# Questions involving polynomials

I have been having problems with questions involving polynomials like asked in olympiads .
A few examples of these type of problems I'm posting here: (please don't give me solutions to these questions, cause I want to learn how to solve them not just remember their solutions)

1. Solve for real $x$: $$\frac1{\lfloor x\rfloor} + \frac1{\lfloor 2x\rfloor}= \langle9x\rangle + \frac13\;,$$ where $\lfloor x\rfloor$ is the greatest integer less than or equal to $x$ and $\langle x\rangle = x − \lfloor x\rfloor$.

2. Show that there are exactly $16$ pairs of integers $(x, y)$ such that $11x + 8y + 17 = xy$.

3. For a polynomial $f(x)$, let $f^{(n)}(x)$ denote the $n$-th–derivative for $n \ge 1$ and $f^{(0)}(x) = f(x)$. Is the following true or false ?

$$f^{(n)}(a) = 0,\text{ for }n = 0, 1, \dots , k\quad \iff\quad (x−a)^{k+1}\text{ divides }f(x)$$

Well, mainly the problems where you are given nothing but a few conditions that are to be applied on a polynomial of degree $n$.

Assuming that I have my basics clear in quadratic equations and polynomials Can anyone please refer me any book or any source from where I can learn to do these types of questions.

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(1) is not really a polynomial problem. (2) calls for rewriting as a product $(x-a)(y-b)=c$. (3) is fairly easy with induction on $k$ and product rule and other standard tools –  Hagen von Eitzen Dec 16 '12 at 23:48
I took the liberty of replacing old-fashioned $[x]$ with $\lfloor x\rfloor$ and adding boldface to the request not to solve these problems. –  Brian M. Scott Dec 16 '12 at 23:52
Ed Barbeau's book on polynomials is nice. However, the problems you give a a somewhat disparate bunch. Variants of (2) are common contest problems. For (1), just need basic detective work. For (3), once typo is corrected, it involves basic theory. –  André Nicolas Dec 16 '12 at 23:55
I find Intermediate Algebra by R. Rusczyk and M. Crawford quite suitable for my purpose but it is too costly. Can anyone recommend me any other book similar to the book above mentioned by me. –  shrey Dec 17 '12 at 0:24
Searching the web for mathematics olympiad preparation should net you a lot of material from all over the world. –  vonbrand Apr 13 '14 at 5:29

## 1 Answer

The first thing that comes to mind is The Art of Problem Solving. This book is used for the AMC. I have not personally used it, but my friend tells me that it is very useful for "contest problems".

Vipul Naik, a representative of India in the IMO, has a pdf here (warning: This is a direct download link) that goes into great detail about preparing for the IMO. As a whole, perhaps that is suitable here.

As a side note, I would personally recommend frequenting Math.SE for any and all mathematical help given that this is a wonderful place to learn.

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I have already mentioned the book by aops and as of vipul naik's article it does not contain any recommendation inalgebra besides mentioning the name of the books by arthur engel ( too hard for a beginner no theory only problems) byt anyway thanks for the reply. –  shrey Dec 17 '12 at 1:04
@shrey I am sorry that my answer doesn't adequately address your question. I hope someone of more knowledge can be of use to you. :-) –  000 Dec 17 '12 at 3:26