)
Right, the first few angles are 160, 155, 150, and so on.

DPatrick (19:35:00)
What's the smallest angle?

HoratioK (19:35:11)


DPatrick (19:35:41)
Right. The smallest is 165-5n, which is the same as 160-5(n-1). (We subtract 5 n-1 times to get the smallest.)

DPatrick (19:35:53)
Now, how can we use these values?

lyra (19:36:14)
we need an equation...

DPatrick (19:36:38)
Right...what equation can we form?

DPatrick (19:36:54)
What other information could we use in conjunction with these angle values?

lyra (19:37:05)
convex polygon

DPatrick (19:37:41)
Well...what do we know about the angles of a convex n-sided polygon?

HoratioK (19:37:51)
The fact that the sum of the angles is ;180*(n-2)

DPatrick (19:38:08)
Yes. We know that their sum must be 180(n-2).

DPatrick (19:38:17)
Once we calculate the sum of the arithmetic progression we can set it equal to 180(n - 2). This will give us an equation that we can solve. Finding two ways to express the same quantity is one of the most important principles in algebraic problem solving.

DPatrick (19:38:53)


DPatrick (19:39:28)


DPatrick (19:39:47)
That came out a little wide...

DPatrick (19:40:02)


HoratioK (19:39:37)


DPatrick (19:40:35)


DPatrick (19:40:54)
(I've skipped some of the algebra steps to save us a bit of time.)

DPatrick (19:41:12)


DPatrick (19:41:32)
What are the solutions to this equation?

galbatorix (19:41:51)
325n-5n^2=360n-720

oldonion (19:41:56)
n=9

DPatrick (19:42:44)
Right, after doing a little bit of algebra, we'll get n=9 as the solution. (n=-16 is also a solution, but it doesn't make sense for a polygon to have -16 sides.)

DPatrick (19:42:57)
So we have our answer, n = 9. (A).

DPatrick (19:43:06)
This problem shows us that we can often draw upon information not directly mentioned in the problem to help us create an equation to solve. Of course, we also needed to sum an arithmetic series to give us the second side of this equation.

DPatrick (19:43:29)
Another important topic that will be covered extensively in the AMC 10 class is COMBINATIONS.

DPatrick (19:43:47)
Combinations lie at the heart of a great many AMC problems and so we will examine a number of different kinds of problems that require us to use combinations.

DPatrick (19:43:54)


DPatrick (19:44:29)
How can we examine the number of line segments formed between 2 vertices of a cube?

felixnguyen (19:44:51)
8 for first point and 7 for second

DPatrick (19:45:31)
The cube has 8 vertices. We can construct such a segment by picking one of the 8 vertices to start, and then picking one of the 7 remaining vertices to be the other end of the segment.

kyyuanmathcount (19:45:18)
We must divide by two because the order does not matter.

DPatrick (19:45:58)
Right. Since the segment PQ is the same as the segment QP, it doesn't matter what order we pick the vertices.

lyra (19:45:56)
so D, 28

kyyuanmathcount (19:45:57)


HoratioK (19:45:58)
That gives (D) 28

DPatrick (19:46:06)
Right.

DPatrick (19:46:12)


DPatrick (19:46:33)
Understanding combinations can directly lead us to quick solutions to some AMC problems.

DPatrick (19:46:37)
Here's another:

DPatrick (19:46:42)


DPatrick (19:47:14)
How can we begin with this problem?

lyra (19:47:25)
simplify

DPatrick (19:47:44)
We can try to simplify that expression.

felixnguyen (19:47:47)
Mulitiply and simplify

DPatrick (19:48:01)


DPatrick (19:48:27)


kyyuanmathcount (19:47:55)
We can split n-2k-1/(k+1) into two parts and try to make it into two combinations

felixnguyen (19:48:35)
You could simplify furthur I think

DPatrick (19:49:20)


DPatrick (19:49:57)


kyyuanmathcount (19:49:55)
Those are C(n+1,k+1) and 2C(n,k)

DPatrick (19:50:09)
They certainly are.

DPatrick (19:50:15)


DPatrick (19:50:35)
This is always an integer.

DPatrick (19:50:44)
So the answer is (A), all values of n and k work.

DPatrick (19:50:53)
The key to this problem was manipulating the original product with an eye toward simplification. Since we started with a binomial coefficient, it was easiest to search for a way to represent our quantity using other binomial coefficients. When dealing with different kinds of quantities, it is often useful to look for ways to manipulate like quantities into like quantities.

