How to take the USAMO

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Facilitator: Richard Rusczyk




rrusczyk (19:29:56)
Before we get started, I'd like to say a few things about our new classroom

rrusczyk (19:30:05)
Like the old classroom, this room is moderated.

rrusczyk (19:30:16)
There are several differences between this and the old room.

rrusczyk (19:30:20)
First, you can use your avatars from the message board as your avatars here. Click on the little person next to the input box to change your avatar. If you have an avatar for the message board, you should be able to access it in that little box (this won't work if you've only uploaded your avatar in the last day or two).

rrusczyk (19:30:26)
Second, when an instructor wishes to correspond privately with you, the instructor will open a private window with you.

rrusczyk (19:30:35)
Third, there's a little clipboard right next to the eraser above the input box. If you click there, you'll get a copy-pasteable transcript of the session. We will still have our regular transcripts.

rrusczyk (19:30:49)
Fourth, there is currently a bug in the classroom that allows some of you to speak privately to others. We will be disabling this soon. It happens when you click on a user in the user list, then type. The resulting message is called a whisper - you, the person you are whispering to, and our chat logs will see the message.

rrusczyk (19:30:55)
Please don't do this; it can be very distracting for other students. PLEASE DO NOT WHISPER TO THE INSTRUCTORS; we cannot post whispers into the classroom, and it is very distracting. If you are 'stuck' in whisper mode, right-click in the user list window and then you should be fine.

rrusczyk (19:31:18)
Are there any questions about how the classroom operates?

rrusczyk (19:31:54)
Who is taking the USAMO next week?

Dnas (19:31:59)
me

zabelman (19:32:02)
me

pi^2 (19:32:05)
me

hilbert (19:32:06)
I am

hmmm (19:32:07)
me

polymorphic (19:32:09)
I am

vela (19:32:20)
I am

z9R4C3 (19:32:23)
i am (going to do very very poorly)

dancer87 (19:32:24)
i am

cold entr?e (19:32:24)
I know I am

Myself (19:32:38)
me

rrusczyk (19:32:48)
Today we will be talking about general strategy for taking the USAMO. There will be little actual math tips in this discussion; we are primarily focusing on how to approach the test.

rrusczyk (19:33:00)
We will discuss three main areas: first, state of mind and preparation for game day, second, how to write a clear solution, third, strategies for attacking problems.

rrusczyk (19:33:14)
After each area we will have a little question and answer session about that area.

BHorseMath (19:33:08)
would this also help us on other contests?

rrusczyk (19:33:42)
This will help on USAMO-like contests - ones with very hard problems for which you are expected to write full solutions.

rrusczyk (19:34:05)
MCrawford will now take over.

MCrawford (19:34:12)
APPROACHING THE TEST

MCrawford (19:34:18)
You could simply sit down and begin working on problem 1. This is usually the easiest problem for most people, but this is not always the case. This approach fails to let you test the waters with the other problems and leaves you thinking from the very beginning about the other problems with less time to solve them.

MCrawford (19:34:39)
A better idea might be to read through all the problems before starting work on any of them. This allows you to work on the problems in the order in which you are confident in your ability to make progress toward the solutions.

MCrawford (19:34:53)
Another benefit of reading all the problems at the beginning is that it gives you a chance to spend a few minutes thinking about each one before you actually go after a solution.

MCrawford (19:35:14)
Often times problem solvers read a problem, think about it, formulate a few general ideas and fail to come up with a solution quickly. Later they come back to the same problem that has been churning in the back of their minds and they find new inspiration based on their initial thoughts.

MCrawford (19:35:48)
After reading through the problems at the beginning it?s a good idea to jot down all ideas that you have on the problem. Spend 5 or 10 minutes per problem brainstorming what techniques might help and list all facts that are immediately apparent. The inspiration that solves a problem is very often made when a thinker matches matches a technique with a set of available facts.

MCrawford (19:36:25)
Keep your thoughts, diagrams, equations, etc. separated by problem so that you can make the most of the information you gather about each problem. Writing a good proof is about organizing your thoughts in a way that leads the reader from a simple beginning to a triumphant ending. Organizing your work helps you organize your thoughts.

MCrawford (19:36:57)
After you have read through the problems and made notes of your original thoughts, pick your target. You should at this point have a good idea about which problems you are likely to make progress on.

MCrawford (19:37:29)
Another good thing to remember when picking your target is that the problems that are most simply stated are almost always the most difficult. Though Fermat?s Last Theorem might look pretty and approachable to the unsuspecting (if illiterate) problem solver, it ain?t easy to prove.

MCrawford (19:38:04)
This point about simply stated problems can also illuminate ways you might approach a problem. Longer, more complicated-looking problems are often long and complicated-looking precisely because they are engineered from simpler ideas in order to create an Olympiad level problem. You can often unravel these problems largely by pondering the way in which the problem is engineered.

MCrawford (19:38:37)
On the other hand, tackling simply-stated problems often involves generating an entire framework in which to attack the problem. You must often decide how to assign variables, what kind of diagram to construct, or simply how to frame the problem mathematically so that you have a model/equation/picture to work with.

MCrawford (19:38:56)
I will now take questions before moving on.

Fermatprime (19:38:34)
What is an example? I'm not sure I quite understand...

MCrawford (19:39:25)
An example of a simply stated problem that is difficult or an example of a highly engineered problem that is not so difficult?

Fermatprime (19:39:39)
highly engineered

MCrawford (19:40:10)
Remember the star problem on the AIME this year?

MCrawford (19:40:20)
You can find it in the Math Jam transcripts.

