## Math Olympiad Corner- CMO,USAMO,IMO, and others

Hello folks,
I am quite new to Olympiad level problem solving , which is why I am considering buying Paul Zeitz's "Art and Craft of Problem Solving". Does anyone have the solutions? I have heard the solutions are in a seperate Instructor's Manual, does anyone know how someone can get that? Please reply as soon as possible. And feel free to use this Sub-forum to discuss any Olympiad level questions, or other related topics!
 Yes, and one more thing! Can anybody suggest me a Geometry book that covers all of Olympiad Geometry. I am asking for this because the Zeitz book does not have any geometry in it. Furthermore, does the Zeitz book (The Art and Craft of Problem Solving) have answers to the problems at the back?
 Can anybody help me with the following problem: The polynomial equation x^3 -6x^2 +5x-1=0 has three real roots a,b, and c. Then determine the value of a^5 + b^5 +c^5. (COMC,PART B #4,2003).

## Math Olympiad Corner- CMO,USAMO,IMO, and others

Ok, I have a solution for you, but I don't know how to use the math symbols here, so it might look a bit bad.

You have x^3 - 6x^2 + 5x - 1 = 0, where a,b,c are real roots. You want a^5 + b^5 + c^5.
Now, the sum of all triplets of a,b,c = a*b*c = 1, the sum of all pairs of a,b,c = 5, and the sum of all single a,b,c = 6. We denote this by
PI_1(a,b,c) = 6
PI_2(a,b,c) = 5
PI_3(a,b,c) = 1
We want S_5(a,b,c) = a^5 + b^5 + c^5.
Notice that S_1 = a^1 + b^1 + c^1 = a+b+c = PI_1 = 6.
Also notice that PI_m = 0 for m>3.
From now on I will only write PI_1 for PI_1(a,b,c) etc.
We need to use the Newton-Girard Formulas.
First one says:
S_1 - PI_1 = 0, so 6-6=0, OK.
Next one:
S_2 - S_1*PI_1 + 2PI_2 = 0
S_2 = S_1*PI_1 - 2PI_2 = 6*6 - 2*5 = 26, so a^2 + b^2 + c^2 = 26.
Next:
S_3 - S_2*PI_1 + S_1*PI_2 - 3PI_1 = 0
S_3 = S_2*PI_1 - S_1*PI_2 + 3PI_3 = 26*6 - 6*5 + 3*1 = 129 = a^3 + b^3 + c^3.
Next:
S_4 - S_3*PI_1 + S_2*PI_2 - S_1*PI_3 + 4*PI_4 = 0
S_4 = S_3*PI_1 - S_2*PI_2 + S_1*PI_3 - 4*PI_4 = 129*6 - 26*5 + 6*1 - 0 = 650.
Do you see a pattern now? Try do S_5 yourself.
 Do u know kalva??? If u dont then go to www.kalva.demon.co.uk It's too good to be true THANKS J SCHOLES!
 I have known kalva for some time now, and indeed it is too good to be true. But the thing is that I am quite new to olympiad level problem solving, so new that I really need to train myself. For that purpose I want to get my hands on Paul Zeitz's "The Art and Craft of Problem Solving". I have also heard from kalva and other sources that the solutions are in a seperate Instructor's manual, which is hard to get hold of. So, #1, do you know how one can get hold of the Instructor's Manual? Or,#2, since, it seems that you are a regular kalva visitor, might I ask what books did you use to develop your olympiad problem solving skills?
 yeah, the art of problem solving is a good book, they have a forum for people who like to problem solving, its at artofproblemsolving.com, the problems are quite motivating. I was way into doing these problems last year (senior year) but now ive slowed down a bit and i think that ive actually gotten worse at math since then
 Thank you Tongos, for your reply. I am a big fan of "The Art of Problem-Solving" series myself. But you and others are missing the Point! I am not talking about "The Art of Problem Solving", but I am talking about "The Art and 'Craft' of Problem Solving" by Paul Zeitz. This book is a recommended follow-up to the Volume 2 of The Art of Problem Solving series. I want to know how someone can get hold of the solutions for this Paul Zeitz book because the solutions are in some kind of an Instructor's Manual.
 Try Wiley's main page http://he-cda.wiley.com/WileyCDA/Hig...pplements.html
 This is where my problem is! The website says that the solutions are included in an instructor's manual, which is obviously for instructors only! So my question is: How does one get hold of it, when one is not an instructor?
 May I suggest paying for and having a math teacher in your school order it for you? I think you'll discover that teachers not familiar with you would be reluctant to "give" you the means to order it yourself. What assurances do teachers have that in reality you aren't looking for the solution manual to aid you unfairly in a class that you are taking? I would suspect that any decent teacher who cares at your school would be willing to take the time to find where it can be ordered from.
 Unless you find an instructor who can get it for you, Wiley will not sell it to you. They are pretty strict in this regard.
 So you guys think that I should just ask my math teacher? There is no course at my high-school that uses this book, so by getting the manual I will not be getting any unfair advantage over anybody. Do you guys know of any other books that are an equavalent to "The Art and Craft of Problem Solving" by Paul Zeitz?

 Do you guys know of any other books that are an equavalent to "The Art and Craft of Problem Solving" by Paul Zeitz?
Yes, I gather that you want a book that includes solutions to most problem? If so, Arthur Engel's Problem-Solving Strategies is quite excellent, though it isn't as approchable as Zeitz's book. If you are able to at least answer one question on the USAMO, then by all means get Engel's book.

Larson's Problem Solving Through Problem remains as one of my favorite problem solving books, but unfortunately, it has no answers to its problems.