C R Pranesachar, B J Venkatachala and, C S Yogananda Solutions for Chapter: Algebra, Exercise 1: Algebra

Determine all functions $f:R-\left\{0,1\right\}\to R$ (here $R$ denotes the set of real numbers) satisfying the functional relation
$f\left(x\right)+f\left(\frac{1}{1-x}\right)=\frac{2\left(1-2x\right)}{x\left(1-x\right)}$ for $x\ne 0,1$.

Let $p\left(x\right)=x^2+ax+b$ be a quadratic polynomial in which $a$ and $b$ are integers. Given any integer $n$, show that there is an integer $M$ such that $p\left(n\right)p\left(n+1\right)=p\left(M\right)$.

If $a_1,a_2,\dots ,a_n$ are $n$ distinct odd natural numbers not divisible by any prime greater than $5$, show that
$\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}+\cdots +\frac{1}{a_n}<2,$

If $p\left(x\right)$ is a polynomial with integer coefficients and $a,b,c$ are three distinct integers, then show that it is impossible to have $p\left(a\right)=b, p\left(b\right)=c$ and $p\left(c\right)=a$.

Let $a,b,c$ denote the sides of a triangle, show that the quantity
$\frac{a}{b+c}+\frac{b}{c+a}+\frac{c}{a+b}$
lies between the limits $\frac{3}{2}$ and $3$ . Can equality hold at either limit?

Let $f$ be a function defined on the set of non-negative integers and taking values in the same set. Suppose we are given that

$\left(\mathrm{i}\right) x-f\left(x\right)=19\left[\frac{x}{19}\right]-90\left[\frac{f\left(x\right)}{90}\right]$ for all non-negative integers.

$\left(\mathrm{ii}\right)$ $1900<f\left(1900\right)<2000$.

Find all the possible values of $f\left(1900\right)$. (Here $\left[z\right]$ denotes the largest integer $\le z$; e.g., $\left[3.145\right]=3$.)

Define a sequence by $a_1=1,a_2=2$ and $a_{n+2}=2a_{n+1}-a_n+2,n\ge 1$.

Prove that for any $m$, $a_m{a}_{m+1}$ is also a term in the sequence.

Suppose $a$ and $b$ are two positive real numbers such that the roots of the cubic equation $x^3-ax+b=0$ are all real. If $\alpha$ is a root of this cubic with minimal absolute value prove that $\frac{b}{a}<\alpha \le \frac{3b}{2a}$.