MCrawford (19:41:48)
An example of a hard problem that is simply stated would be the following number 15 from an AIME long ago (this might not be exactly right):

MCrawford (19:42:21)
Find the smallest even natural number that is not the sum of two primes.

rrusczyk (19:43:04)
An example of a highly 'contrived' problem that you might be able to see the solution from the form of the question is from the 2002 USAMO.

rrusczyk (19:43:06)
http://www.kalva.demon.co.uk/usa/usa02.html

rrusczyk (19:43:11)
Look at question A2.

rrusczyk (19:43:36)
The triangle ABC satisfies the relation cot^2A/2 + 4 cot^2B/2 + 9 cot^2C/2 = 9(a+b+c)2/(49r^2), where r is the radius of the incircle (and a = |BC| etc, as usual). Show that ABC is similar to a triangle whose sides are integers and find the smallest set of such integers.

rrusczyk (19:43:53)
To an experienced problem solver that left-hand side yells:

MysticTerminator (19:43:48)
Cauchy?

rrusczyk (19:44:01)
Yep.

MCrawford (19:44:17)
Any other questions?

hilbert (19:44:44)
What is Cauchy?

MCrawford (19:45:44)
The Cauchy-Schwartz inequality is one of the most useful inequalities. We won't get into it in this Math Jam, but it would be very useful to look up or discuss in one of the forums.

MCrawford (19:46:06)
STATE OF MIND

MCrawford (19:46:13)
Before you look at the problems, before you enter the test room, and before you go to school in the morning, your state of mind matters. If you are tired, bored, angry, or otherwise distracted, you are not likely to perform as well on the USAMO.

MCrawford (19:46:28)
You are all thinkers. Your achievement in making it to the USAMO is that you have demonstrated the ability to think hard about math problems that frighten most of your peers. You approach problems with confidence and, I should hope, a sense of adventure and enjoyment. Simply put, your attitude makes you a problem solver.

MCrawford (19:46:47)
So bring a problem solvers attitude to the test.

MCrawford (19:46:54)
First, as simple as it sounds, go about life in a completely normal way leading up to the test. Whatever state of mind suits you on a daily basis is the state of mind that prepared you for the USAMO.

MCrawford (19:47:10)
Food fuels you. Don't go into the USAMO starving. 4 1/2 hours is the mental equivalent of running 10 miles. It is something you are trained to do, but you're not going to do it well without fuel. That said, don't fill yourself up so much that the blood that normally drives your brain is driving your stomach.

MCrawford (19:47:29)
It is likewise a good idea to be hydrated and to keep water around during the exam.

MCrawford (19:47:44)
Second, be prepared for the exam. If running around frantically looking for car keys kept drivers in a good mental state for the commute to work, they would enjoy tossing their keys randomly over their shoulder each time they arrived home. It's important to be equipped with your own tools.

MCrawford (19:48:04)
Set yourself up ahead of the test day with all of the necessities: plenty of pencils, paper, a protractor, a compass, and a straight edge. Don't look for these half an hour before the test. Create peace of mind and confidence by taking care of the little things ahead of time.

MCrawford (19:48:31)
Third, set yourself up with a good space for taking the test. Space is a resource. Make sure you have plenty of it. Find a large table where you can spread your work out and stay organized without having to flip through a dozen pieces of scratch work every time you want to go back to a previous idea.

MCrawford (19:48:46)
Fourth, don't go into the USAMO with unreasonable expectations. It isn't the AMC and it's beyond the AIME. Only a few students have a shot at 3 perfect solutions on any given day.

MCrawford (19:49:12)
The USAMO is a different test from short answer tests or any test where each problem challenges you for 10 seconds to 10 minutes at a time. USAMO problems are solved through creative exploration. While taking the USAMO, you are no longer merely a hiker in the world of mathematics. You are an explorer with new ground to cover. It should not be disappointing to work for 2 hours without a solution. That happens to nearly everyone. You should have the mindset of an explorer. Steady your pace and stick to exploring.

MCrawford (19:49:52)
Finally, put all your worries behind you. Worrying does nothing positive. It's a waste of mental energy. Don't worry about where you're going to college. Don't worry about that annoying thing one of your parents said before they dropped you off at school. Don't worry about your date to the prom. Mostly importantly, don't worry about your USAMO score. Worrying doesn't get you points and it takes you out of the state of mind that makes you a problem solver.

MCrawford (19:50:17)
You are a problem solver. Playing with cool math problems is what you're going to do for 4 1/2 hours each of two days in a row. Enjoying those hours the most means having all your energy focused on the test problems. It definitely won't hurt your score.

cold entr'e (19:48:32)
should you bring food with you

MCrawford (19:51:07)
That's up to you. I wouldn't bring a pizza in with me, but Richard's eating cookies now because he's hungry and having energy is good when working.

hilbert (19:51:01)
I'm taking it from 10:30 - 3:00. Lunch lands right in the middle. I'm going to bring food and a snack.

MCrawford (19:51:32)
That makes sense.

hilbert (19:49:49)
I think that most people get "binary scores" (a 1 or a 0). If you can get one problem you're doing well.

MCrawford (19:52:43)
Yes, don't panic if you work for two days and solve one problem or even none. The problems are meant to be a very high level filtering device for high level thinkers. They are supposed to be hard -- even for extremely bright students.

MCrawford (19:53:01)
So long as you maintain focus, you are more likely to achieve your potential.

MCrawford (19:53:17)
Are there any other questions?

justdudxit (19:53:32)
What is the best way to prepare

MCrawford (19:53:54)
Maintain your sense of curiosity and fun.

