Found problems: 108
Let $n$ be a given positive integer. Alice and Bob play a game. In the beginning, Alice determines an integer polynomial $P(x)$ with degree no more than $n$. Bob doesn’t know $P(x)$, and his goal is to determine whether there exists an integer $k$ such that no integer roots of $P(x) = k$ exist.
In each round, Bob can choose a constant $c$. Alice will tell Bob an integer $k$, representing the number of integer $t$ such that $P(t) = c$. Bob needs to pay one dollar for each round. Find the minimum cost such that Bob can guarantee to reach his goal.
[i]Proposed by ltf0501[/i]
Let $a,b,c,d$ be distinct integers such that
$$(x-a)(x-b)(x-c)(x-d) -4=0$$
has an integer root $r.$ Show that $4r=a+b+c+d.$
For a non-negative integer $n$, call a one-variable polynomial $F$ with integer coefficients $n$-[i]good [/i] if:
(a) $F(0) = 1$
(b) For every positive integer $c$, $F(c) > 0$, and
(c) There exist exactly $n$ values of $c$ such that $F(c)$ is prime.
Show that there exist infinitely many non-constant polynomials that are not $n$-good for any $n$.
Let $P (x)$ be a non constant integer polynomial and positive integer $n$. The sequence $a_0, a_1, ...$ is defined by $a_0 = n$ and $a_k = P (a_{k-1})$ for $k \ge 1$. Given that for each positive integer $b$, the sequence contains a $b$-th power of some positive integer greater than $1$. Prove that deg $P = 1$
It is known that $1$ and $2$ are roots of a polynomial with integer coefficients. Prove that the polynomial has a coefficient with value less than $-1$ .
Let $P(x)$ be a polynomial with integer coefficients of degree $n \ge 3$.
If $x_1,...,x_m$ ($n\ge m\ge3$) are different integers such that $P(x_1) = P(x_2) = ... = P(x_m) = 1$, prove that $P$ cannot have integer roots$.
Equation $$x^n + a_1x^{n-1} + a_2x^{n-2} +...+ a_{n-1}x + a_n = 0$$ with integer non-zero coefficients $a_1$, $a_2$, $...$ , $a_n$ has $n$ different integer roots. Prove that if any two roots are relatively prime, then the numbers $a_{n-1}$ and $a_n$ are coprime.
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 $m>1$.