Does there exist a nonzero polynomial $P(x)$ with integer coefficients satisfying both of the following conditions? [list] [*]$P(x)$ has no rational root; [*]For every positive integer $n$, there exists an integer $m$ such that $n$ divides $P(m)$. [/list]

Prove that for every $n$ nonzero integer , there are infinite triples of nonzero integers $a, b$ and $c$ that satisfy the conditions: 1. $a + b + c = n$ 2. $ax^2 + bx + c = 0$ has rational roots.

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Prove that $ \forall n > 1, n \in \mathbb{N}$ the equation \[ \sum^n_{k\equal{}1} \frac{x^k}{k!} \plus{} 1 \equal{} 0\] has no rational roots.

Find all pairs $(p,q)$ of positive primes such that the equation $3x^2 - px + q = 0$ has two distinct rational roots.

Assume natural number $n\ge2$. Amin and Ali take turns playing the following game: In each step, the player whose turn has come chooses index $i$ from the set $\{0,1,\cdots,n\}$, such that none of the two players had chosen this index in the previous turns; also this player in this turn chooses nonzero rational number $a_i$ too. Ali performs the first turn. The game ends when all the indices $i\in\{0,1,\cdots,n\}$ were chosen. In the end, from the chosen numbers the following polynomial is built: $$P(x)=a_nx^n+\cdots+a_1x+a_0$$ Ali's goal is that the preceding polynomial has a rational root and Amin's goal is that to prevent this matter. Find all $n\ge2$ such that Ali can play in a way to be sure independent of how Amin plays achieves his goal.

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

We are given the equation $x^3-cx^2+(c-3)x+1=0$, where $c$ is an arbitrary number. Prove that, if the equation has at least one rational root, then all of its roots are rational.

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

Let $f$ and $g$ be two nonzero polynomials with integer coefficients and $\deg f>\deg g$. Suppose that for infinitely many primes $p$ the polynomial $pf+g$ has a rational root. Prove that $f$ has a rational root.

If $p$ and $q$ are odd integers, prove that the equation $$x^2 + 2px + 2q = 0$$ has no rational roots.