MCrawford (19:54:04)
Preparing for the USAMO is not a weekend task.

MCrawford (19:54:40)
The rare students who crush the USAMO are those who spend years exploring challenging problems. They get there because they love the journey.

hilbert (19:54:09)
Should you be practicing problems?

MCrawford (19:55:02)
I advocate working on problems any time that it's fun.

MCrawford (19:55:40)
On the other hand, I already made the analogy of the USAMO being like two days of 10 mile runs. I wouldn't overdo your studying time the day or two before the USAMO. You want to be well-rested.

zondruviak (19:54:31)
What do you do when you're stuck? What ways are there to get out of mindsets/ruts on hard problems?

MCrawford (19:56:35)
Richard will talk a bit about this later, but it's important to write down everything you know and continue to explore new ways to piece together the information you have.

hilbert (19:56:43)
What theorems should you know that most people normally don't?

MCrawford (19:57:07)
Richard will discuss techniques later.

cold entr'e (19:57:32)
How is the USAMO scored?

MCrawford (19:57:54)
Each problem is worth a maximum of 7 points.

MCrawford (19:59:14)
It is difficult to explain how points are awarded, but in general, having a great insight into the problem will get you 1 or 2 points, having all the work but a little step gets you 3 or 4, a complete solution with a careless arithmetic or obviously small mistake gets you 5 or 6, and a perfect solution gets 7.

Fierytycoon (19:58:43)
How important is neatness in handwriting? Would writing in cursive be acceptable?

MCrawford (19:59:55)
You want to make it as easy on the grader as possible.

MCrawford (20:00:05)
Write nearly and I wouldn't suggest cursive.

Nukular (19:59:37)
I thought scores of 3-4 were really rare.

MCrawford (20:00:16)
They are, but they happen.

MCrawford (20:00:38)
Any other questions?

rrusczyk (20:01:02)
Once you've solved the problem, you're far from finished.

rrusczyk (20:01:08)
You have to write up your solution.

rrusczyk (20:01:17)
Sounds insignificant, but it is extremely important.

rrusczyk (20:01:30)
My senior year there were around 10 people who essentially got 4 questions right.

rrusczyk (20:01:44)
There were 3 who got more right (there were only 5 questions then).

rrusczyk (20:01:50)
So there was big tie for 4th.

rrusczyk (20:02:04)
They essentially split up the knot at 4th by clarity of solutions.

rrusczyk (20:02:14)
4th-8th were winners, the rest weren't.

rrusczyk (20:02:25)
I came in 4th.

rrusczyk (20:02:29)
So pay attention.

rrusczyk (20:02:50)
Outline your solution. Scratch it out on scrap paper - just a few words for each step. That way, you'll know what order you want to write your steps. If you don't outline your steps, you may get half-way through your proof and hit something that you should have proved earlier. An outline will also help you catch any flaws or missing steps. Once you have your outline, start writing.

rrusczyk (20:03:03)
Once you start writing, there are few important guidelines.

rrusczyk (20:03:15)
First and foremost, you must remember that you lose whenever you force a grader to think. (Unless you make them think because you've come up with a very clever or novel solution.)

rrusczyk (20:03:21)
You must write clearly. Writing neatly is important. If you use cursive, it better be easy to read.

rrusczyk (20:03:24)
You must stay out of the margins.

rrusczyk (20:03:47)
Don't turn when you get to the end of a line and start writing vertically.

rrusczyk (20:03:55)
If you write in pencil, don't erase big blocks of text. You shouldn't scribble out things you want to omit - draw a single line through what you want omitted and move on. If it's a whole block of text you want gone, put a big X through it. Don't scribble.

rrusczyk (20:04:07)
You have all the paper you want - leave lots of space. Don't wedge important equations into paragraphs; given them their own lines. If you turn in just a dense thicket of words, you will not be happy with the results.

rrusczyk (20:04:14)
Write clearly and use complete sentences.

rrusczyk (20:04:22)
In many problems, you will work backwards to get to the solution. Geometry problems and inequalities are often solved this way.

rrusczyk (20:04:37)
sdrawkcaB knihT, Write Forwards.

rrusczyk (20:04:49)
Even if you solved it by working backwards, write your solution 'forwards'.

rrusczyk (20:05:06)
You should point to the fences in your solutions - explain the general attack you are going to take at the beginning. This is particularly true of contradiction and induction problems. Simply starting with 'We will prove the result by induction' is a fine way to do so.

rrusczyk (20:05:28)
Clearly define your notation at the beginning; separate it from the rest of the text; give each variable or function its own line. Don't define your variables in a dense paragraph at the beginning, and don't just define them in text as you go along.

rrusczyk (20:05:39)
In your geometry problems, you must include a diagram. Don't just assume the grader can look at the problem or use someone else's diagram. Also, draw your diagram precisely.

rrusczyk (20:05:51)
Many of you will get to a point where you're not sure if you have to prove something or if you can assume it is true. Here are a few rules of thumb you can follow:

rrusczyk (20:05:58)
If you can name it, you don't have to prove it.

rrusczyk (20:06:05)
If you can't name it and will just take you one line to prove it, go ahead and prove it.

rrusczyk (20:06:14)
If you have no idea how to prove it and are running out of time, just stick it in as if it's obvious. Maybe you'll get lucky.