DPatrick (19:51:17)
Note that I worked through this problem much faster than we would actually do it in class (to save time here tonight).

DPatrick (19:51:32)
Are there any questions about the AMC 10 class?

HoratioK (19:52:19)
Will we study for the AMC 12 too?

DPatrick (19:52:48)
Sort of. The class is geared for the AMC 10, but there's a lot of overlap between the 10 and the 12.

DPatrick (19:53:06)
The main thing is that we won't cover topics (such as trigonometry) that only appear on the AMC 12.

lyra (19:53:30)
AMC 12 class cover trig?

DPatrick (19:53:45)
Yes, there are trig problems on the AMC 12 (and in the AMC 12 class).

DPatrick (19:54:06)
I should mention that we'll have an AMC 12 class this fall as well, starting in late October; we'll have a Math Jam for that class in mid-October sometime.

mathgeek117 (19:53:15)
[i][/i][img id=em-10]Where can I learn Latex

rrusczyk (19:54:14)
Start here: http://www.artofproblemsolving.com/LaTeX/AoPS_L_About.php

hadasah (19:54:48)
When is the usual time for taking AMC10

DPatrick (19:55:13)
The contest is in February...is that what you mean?

mathgeek117 (19:54:42)
Is it free[img id=em-6]

rrusczyk (19:55:26)
The class, no. That link about LaTeX, yes. LaTeX in general, yes.

hadasah (19:55:31)
no, how old should you be

DPatrick (19:56:07)
Most people who take the AMC 10 contest are in 9th or 10th grade, but there are also some 8th and even 7th grade students who take it.

DPatrick (19:56:32)
If you've got a decent background in Algebra, and some basic Geometry, you should be fine in the AMC 10 class.

DPatrick (19:56:44)
You cannot take the AMC 10 contest if you're in 11th or 12th grade.

lyra (19:55:02)
are we going on the the Intro to Geometry class?

DPatrick (19:57:14)
Yes, right now!

DPatrick (19:57:22)
I'll turn things over to Richard Rusczyk to talk about the Intro to Geometry class.

rrusczyk (19:57:27)
Introduction to Geometry

rrusczyk (19:57:40)
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. The class will follow the text Introduction to Geometry.

rrusczyk (19:57:49)
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.

rrusczyk (19:57:57)
The Introduction to Geometry course is on Thursdays from 7:30 to 9 PM ET (4:30 to 6 PM PT) starting October 5. I am the instructor.

rrusczyk (19:58:12)
We'll do a couple sample problems, then take some questions.

rrusczyk (19:58:19)
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.

rrusczyk (19:58:26)


rrusczyk (19:58:29)
http://www.artofproblemsolving.com/Classes/IntroGeom/Images/18363682.gif

rrusczyk (19:58:31)
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.

rrusczyk (19:58:46)
In order to find AC, what will we need?

lyra (19:58:55)
OC

galbatorix (19:59:04)
Co

rrusczyk (19:59:13)
We need AO, which we already have, and OC. How can we find OC?

kyyuanmathcount (19:58:39)
OD=AO=6

felixnguyen (19:59:14)
OD is also radius so it is 6

felixnguyen (19:59:32)
You can find OC with OD using pythagorean theorm

hadasah (19:59:44)
well, OD is 6

notehead (19:59:50)
OD=6

rrusczyk (20:00:30)
OD is a radius, so it equals AO, which gives us OD = 6.

rrusczyk (20:00:34)


rrusczyk (20:00:35)
http://www.artofproblemsolving.com/Classes/IntroGeom/Images/18363682.gif

rrusczyk (20:00:37)
So, what is CO?

felixnguyen (20:00:07)
if OC = x then 2x^2=36

felixnguyen (20:00:26)
x = 3*sqrt2

galbatorix (20:00:40)
CO = 3 * sqrt2

oldonion (20:01:21)
oc is the sq.rt of 18

hadasah (20:01:03)
sq rt of18

rrusczyk (20:01:33)
From 45-45-90 triangle OCD, we have OC = OD/sqrt(2) = 6/sqrt(2) = 3*sqrt(2).

rrusczyk (20:01:40)
Now what?

