Olympiad problems

1976 Euclid, 4

Source: 1976 Euclid Part B Problem 4 ----- The remainder when $f(x)=x^5-2x^4+ax^3-x^2+bx-2$ is divided by $x+1$ is $-7$. When $f(x)$ is divided by $x-2$ the remainder is $32$. Determine the remainder when $f(x)$ is divided by $x-1$.

1992 Nordic, 2

Tags: integer polynomial , polynomial division , algebra

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.

1994 Poland - Second Round, 1

Tags: polynomial , algebra , polynomial division

Find all real polynomials $P(x)$ of degree $5$ such that $(x-1)^3| P(x)+1$ and $(x+1)^3| P(x)-1$.

1979 Brazil National Olympiad, 2

Tags: polynomial , algebra , quadratic , polynomial division

The remainder on dividing the polynomial $p(x)$ by $x^2 - (a+b)x + ab$ (where $a \not = b$) is $mx + n$. Find the coefficients $m, n$ in terms of $a, b$. Find $m, n$ for the case $p(x) = x^{200}$ divided by $x^2 - x - 2$ and show that they are integral.

2007 Singapore MO Open, 2

Tags: algebra , integer polynomial , polynomial division

Let $n > 1$ be an integer and let $a_1, a_2,... , a_n$ be $n$ different integers. Show that the polynomial $f(x) = (x -a_1)(x - a_2)\cdot ... \cdot (x -a_n) - 1$ is not divisible by any polynomial with integer coefficients and of degree greater than zero but less than $n$ and such that the highest power of $x$ has coefficient $1$.