Nukular (20:05:19)
will lemmas suffice as well?

rrusczyk (20:06:32)
Many times your proof will consist of several different items you prove separately then use for later results. You can separate these preliminary proofs by identifying them as 'Lemmas', proving them, then referring to them later when you need the result. For example, you would write:

rrusczyk (20:06:44)
Lemma 1: first important thing

rrusczyk (20:06:47)
Proof: Proof of this important thing.

rrusczyk (20:06:58)
After the proof of the lemma, you can separate it from the rest of your solution by a line or whatever. When you need to invoke the result later, you can just say, 'By Lemma 1, we have . . . '

rrusczyk (20:07:08)
Just as you should separate your Lemmas clearly, if you have to use Cases, separate them clearly.

rrusczyk (20:07:23)
Case 1: stuff for case 1.

rrusczyk (20:07:27)
Case 2: stuff for case 2.

rrusczyk (20:07:38)
Make these distinct paragraphs, or even sections separated by lines.

rrusczyk (20:07:47)
Finally, summarize your solution. This will usually be just one line, such as, 'Thus, we have shown by contradiction that prime factorization is unique.'

Fierytycoon (20:07:14)
So the lemmas are placed at the beginning of the solution?

Nukular (20:07:25)
Do lemmas be proven in the beginning, or can they be proven after the body of the proof?

rrusczyk (20:08:22)
I would generally stick them at the start, but you can put them in the middle as long as you clearly separate them. I'm not sure what's standard.

zabelman (20:07:26)
what if you can give a source but can't name it?

rrusczyk (20:08:47)
If you can cite a page of a book, you're set (or probably even a book).

rrusczyk (20:08:56)
If it's easy to prove, then I would go ahead and prove it.

Henjinchi (20:06:06)
Is scale important?

rrusczyk (20:09:17)
Try to be precise in your diagrams, but it doesn't have to be exactly right.

rrusczyk (20:09:44)
MCrawford notes that you can say 'By a well known theorem' for those common theorems that you don't prove.

rrusczyk (20:09:58)
(For example, if the Pythagorean Theorem didn't have a name)

rrusczyk (20:10:11)
Yes.

Henjinchi (20:10:24)
What if you do not know the name of a theorem or property?

cold entr'e (20:10:37)
Is there actually a common theorem that is nameless?

rrusczyk (20:11:01)
If you're pretty sure about the theorem, you can just say 'by a well known Theorem'.

rrusczyk (20:11:13)
An example might be, area of a triangle = rs.

Fierytycoon (20:11:09)
How about something like...say... tan(A)tan(B)tan(C) = tan(A)+tan(B)+tan(C) if A+B+C=180 degrees? Would one need to prove that?

rrusczyk (20:12:51)
I'm not sure - that would be one that I would prove if I knew how, and just cite if I didn't know how to prove and hope you get lucky.

Henjinchi (20:11:49)
Must the proofs be written in standard notation?

rrusczyk (20:13:15)
They don't have to be, but they should be.

hilbert (20:12:13)
Should you include your scratchwork and failed ideas if you don't have a complete solution in the hope that one of them might contain an idea worth 1-2 points?

rrusczyk (20:13:22)
Yes.

riposte (20:12:47)
In the geometry course we did proofs by homothetic (sp?) methods. Are there any named theorems for these?

rrusczyk (20:13:52)
You can just say 'Since ABC and DEF are homothetic, AB = DE and AB || DE', etc.

justdudxit (20:13:32)
What's "standard notation"

sonic (20:13:42)
what is standard notation for proofs

rrusczyk (20:14:26)
Hard to say what's really standard notation. I wouldn't define things differently than mathematically standard, for examlpe.

rrusczyk (20:14:31)
Examples are:

rrusczyk (20:14:46)
Use G for centroid, I for incenter, O for circumcenter, H for orthocenter.

rrusczyk (20:14:53)
Use r for inradius, R for circumradius.

rrusczyk (20:15:01)
Use x for problems involving reals.

rrusczyk (20:15:17)
Use m and n for general integers.

rrusczyk (20:15:20)
And so on.

Dnas (20:11:00)
If, in our diagrams, there are too many auxilirary(sp?) lines, can you omit them in the drawing?

rrusczyk (20:15:41)
Include the lines you use in your solution.

zondruviak (20:14:00)
What about notations like "WLOG" and cyclic-sum notation?

rrusczyk (20:16:07)
Anything you've seen commonly in a text, you can assume a grader knows. Both of these fall in that category.

rrusczyk (20:16:14)
Anything you're not sure about, you better define.

fanzha (20:12:42)
If you've found an approach that you think would work, but may take some time, how do you decide whether to write the proof you have or try to think of others?

rrusczyk (20:16:30)
This depends on time.

rrusczyk (20:16:41)
There are two issues here:

rrusczyk (20:17:05)
When you get short on time, you should throw out the 'outline solution before writing'

rrusczyk (20:17:19)
If you have an idea, start writing it up, maybe it will turn into something.

rrusczyk (20:17:35)
If you have plenty of time, then you probably want to explore other options.

rrusczyk (20:17:58)
For example, if you see a geometry problem and you think coordinates + 5 pages of algebra will get you there, I'd look for something else.

rrusczyk (20:18:45)
If you can't get anywhere on the other problems, or are finished with them, then you can feel a little more comfortable slugging through a ton of algebra (and in such case, I would just start writing it up as a solution rather than 'outline' and do it again).

rrusczyk (20:18:58)
(But that only goes for long ugly algebra solutions.

zabelman (20:16:28)
should we use "we" or "I" (or neither)?

rrusczyk (20:19:07)
I usually use we.

3cnfsat (20:13:51)
How long shoulod it take for me to write up a solution?

rrusczyk (20:19:25)
Depends on the problem.

rrusczyk (20:19:48)
Once you're confortable writing solutions, you should expect to spend 10-15 minutes writing a clean solution.