rrusczyk (20:01:50)


felixnguyen (20:01:36)
6^2+3sqrt2^2 = 36+18 = 54

HoratioK (20:01:51)
AC = 3sqrt(6)

meenamathgirl (20:01:55)
pythagorean theorem

hadasah (20:01:58)
use pythagorean theorem

oldonion (20:02:00)
ac is sqrt (54)

rrusczyk (20:02:15)
Right triangle -> Pythagoras:

rrusczyk (20:02:19)


rrusczyk (20:02:33)
The Pythagorean Theorem is one of our main tools for finding lengths in geometry problems.

rrusczyk (20:02:42)
Let's try a more challenging problem.

rrusczyk (20:02:47)
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.

rrusczyk (20:02:52)


rrusczyk (20:02:53)
http://www.artofproblemsolving.com/Classes/IntroGeom/Images/03624113.gif

rrusczyk (20:03:09)
What do we want to find in this problem?

lyra (20:03:27)
the radis of the inscribed circle

hadasah (20:03:30)
radius of the circle

mathgeek117 (20:03:35)
the radius

galbatorix (20:03:42)
radius of the circle

rrusczyk (20:04:09)
We want the radius, so what should we add to the diagram?

HoratioK (20:04:21)
The radius

hadasah (20:04:26)
a radius

rrusczyk (20:04:42)
What radii of the little circle should we draw?

rrusczyk (20:04:51)


rrusczyk (20:04:53)
http://www.artofproblemsolving.com/Classes/IntroGeom/Images/03624113.gif

lyra (20:04:59)
the one from the outside tangent point

HoratioK (20:05:01)
The one perpindicular to AB

hadasah (20:05:47)
towards oA

rrusczyk (20:05:56)
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.

rrusczyk (20:06:02)


rrusczyk (20:06:03)
http://www.artofproblemsolving.com/Classes/IntroGeom/Images/34258208.gif

rrusczyk (20:06:09)
We like right angles.

rrusczyk (20:06:14)
Now what?

rrusczyk (20:06:53)
What do we see in the diagram?

lyra (20:07:03)
a square

HoratioK (20:07:03)
a square

rrusczyk (20:07:20)
We see OHPG is a square.

rrusczyk (20:07:31)
How can we use this to help?

galbatorix (20:07:46)
HO and GO also =r

rrusczyk (20:08:11)


rrusczyk (20:08:13)
http://www.artofproblemsolving.com/Classes/IntroGeom/Images/34258208.gif

rrusczyk (20:08:18)
Indeed; what other length do we know?

jinye (20:08:03)
OP = sqrt(2)*r

rrusczyk (20:08:37)


rrusczyk (20:08:38)
http://www.artofproblemsolving.com/Classes/IntroGeom/Images/09772391.gif

rrusczyk (20:08:49)
OP is a diagonal of the square, so it equals r*sqrt(2).

rrusczyk (20:08:51)
Now what?

jinye (20:09:44)
OF=FP+PO

rrusczyk (20:10:21)
Note that OP is not half of OF because OP is greater than PF.

rrusczyk (20:10:30)
However, we do have OF = FP + PO.

rrusczyk (20:10:34)
How does this help?

HoratioK (20:10:52)
FP + PO = r + r*sqrt(2)

rrusczyk (20:11:05)
And what is OF?

oldonion (20:11:03)
because OF is 3

jinye (20:11:12)
OF= is the radius of the big circle

lyra (20:11:13)
3

rrusczyk (20:11:26)
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.

rrusczyk (20:11:34)
Now we have an equation and we can solve for r:

oldonion (20:07:27)
r*sqrt(2)+r=3

rrusczyk (20:11:48)


rrusczyk (20:12:08)
How can we write this such that there isn't a square root in the denominator?

galbatorix (20:12:22)
mult by conjugate

lyra (20:12:25)
multiply by 1-sqrt(2)

rrusczyk (20:12:35)
Multiply both the top and bottom by 1-sqrt2.

rrusczyk (20:12:36)


rrusczyk (20:12:47)
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.

rrusczyk (20:13:16)
In the geometry class we'll talk a great deal about how to add extra lines to solve geometry problems as we did in this example.

lyra (20:13:25)
in the class, what levle of problem is this considered

rrusczyk (20:13:44)
The first of those two problems is on the easy end of problems we will discuss. The second is a bit easier than average, though the earlier weeks of the course will be considerably easier for the most part.

rrusczyk (20:13:53)
Are there any questions about the geometry course?

jinye (20:14:07)
how do we know PO and FP are on the same line?

rrusczyk (20:14:25)


rrusczyk (20:15:10)
That's a fantastic question. We'll talk about that in more detail in the Intro Geometry course.

rrusczyk (20:15:31)
If you have the Intro Geometry book, read a bit about the triangle inequality and see if you can prove it on your own.

