Found problems: 3597
Determine all polynomials $P(x)$ with integer coefficients such that, for any positive integer $n$, the equation $P(x)=2^n$ has an integer root.
Determine all integers $n$ such that both of the numbers:
$$|n^3 - 4n^2 + 3n - 35| \text{ and } |n^2 + 4n + 8|$$
are both prime numbers.
Consider a sequence $x_1,x_2,\cdots x_{12}$ of real numbers such that $x_1=1$ and for $n=1,2,\dots,10$ let \[ x_{n+2}=\frac{(x_{n+1}+1)(x_{n+1}-1)}{x_n}. \] Suppose $x_n>0$ for $n=1,2,\dots,11$ and $x_{12}=0$. Then the value of $x_2$ can be written as $\frac{\sqrt{a}+\sqrt{b}}{c}$ for positive integers $a,b,c$ with $a>b$ and no square dividing $a$ or $b$. Find $100a+10b+c$.
[i]Proposed by Michael Kural[/i]
The complex numbers $\alpha_1$, $\alpha_2$, $\alpha_3$, and $\alpha_4$ are the four distinct roots of the equation $x^4+2x^3+2=0$. Determine the unordered set \[\{\alpha_1\alpha_2+\alpha_3\alpha_4,\alpha_1\alpha_3+\alpha_2\alpha_4,\alpha_1\alpha_4+\alpha_2\alpha_3\}.\]
Find all positive integers $ k$ for which the following statement is true: If $ F(x)$ is a polynomial with integer coefficients satisfying the condition $ 0 \leq F(c) \leq k$ for each $ c\in \{0,1,\ldots,k \plus{} 1\}$, then $ F(0) \equal{} F(1) \equal{} \ldots \equal{} F(k \plus{} 1)$.
Let $ p$ be a prime number and $ f$ an integer polynomial of degree $ d$ such that $ f(0) = 0,f(1) = 1$ and $ f(n)$ is congruent to $ 0$ or $ 1$ modulo $ p$ for every integer $ n$. Prove that $ d\geq p - 1$.
Let $P$ be a polynomial of degree $n$ satisfying
\[P(k) = \binom{n+1}{k}^{-1} \qquad \text{ for } k = 0, 1, . . ., n.\]
Determine $P(n + 1).$
Let $p$ be a prime number gretater then $3$. What is the number of pairs $(m, n)$ of integers with $0 <m <n <p$, for which the polynomial $x^p + px^n + px^m +1$ is not a product of two non-constant polynomials with integer coefficients can be written?
Let $n \ge 3$ be a positive integer. Find all nonconstant real polynomials $f_1(x), f_2(x), ..., f_n(x)$ such that $f_k(x)f_{k+1}(x) = f_{k+1}(f_{k+2}(x))$, $1 \le k \le n$ for all real x. [All suffixes are taken modulo $n$.]
Let $p(x)$ be a polynomial with integer coefficients. Suppose that there exist different integers $a$ and $b$ such that $f(a) = b$ and $f(b) = a$. Show that the equation $f(x) = x$ has at most one integer solution.
Let $\delta$ be a symbol such that $\delta \neq 0$ and $\delta^2 = 0$. Define $\mathbb R[\delta] = \{a + b \delta | a, b \in \mathbb R\}$, where $a+ b \delta = c+ d \delta$ if and only if $a = c$ and $b = d$, and define
\[(a + b \delta) + (c + d \delta) = (a + c) + (b + d) \delta,\]\[(a + b \delta) \cdot (c + d \delta) = ac + (ad + bc) \delta.\]
Let $P(x)$ be a polynomial with real coefficients. Show that $P(x)$ has a multiple real root if and only if $P(x)$ has a non-real root in $\mathbb R[\delta].$
Given $n$, let $k = k(n)$ be the minimal degree of any monic integral polynomial
$$f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0$$
such that the value of $f(x)$ is exactly divisible by $n$ for every integer $x.$ Find the relationship between $n$ and $k(n)$. In particular, find the value of $k(n)$ corresponding to $n = 10^6.$
A Mediterranean polynomial has only real roots and it is of the form
\[ P(x) = x^{10}-20x^9+135x^8+a_7x^7+a_6x^6+a_5x^5+a_4x^4+a_3x^3+a_2x^2+a_1x+a_0 \] with real coefficients $a_0\ldots,a_7$. Determine the largest real number that occurs as a root of some Mediterranean polynomial.