Nukular (20:17:45)
Do they tend to favor elegant solutions as opposed to brute-force solutions?

rrusczyk (20:20:12)
A grader will naturally like the elegant solution more.

rrusczyk (20:20:39)
As for the grading, the brute-force is more likely to miss little things and lose a point here or there.

Henjinchi (20:17:18)
By standard notation, I mean a proof such as:\nx is a real number hypothesis\nx+1 is a real number addition property of real numbers

rrusczyk (20:20:58)
No - you want to avoid that.

rrusczyk (20:21:09)
Write something you expect another human to read.

rrusczyk (20:21:30)
Use sentences and paragraphs and such, not the statement - reason format of geometry texts or formal logic.

cold entr'e (20:18:14)
How would "short of time" and "plenty of time" be defined?

rrusczyk (20:21:53)
Last 30 minutes, start scribbling right on the answer sheets.

fanzha (20:20:38)
Is it possible to get a 7 with a ugly but rigorous solution?

rrusczyk (20:22:01)
Sure. If it's perfect.

zondruviak (20:20:52)
Then can an ugly (but correct and carefully written) 5 page solution still get a 7?

rrusczyk (20:22:07)
Yes.

MysticTerminator (20:20:27)
If you do a perfect solution (even using calculus), they have to give you 7, right?

rrusczyk (20:22:16)
Yep

MCrawford (20:23:10)
The first year I took the USAMO I proved a very simple problem using 6 pages of hideous work. I probably got full credit because I only solved 1 of the other 4 and went to the MOP.

MCrawford (20:23:25)
If ugly gets it done, it gets it done.

Nukular (20:21:46)
And should we use directed angles to avoid the need for multiple diagrams when possible?

rrusczyk (20:24:00)
Any time you can collapse cases, it's good. For those of you that don't know about 'directed angles', ask on the board & hopefully Nukular will explain.

Fierytycoon (20:23:02)
About how many different graders are there usually?

rrusczyk (20:24:22)
Not sure. I would guess 3-4 per problem. I've never seen it, so I don't know.

hilbert (20:23:03)
What if you use advanced math in a clever way; that is, calc and up. Would that count as "Elegant" or "Ugly"

rrusczyk (20:24:36)
That sounds elegant to me.

Fermatprime (20:22:24)
How many "words" are good in a section of proof where you're manipulating algebra or inequalities or such?

rrusczyk (20:24:47)
As long as your steps are clear, you're ok.

rrusczyk (20:25:17)
You can put a stack of equations after equations, one row after another, as long as the manipulations are clear.

rrusczyk (20:25:25)
I would not put these in paragraph form.

rrusczyk (20:25:37)
First equation first line, then second on second line, and so on.

Fierytycoon (20:25:48)
If you screw up a small notation in an otherwise perfect solution, such as writing = > instead of < = > , will that cost you a point? (I know, stupid question, but I was just curious)

rrusczyk (20:26:09)
I'm not really sure. I would guess something that tiny wouldn't cost you.

zabelman (20:20:58)
if we do encounter (large amounts of ugly) algebra, how thourough should we be with the steps we show? could we just say "and by rounine algebraic manipulation....."?

rrusczyk (20:26:39)
Use common sense - Step 2 shouldn't be so far from Step 1 that the grader has to think about how you got there.

rrusczyk (20:26:45)
Thinking graders are bad for you.

justdudxit (20:18:14)
Is time as large a factor as it is in, say, the AMC or a Sprint round?

rrusczyk (20:26:54)
Not nearly.

hilbert (20:14:12)
Should you bother to attach your scratchwork if you think that you have a complete solution, or would it detract?

rrusczyk (20:27:14)
If you have a complete solution, don't attach scratch.

Nukular (20:26:52)
are we allowed to use things like "by analogy, [CLAIM]" in a geometry problem (or any other problem) where cases are very similar?

rrusczyk (20:27:32)
If they are equivalent, yes.

Nukular (20:27:53)
my favorite assumption for proofs is "Assume the grader is mean and dumb."

rrusczyk (20:28:12)
Yes, that's a good way to approach writing solutions.

cold entr?e (20:27:57)
You get unlimited paper for writing solutions, right?

rrusczyk (20:28:27)
Yes.

Fierytycoon (20:27:58)
So can we say something like "Solving for x in ((x-3) + 7(x-2))^2 = 9x - 1, we find that x=1 or x=7" without showing the algebra?

rrusczyk (20:28:56)
I'd stick a step or two in there. Can you glance at that and see that 1 and 7 are the only solutions? I can't.

zabelman (20:28:22)
for partial solutions, should we begin to write up a formal solution and stop dead in our tracks, or hand in a list of possible ideas and approaches that we have considered?

rrusczyk (20:29:27)
Make all the observations you can.

justdudxit (20:24:40)
What books do you recommend, other than previous IMO problems off the kalva website

rrusczyk (20:29:55)
The Zeitz and Engel books are good.

rrusczyk (20:30:02)
Most of Titu's are as well.

MysticTerminator (20:30:03)
bit late now, eh?

rrusczyk (20:30:21)
Pretty much. Cramming for the next 4 days isn't going to get you too far.

justdudxit (20:30:25)
next year man, next year

Nukular (20:30:26)
Would you suggest staying away from math on Monday?

rrusczyk (20:31:07)
For the most part - mostly, I wouldn't look at any problem to which I didn't have the solution.

rrusczyk (20:31:38)
(In general, I wouldn't do this while training, but particularly on the day before.)

polymorphic (20:31:36)
why not

hilbert (20:32:07)
You could get it in your head.

rrusczyk (20:32:31)
Pretty much.

rrusczyk (20:32:48)
If you start on a problem you don't have the solution to, you could get sucked into thinking too much about it.

rrusczyk (20:32:57)
Bad thing to do the night before a big test.

