Olympiad problems

2012 NZMOC Camp Selection Problems, 4

Let $p(x)$ be a polynomial with integer coefficients, and let $a, b$ and $c$ be three distinct integers. Show that it is not possible to have $p(a) = b$, $p(b) = c$, and $p(c) = a$.

1998 Austrian-Polish Competition, 5

Tags: Integer Polynomial , Integers , polynomial , algebra

Determine all pairs $(a, b)$ of positive integers for which the equation $x^3 - 17x^2 + ax - b^2 = 0$ has three integer roots (not necessarily different).

2011 Tuymaada Olympiad, 4

Tags: quadratics , algebra , polynomial , Analytic Number Theory , Integer sequence , Integer Polynomial

Let $P(n)$ be a quadratic trinomial with integer coefficients. For each positive integer $n$, the number $P(n)$ has a proper divisor $d_n$, i.e., $1<d_n<P(n)$, such that the sequence $d_1,d_2,d_3,\ldots$ is increasing. Prove that either $P(n)$ is the product of two linear polynomials with integer coefficients or all the values of $P(n)$, for positive integers $n$, are divisible by the same integer $m>1$.

2016 Thailand TSTST, 1

Tags: algebra , polynomial , number theory , Integer Polynomial

Find all polynomials $P\in\mathbb{Z}[x]$ such that $$|P(x)-x|\leq x^2+1$$ for all real numbers $x$.