[i](Proposed by Gerhard Woeginger, Austria)[/i]
Find all pairs $ (m, n)$ of positive integers that have the following property:
For every polynomial $P (x)$ of real coefficients and degree $m$, there exists a polynomial $Q (x)$ of real coefficients and degree $n$ such that $Q (P (x))$ is divisible by $Q (x)$.
Let $x+y=a$ and $xy=b$. The expression $x^6+y^6$ can be written as a polynomial in terms of $a$ and $b$. What is this polynomial?
Given arbitrary positive integer $ a$ larger than $ 1$, show that for any positive integer $ n$, there always exists a n-degree integral coefficient polynomial $ p(x)$, such that $ p(0)$, $ p(1)$, $ \cdots$, $ p(n)$ are pairwise distinct positive integers, and all have the form of $ 2a^k\plus{}3$, where $ k$ is also an integer.
i thought that this problem was in mathlinks but when i searched i didn't find it.so here it is:
Find all positive integers m for which for all $\alpha,\beta \in \mathbb{Z}-\{0\}$
\[ \frac{2^m \alpha^m-(\alpha+\beta)^m-(\alpha-\beta)^m}{3 \alpha^2+\beta^2} \in \mathbb{Z} \]
We are given a natural number $d$. Find all open intervals of maximum length $I \subseteq R$ such that for all real numbers $a_0,a_1,...,a_{2d-1}$ inside interval $I$, we have that the polynomial $P(x)=x^{2d}+a_{2d-1}x^{2d-1}+...+a_1x+a_0$ has no real roots.
A polynomial $P(x)$ with real coefficients and degree $2021$ is given. For any real $a$ polynomial $x^{2022}+aP(x)$ has at least one real root.
Find all possible values of $P(0)$
On a blackboard, several polynomials of degree $37$ are written, each of them has the leading coefficient equal to $1$. Initially all coefficients of each polynomial are non-negative. By one move it is allowed to erase any pair of polynomials $f, g$ and replace it by another pair of polynomials $f_1, g_1$ of degree $37$ with the leading coefficients equal to $1$ such that either $f_1+g_1 = f+g$ or $f_1g_1 = fg$. Prove that it is impossible that after some move each polynomial
on the blackboard has $37$ distinct positive roots. [i](8 points)[/i]
[i]Alexandr Kuznetsov[/i]
Let $a, b, c$ be three non-zero integers. It is known that the sums $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}$ and $\frac{b}{a}+\frac{c}{b}+\frac{a}{c}$ are integers. Find these sums.
For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \in \mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ [i]excludes[/i] $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?
[i]The following problem is open in the sense that the answer to part (b) is not currently known. A proof of part (a) will be awarded 5 points. Up to 7 additional points may be awarded for progress on part (b).[/i]
Let $p(x)$ be a polynomial of degree $d$ with coefficients belonging to the set of rational numbers $\mathbb{Q}$. Suppose that, for each $1 \le k \le d-1$, $p(x)$ and its $k$th derivative $p^{(k)}(x)$ have a common root in $\mathbb{Q}$; that is, there exists $r_k \in \mathbb{Q}$ such that $p(r_k) = p^{(k)}(r_k) = 0$.
(a) Prove that if $d$ is prime then there exist constants $a, b, c \in \mathbb{Q}$ such that
\[ p(x) = c(ax + b)^d. \]
(b) For which integers $d \ge 2$ does the conclusion of part (a) hold?
Find all injective functions $f\colon \mathbb{R}^* \to \mathbb{R}^* $ from the non-zero reals to the non-zero reals, such that \[f(x+y) \left(f(x) + f(y)\right) = f(xy)\] for all non-zero reals $x, y$ such that $x+y \neq 0$.
A sequence of polynomials $\{f_n\}_{n=0}^{\infty}$ is defined recursively by $f_0(x)=1$, $f_1(x)=1+x$, and
\[(k+1)f_{k+1}(x)-(x+1)f_k(x)+(x-k)f_{k-1}(x)=0, \quad k=1,2,\ldots\]
Prove that $f_k(k)=2^k$ for all $k\geq 0$.