Fermatprime (20:32:43)
distraction...I have a friend who's still working on a USAMO problem we went over together a week or so back.

rrusczyk (20:33:26)
This is why, for training, you generally want to stick to problems you have solutions for.

rrusczyk (20:33:40)
You get most of the learning in the first 1-2 hours of working on an olympiad problem.

rrusczyk (20:33:50)
Hours five and six are pretty wasted.

rrusczyk (20:34:05)
You'd be better off limiting yourself to 90 minutes a problem or something when training.

rrusczyk (20:34:16)
Give up at that point, look up the solution, learn what you can, and move on.

MysticTerminator (20:32:53)
but what about Kedlaya's packets? I'm sure they're great resources but they don't have solutions. Same for Zeitz unless you're lucky enough to get the soln packet.

rrusczyk (20:34:27)
Wouldn't look at them right before the test.

rrusczyk (20:34:38)
I would use them in general,

rrusczyk (20:34:46)
and depend on the message board for solutions.

rrusczyk (20:34:54)
Give up at 90 minutes, post the problem on the board.

rrusczyk (20:34:58)
Someone will solve it.

Henjinchi (20:03:47)
Has there ever been a perfect score?

rrusczyk (20:35:31)
Yes.

MysticTerminator (20:35:10)
= lots of problems on the board :-)

rrusczyk (20:35:43)
That's a good thing.

hilbert (20:35:39)
There were 5 in 2002

rrusczyk (20:35:49)
(Perfect scores)

rrusczyk (20:36:00)
Are there any more questions about solution-writing strategy?

hmmm (20:36:07)
The directions say graph and carbon paper can be used, but should they be sent in as well if part of a solution is on them?

rrusczyk (20:36:38)
Carbon paper is for you to have a copy of your work, I think.

rrusczyk (20:36:57)
Graph paper as part of solutions I would try to avoid.

MysticTerminator (20:36:15)
btw, are we just chilling now or do you have some important stuff to say? either way's ok, j/w

rrusczyk (20:37:17)
We will discuss general math techniques next

Nukular (20:36:40)
I seem to find myself running on and on during proofs when im trying to be formal... is that ok, or should I try to keep it concise...?

rrusczyk (20:37:37)
Concise and precise. Prove everything, try to avoid wordiness.

rrusczyk (20:38:00)
Generally, your solutions on the board have been fine.

zondruviak (20:37:19)
How much detail do you need when justifying calculus (say showing that IF a max exists then it occurs at a certain pt)?

rrusczyk (20:38:21)
Not sure what you mean.

rrusczyk (20:38:47)
You don't have to prove that using derivatives can get you max and min.

rrusczyk (20:39:00)
All the basics of calculus can be used just as the basics of algebra.

rrusczyk (20:39:13)
However, I'll note that reaching for calculus is usually asking for trouble.

rrusczyk (20:39:20)
Many students like doing that for inequalities.

rrusczyk (20:39:26)
There's almost always a better way.

rrusczyk (20:39:42)
Are we ready to talk about general techniques?

zabelman (20:39:45)
Is calc ok for verifying concavity for jensons?

rrusczyk (20:40:01)
Sure.

rrusczyk (20:41:14)
Here are some general techniques to keep in your mind as you do the test:

rrusczyk (20:41:15)
Assign variables - don't just think that a simple formula or theorem is going to pop up and obliterate the problem. Assign variables for lengths or angles. Define functions in terms of quantities in the problem. Make clever substitutions.

rrusczyk (20:41:22)
Play. You must have a playful approach to good problems to solve them. You can't just expect a simple step to pop out and be done. There may be a problem on the test you can 1-step. There won't be many, though.

rrusczyk (20:41:32)
Try to choose your variables or functions in a way that simplifies your work.

rrusczyk (20:41:36)
Here's a very simple example:

rrusczyk (20:41:42)
A number times 6 more than that number equals 2016. We could do n(n+6) = 2016, but instead we might write (n-3)(n+3) = 2016, where we think to do this because manipulating differences of squares is easy.

rrusczyk (20:41:53)
Similarly, when you define functions or make substitutions in algebraic problems, try to force forms you know how to work with. Aim for perfect squares, differences of squares, expressions that might telescope in summation problems, etc.

rrusczyk (20:42:02)
If there's something that can be varied in a problem (for example, a locus problem in geometry), there are two general tactics that are often useful:

rrusczyk (20:42:08)
First, consider extreme cases.

rrusczyk (20:42:11)
Second, look for things that stay fixed as you vary whatever changes.

rrusczyk (20:42:17)
The latter is also very useful for 'algorithmic' problems - i.e. those in which you perform some procedure over and over again and try to prove something about the result.

rrusczyk (20:42:27)
Contradiction. If you can't prove a statement, try assuming it is false and aiming for a contradiction.

rrusczyk (20:42:33)
If you are trying to prove something for all positive integers n, try induction.

rrusczyk (20:42:41)
Solve a simpler problem and see if that helps guide you to the solution to the more challenging problem. In relevant problems, stick in small numbers and look for patterns.

rrusczyk (20:42:47)
If the problem involves proving something for an absurd number like 2004!, then try generalizing.

