# 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