Found problems: 108
Let $n$ be a positive integer and let $k_0,k_1, \dots,k_{2n}$ be nonzero integers such that $k_0+k_1 +\dots+k_{2n}\neq 0$. Is it always possible to a permutation $(a_0,a_1,\dots,a_{2n})$ of $(k_0,k_1,\dots,k_{2n})$ so that the equation
\begin{align*}
a_{2n}x^{2n}+a_{2n-1}x^{2n-1}+\dots+a_0=0
\end{align*}
has not integer roots?
Determine all pairs $(P, d)$ of a polynomial $P$ with integer coefficients and an integer $d$ such that the equation $P(x) - P(y) = d$ has infinitely many solutions in integers $x$ and $y$ with $x \neq y$.
Find the free coefficient of the polynomial $P(x)$ with integer coefficients, knowing that it is less than $1000$ in absolute value and that $P(19) = P(94) = 1994$.
Let $P(x) = x^4 + x^3 + x^2 + x + 1$. Show that there exist two non-constant polynomials $Q(y)$ and $R(y)$ with integer coefficients such that for all $Q(y) \cdot R(y)= P(5y^2)$ for all $y$ .
Polynomial P(x) with integer coefficient is called [i]cube-presented[/i] if it can be represented as sum of several cube of polynomials with integer coefficients.
Examples: $3x + 3x^2$ is cube-represented because $3x + 3x^2 = (x + 1)^3 +(-x)^3 + (-1)^3$.
a) Is $3x^2$ a cube-represented polynomial?
b). How many quadratic polynomial P(x) with integer coefficients belong to the set $\{1,2, 3, ...,2017\}$ which is cube-represented?
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).
Find all polynomials $P$ with integer coefficients such that wherever $a, b \in N$ and $a+b$ is a square we have $P(a) + P(b)$ is also a square.
Let $f(x)$ be a polynomial with integer coefficients. We assume that there exists a positive integer $k$ and $k$ consecutive integers $n, n+1, ... , n+k -1$ so that none of the numbers $f(n), f(n+ 1),... , f(n + k - 1)$ is divisible by $k$.
Show that the zeroes of $f(x)$ are not integers.
Consider the trinomial $f(x) = x^2 + 2bx + c$ with integer coefficients $b$ and $c$.
Prove that if $f(n) \ge 0$ for all integers $n$, then $f(x) \ge 0$ even for all rational numbers $x$.
Suppose $f(x)$ is a polynomial with integer coefficients satisfying the condition $0 \le f(c) \le 1997$ for each $c \in \{0, 1, ..., 1998\}$. Is is true that $f(0) = f(1) = ... = f(1998)$?
(variation of [url=https://artofproblemsolving.com/community/c6h49788p315649]1997 IMO Shortlist p12[/url])
Find the largest integer $N \in \{1, 2, \ldots , 2019 \}$ such that there exists a polynomial $P(x)$ with integer coefficients satisfying the following property: for each positive integer $k$, $P^k(0)$ is divisible by $2020$ if and only if $k$ is divisible by $N$. Here $P^k$ means $P$ applied $k$ times, so $P^1(0)=P(0), P^2(0)=P(P(0)),$ etc.
A polynomial $P(x)$ of degree $n \ge 5$ with integer coefficients has $n$ distinct integer roots, one of which is $0$. Find all integer roots of the polynomial $P(P(x))$.
Let $m,n$ be positive integers and $p$ be a prime number.
Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.
Prove that the polynomial $P(X) = (X^2-12X +11)^4+23$ can not be written as the product of three non-constant polynomials with integer coefficients.
(Le Anh Vinh)
Decide whether for every polynomial $P$ of degree at least $1$, there exist infinitely many primes that divide $P(n)$ for at least one positive integer $n$.
[i](Walther Janous)[/i]
Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.
Find all positive integers $n$ such that there exist non-constant polynomials with integer coefficients $f_1(x),...,f_n(x)$ (not necessarily distinct) and $g(x)$ such that $$1 + \prod_{k=1}^{n}\left(f^2_k(x)-1\right)=(x^2+2013)^2g^2(x)$$
Determine all polynomials $p(x)$ with non-negative integer coefficients such that $p (1) = 7$ and $p (10) = 2014$.
Let $P(x)$ be a polynomial with integer coefficients. It is known that the number $\sqrt2+\sqrt3$ is its root. Prove that the number $\sqrt2-\sqrt3$ is also its root.
A polynomial $P(x)$ with integer coefficients and a positive integer $a>1$, are such that for all integers $x$, there exists an integer $z$ such that $aP(x)=P(z)$. Find all such pairs of $(P(x),a)$.
Let $S = \{f(a, b) | a, b = 1,2,3, 4$ and $a \ne b\}$, and consider all nonzero polynomials $p(X,Y )$ with integer coefficients such that $p(a, b) = 0$ for every element $(a,b)$ in $S$.
(a) What is the minimal degree of such polynomial $p(X, Y )$ ?
(b) Determine all such polynomials $p(X, Y )$ with minimal degree.
Find a polynomial $P$ of lowest possible degree such that
(a) $P$ has integer coefficients,
(b) all roots of $P$ are integers,
(c) $P(0) = -1$,
(d) $P(3) = 128$.
Let $P(x)=x^{2016}+2x^{2015}+...+2017,Q(x)=1399x^{1398}+...+2x+1$. Prove that there are strictly increasing sequances $a_i,b_i, i=1,...$ of positive integers such that $gcd(a_i,a_{i+1})=1$ for each $i$. Moreover, for each even $i$, $P(b_i) \nmid a_i, Q(b_i) | a_i$ and for each odd $i$, $P(b_i)|a_i,Q(b_i) \nmid a_i$
Proposed by [i]Shayan Talaei[/i]
Let $a,b$ be integers and $p$ be a prime number such that:
(i) $p$ is the greatest common divisor of $a$ and $b$;
(ii) $p^2$ divides $a$.
Prove that the polynomial $x^{n+2}+ax^{n+1}+bx^{n}+a+b$ cannot be decomposed into the product of two polynomials with integer coefficients and degree greater than $1$.
Find all $f(x)\in \mathbb Z (x)$ that satisfies the following condition, with the lowest degree.
[b]Condition[/b]: There exists $g(x),h(x)\in \mathbb Z (x)$ such that $$f(x)^4+2f(x)+2=(x^4+2x^2+2)g(x)+3h(x)$$.