rrusczyk (20:42:56)
If there are quantities that can be ordered in the problem, consider the smallest or largest. An example of this is 'In each convex pentagon, we can choose three diagonals from which a triangle can be constructed.' We can order the lengths of the diagonals, then focus on the largest or smallest.

rrusczyk (20:43:09)
Sometimes we use this ordering technique in conjunction with contradiction. If we are trying to prove that no finite set of numbers satisfies some condition, we might say 'assume n is the largest of the numbers', then try to prove there's a larger n in the set, thus providing our contradiction.

rrusczyk (20:43:45)
When you are doing geometry problems, draw the diagram a few different ways if the set-up of the problem permits it. Sometimes midpoints or parallel lines or perpendicular lines will pop out at you. Also, you won't waste time trying to prove something that's not true when Diagram 2 disproves something you thought true about Diagram 1.

rrusczyk (20:43:53)
When you're totally stuck on a problem, there are few general things you can ask yourself.

rrusczyk (20:43:59)
Often, the most useful one is 'what haven't we used yet?' Look through the info given and see which piece you haven't used.

rrusczyk (20:44:06)
Another is 'what makes this problem hard?' Figure out what makes it ugly and focus on why that item makes the problem tough.

rrusczyk (20:44:17)
Another is 'What if I remove a restriction of the problem?'

rrusczyk (20:44:32)
Another: Have I seen a similar problem? If you can solve a similar problem, maybe you can carry over that technique to this one. Don't get too carried away with this - if you start spending lots of time trying to remember solutions, you tend to stop thinking.

rrusczyk (20:44:45)
One nice thing about USAMO problems as opposed to, say, AIME problems, is that you can work in 2 directions. You can try picking starting points and working towards the answer, or you can start from what you're trying to prove and work backwards - ask yourself 'what could I prove to show (whatever it is you want to show)' and keep going backwards.

Nukular (20:45:00)
Also I found writing a simpler problem and solving that gives insight to solving the harder problem.

rrusczyk (20:45:27)
Absolutely - if you can simplify the problem in any way and solve that, you'll often get a key insight.

rrusczyk (20:45:57)
That's pretty much a laundry list of the very general techniques I use. Most seem pretty obvious, but they're easy to overlook when you're taking the test.

rrusczyk (20:46:02)
Are there any questions?

justdudxit (20:46:30)
are there any guidelines for finding clever substitutions

MCrawford (20:47:41)
It would be hard to come up with any small set of guidelines, but in general it's good to be able to think about problems in different frameworks such as using binary numbers to represent places in a sequence.

MCrawford (20:47:54)
Sorry, I misread the question.

MCrawford (20:47:57)
Nevermind.

MCrawford (20:47:59)
Hmmm.

MCrawford (20:48:21)
As for clever substitutions, look for the part of an algebraic expression that is ugly put "portable".

MCrawford (20:48:46)
By portable I mean that once you substitute, the entire expression loses a big chuck of ugliness.

MCrawford (20:48:50)
chunk.

hilbert (20:46:59)
What theorems should I know?

MCrawford (20:49:21)
That's too long a list to really cover here. I would suggest taking a look at old USAMO problems for those that are commonly employed.

hilbert (20:47:40)
Is there anything I can do to get some techniques for tehse problems?

MCrawford (20:49:58)
For inequality problems: Practing using AM-GM and Cauchy.

MCrawford (20:50:29)
For Geometry: practice seeing all the ways angles and segments work in triangles in circles.

MCrawford (20:50:46)
It's also good to know various synthetic geometry techniques such as homothecy.

dancer87 (20:49:31)
how would you compare USAMO problems and USAMTS problems in terms of difficulty? I don't have much experience with USAMO problems

MCrawford (20:51:31)
USAMO problems tend to be harder. The USAMTS aims for exploratory problems that many students will solver and learn from. The USAMO attempts to separate the IMO contenders from the pack.

zondruviak (20:50:47)
On a "game" problem (EG usamo 2003 #6) how much time should you spend playing the game?\nHow do you divide your time between gathering info and formulating an argument?

MCrawford (20:51:58)
Until you have a solution, you are always gathering info.

zabelman (20:51:55)
one quick, simple question... does inversion preserve angles?

rrusczyk (20:53:11)
We define the angle between intersection circles to be the angle formed by their tangents at the point of intersection.

rrusczyk (20:53:22)
With that definition, yes, inversion preserves angles.

rrusczyk (20:53:26)
Now see if you can prove it.

Fermatprime (20:48:04)
Are there any "giveaways" that can tell you what one general method might be to solve problem "x"?

MCrawford (20:53:47)
That's what we teach in our courses :)

MCrawford (20:54:13)
But seriously, the list is long. If you see an inequality involving constants and means, use some part of the power mean inequality chain.

MCrawford (20:54:22)
Like AM-GM or AM-HM.

MCrawford (20:54:47)
Some number theory problems scream to be framed in very particularly mods.

Fierytycoon (20:52:18)
On the second day, before solving problems 4 and 5, should most people even try problem 6 at all?

MCrawford (20:55:56)
If you are unsure that you will solve one problem over any two days, it will rarely if ever be the last of the three problems on a particular day unless that is your extreme strong suit.

MCrawford (20:56:11)
In that case, you can figure that out at the beginning when feeling out the questions for your level of confidence.

zabelman (20:55:56)
which is generally easier: 2 or 4?

MCrawford (20:56:44)
2 is a pickey eater, but 4 always nags about the service.

MCrawford (20:56:54)
Really, I have no idea. I would guess 4 is easier.