HoratioK (20:14:19)
Are there harder problems than that?

rrusczyk (20:15:48)
Yes, there are many problems that are harder than the problems we just discused.

lyra (20:14:47)
how does this class compare to the average school geometry class (other than the fact that it is taught by an AoPS instructer [img id=em-8])

rrusczyk (20:16:09)
The class will cover the standard 'average class' geometry quickly and spend most of the time on harder, more interesting problems.

oldonion (20:14:53)
how much time will be for a lecture and how much time for problem solving?

rrusczyk (20:16:58)
Almost all of the course will be focused on problem solving. In the early weeks, there will be a little more time spent on standard theorems and such, but even then, we will spend most of each class working on problems.

notehead (20:15:29)
I will be just learning Geometry from a good textbook as I start this course. Will this be a little ahead of me?

rrusczyk (20:17:45)
This class will parallel our Introduction to Geometry course. If you participate fully in this class, you'll likely find the other textbook very easy.

oldonion (20:17:41)
will you be proving the geometry theorems?

rrusczyk (20:18:07)
No. The students will be :)

lyra (20:18:05)
this sounds like a terrific class, I cannot wait for it to start!

rrusczyk (20:18:15)
We agree.

oldonion (20:18:30)
I concur[img id=em-10]

rrusczyk (20:18:41)
Are there any more questions?

HoratioK (20:18:49)
What does concur mean?

rrusczyk (20:18:56)
Agree.

mathgeek117 (20:18:50)
how many people are currently signed up for your class?[img id=em-10]

rrusczyk (20:19:11)
There are on the order of 35-40 signed up.

notehead (20:19:27)
But if I have no experience with geometry and I'm learning it at the same time...I just don't want to feel lost.

rrusczyk (20:19:59)
If you ask lots of questions and are willing to read ahead before the lectures, you should be fine.

lyra (20:20:48)
will we learn about 3 d volumes and surface area of weidrd objects?

rrusczyk (20:21:15)
We'll do 3D volumes and surface area.

notehead (20:21:04)
Can't wait for this class! I already took your Intro to number theory, and it was great.

rrusczyk (20:21:26)
If you could keep up with that, you should be ok with the geometry.

yifangzhao (20:21:16)
do we have to learn geometry theorem before take this course

rrusczyk (20:22:02)
You should have the Intro Geometry text to study along with the course. The course does not assume prior geometry knowledge.

avec_une_h (20:21:43)
How much time should this class take?

rrusczyk (20:22:27)
You should spend 3-4 hours a week on the class, at least.

lyra (20:22:09)
I hope (and am sure it will be ) as good as DPatricks intro to counting and prob, I can't wait

oldonion (20:23:00)
will you follow your text book directly, or close to that?

rrusczyk (20:23:12)
It will stay close to the text.

mathgeek117 (20:23:06)
[img id=em-7]how much independent work is there[img id=em-6]

notehead (20:23:12)
Will there be lots of problems on the message board?

rrusczyk (20:23:27)
There will be 10-15 problems each week on the message board.

rrusczyk (20:23:44)
There will be plenty of opportunity for independent work, and to work together.

rrusczyk (20:24:18)
Are there any more questions?

HoratioK (20:24:31)
What is the message board?

rrusczyk (20:24:40)
There is a message board for each course.

rrusczyk (20:24:50)
We post problems on it after each class.

jinye (20:25:09)
Is there audio in real class?

rrusczyk (20:25:16)
No.

HoratioK (20:25:14)
Including this one?

rrusczyk (20:25:35)
No. This is just a Math Jam, not part of the Geometry course.

mathgeek117 (20:25:29)
Can we post questions from the class in the forum

rrusczyk (20:25:45)
In the class forum, yes.

rrusczyk (20:26:24)
Any more questions?

rrusczyk (20:26:50)
Thanks for coming to the Math Jam!

rrusczyk (20:27:04)
If you think of questions later, you can email them to classes@artofproblemsolving.com