Nukular (20:55:13)
Oh yeah and a lot of number theory makes good use of well-ordering principle.

hilbert (20:52:25)
Once you gather enough info that you think that you have something useful, try to disprove it. Then try to prove it.

MCrawford (20:57:18)
Any other questions?

justdudxit (20:54:04)
Fast question: does anyone have solutions for kiran kedlaya's MOP packets

rrusczyk (20:58:10)
If you can't solve one, post it on the board.

zabelman (20:57:57)
kind of a non seq, but why is there a hammer next to Mr. R's name on my users list?

rrusczyk (20:58:16)
I'm a moderator.

rrusczyk (20:59:12)
Good luck all of you!

justdudxit (20:59:14)
How did Nukular get a picture of a terran science vessel guy as his icon?

rrusczyk (20:59:29)
That's his icon on the message board.

rrusczyk (20:59:41)
You can use your message board icons in clss.

hilbert (20:59:28)
Good luck all.

MysticTerminator (21:00:00)
gluck?

zondruviak (20:59:24)
On, say, a combinatorial geometry problem, when can you state something as geometrically obvious/ obvious from the diagram? For example, if you have a convex pentagon x_1..x_5 then the union of the quadrilaterals x_ix_i+1x_i+2x_i+3 is the whole thing?

rrusczyk (21:00:56)
If there's a 1-line proof, include it (there is for that one)

zondruviak (21:00:27)
Nice new classroom! Will this be used for later classes? In particular, is there another geomerty class on Mon.?

hmmm (21:00:32)
thanks to the moderators for all the help today, and good luck everyone.

rrusczyk (21:01:26)
The olympiad geometry class meets on Mondays; that's not a Math Jam.

rrusczyk (21:01:44)
This is the classroom we will be using in the future.

rrusczyk (21:01:53)
Here's one reason we really like it; we can do this:

rrusczyk (21:02:24)


Nukular (21:02:16)
Is this hosted on the AoPS server now?

rrusczyk (21:02:39)
Yes.

Nukular (21:02:35)
we can use tex?

rrusczyk (21:02:58)
Not yet. I have a request into the software company that made this to allow you to.

rrusczyk (21:02:59)
We can.

fanzha (21:02:57)
how do we use tex here?

rrusczyk (21:03:14)
You can't yet (I can, because of that hammer)

MysticTerminator (21:02:46)
except that still turns out horrible on the autotranscript

rrusczyk (21:03:36)
Yeah, the images don't show on that clipboard.

rrusczyk (21:03:45)
But they show up in the regular transcripts.

MysticTerminator (21:03:27)
awesome. can I have a hammer?

rrusczyk (21:04:03)
Nope. Maybe in a few years.

rrusczyk (21:04:15)
Have to earn that puppy like Zeta did.

MysticTerminator (21:04:31)
he has a hammer?

rrusczyk (21:04:46)
Yes, or at least the powers that come with it.

justdudxit (21:04:40)
How do we do that

rrusczyk (21:05:24)
Get hired as an assistant with us. As some of you get to college, we'll might give you a yell should we need assistants.

rrusczyk (21:05:28)
No guarantees, of course.

mathtennisrc (21:04:21)
[img id=em-3]

MysticTerminator (21:05:40)
encore problems

rrusczyk (21:05:57)
Not tonight.

rrusczyk (21:06:50)
(Incidentally, if any of you know really good camp counselors or whatever who would be good at this, we're looking for a full-time hire.)

justdudxit (21:05:57)
well, you've got my email

justdudxit (21:06:35)
Later, dudes. Thanks for the tips

hilbert (21:06:47)
Good class, and good luck all.

chubixqube (21:06:50)
just checking...isn't it only 6:00 PM in California?

rrusczyk (21:07:12)
Yeah, but I've still got a couple hours to go.

hilbert (21:07:13)
Reid Barton, maybe?

rrusczyk (21:07:36)
Has he graduated yet? Also, he's probably headed off to be Prof. Barton.

zondruviak (21:08:02)
How about Gabriel Carroll?

rrusczyk (21:08:14)
I'm guessing Professor Carroll.

rrusczyk (21:08:37)
(He knows we're hiring, but I'm guessing he's on an academic track)

zabelman (21:08:29)
At the bottom of my classroom window, it says "Warning: Applet Window." Any thoughts?

rrusczyk (21:08:45)
dunno.

3cnfsat (21:08:55)
what do you do when you're out of ideas?

rrusczyk (21:09:11)
Try a different problem.

rrusczyk (21:09:22)
If I'm out of ideas on everything, go for a little walk.

zabelman (21:10:28)
and if we finish early (i.e. no possible way to make any more progress) which is a better use of the remaining time..... sketching or sleeping?

rrusczyk (21:10:50)
If you have all 3 problems correct, you can sketch or sleep.

rrusczyk (21:10:55)
If you don't, you keep working.

3cnfsat (21:10:54)
Any good ways of checking proofs?

rrusczyk (21:11:10)
Read it again, check each step.

rrusczyk (21:11:13)
Does A follow from B.

rrusczyk (21:11:17)
And so on.

zabelman (21:11:56)
They say that genius is going from A to D without going through B and C. When you have landed at D, do you have any tips on how to find B and C?

rrusczyk (21:12:30)
I'd say that definition is wishful thinking.

rrusczyk (21:12:48)
Find B and C the same way you found D (which may be wishful thinking, as well)

zabelman (21:14:03)
OK, thanks!

rrusczyk (21:14:44)
That's it for tonight. Good luck on Tuesday!

zondruviak (21:16:30)
thanks!

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