Found problems: 3597
The polynomial $ f(x) \equal{} x^{4} \plus{} ax^{3} \plus{} bx^{2} \plus{} cx \plus{} d$ has real coefficients, and $ f(2i) \equal{} f(2 \plus{} i) \equal{} 0.$ What is $ a \plus{} b \plus{} c \plus{} d?$
$ \textbf{(A)}\ 0 \qquad \textbf{(B)}\ 1 \qquad \textbf{(C)}\ 4 \qquad \textbf{(D)}\ 9 \qquad \textbf{(E)}\ 16$
Find all positive integer $n$ such that there exists a $P\in\mathbb{R}[x]$ satisfying $P(x^{1998}-x^{-1998})=x^{n}-x^{-n}\forall x\in\mathbb{R}-\{0\}$.
A polynomial is called positivist if it can be written as a product of two non-constant polynomials with non-negative real coefficients. Let $f(x)$ be a polynomial of degree greater than one such that $f(x^n)$ is positivist for some positive integer $n$. Show that $f(x)$ is positivist.
Prove there exists no polynomial $f(x)$, with integer coefficients, such that $f(7) = 11$ and $f(11) = 13$.
Let $p(x)$ be a polynomial with real coefficients such that $p(0)=p(n)$. Prove that there are at least $n$ pairs of real numbers $(x,y)$ where $p(x)=p(y)$ and $y-x$ is a positive integer
Let coefficients of the polynomial$ P (x) = a_dx^d + ... + a_2x^2 + a_0$ where $d \ge 2$, are positive integers. The sequences $(b_n)$ is defined by $b_1 = a_0$ and $b_{n+1} = P (b_n)$ for $n \ge 1$. Prove that for any $n \ge 2$, there exists a prime number $p$ such that $p|b_n$ but it does not divide $b_1, b_2, ..., b_{n-1}$.
Let $P(x)$ be a polynomial of degree $n$ such that $P(x)=3^k$ for $0\le k \le n$. Find $P(n+1)$
Find all polynomials $p$ with real coefficients such that for all real $x$
$$(x-8)p(2x)=8(x-1)p(x).$$
Let $n$ be a fixed integer with $n \ge 2$. We say that two polynomials $P$ and $Q$ with real coefficients are [i]block-similar[/i] if for each $i \in \{1, 2, \ldots, n\}$ the sequences
\begin{eqnarray*}
P(2015i), P(2015i - 1), \ldots, P(2015i - 2014) & \text{and}\\
Q(2015i), Q(2015i - 1), \ldots, Q(2015i - 2014)
\end{eqnarray*}
are permutations of each other.
(a) Prove that there exist distinct block-similar polynomials of degree $n + 1$.
(b) Prove that there do not exist distinct block-similar polynomials of degree $n$.
[i]Proposed by David Arthur, Canada[/i]
Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that
a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and
b) $ I$ contains of a polynomial with constant term $ 1$.
Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$.
[i]Gy. Szekeres[/i]
The polynomial $1976(x+x^2+ \cdots +x^n)$ is decomposed into a sum of polynomials of the form $a_1x + a_2x^2 + \cdots + a_nx^n$, where $a_1, a_2, \ldots , a_n$ are distinct positive integers not greater than $n$. Find all values of $n$ for which such a decomposition is possible.
A polynomial $P$ with integer coefficients has at least $13$ distinct integer roots. Prove that if an integer $n$ is not a root of $P$, then $|P(n)| \geq 7 \cdot 6!^2$, and give an example for equality.
Let $n \geq 3$ and $a_1,a_2,...,a_n$ be complex numbers different from $0$ with $|a_i| < 1$ for all $i \in \{1,2,...,n-1 \}.$ If the coefficients of $f = \prod_{i=1}^n (X-a_i)$ are integers, prove that
$\textbf{a)}$ The numbers $a_1,a_2,...,a_n$ are distinct.
$\textbf{b)}$ If $a_j^2 = a_ia_k,$ then $i=j=k.$
Given an integer $n \geqslant 3$ the polynomial $f(x_1, \ldots, x_n)$ with integer coefficients is called [i]good[/i] if $f(0,\ldots, 0) = 0$ and \[f(x_1, \ldots, x_n)=f(x_{\pi_1}, \ldots, x_{\pi_n}),\]for any permutation of $\pi$ of the numbers $1,\ldots, n$. Denote by $\mathcal{J}$ the set of polynomials of the form \[p_1q_1+\cdots+p_mq_m,\]where $m$ is a positive integer and $q_1,\ldots , q_m$ are polynomials with integer coefficients, and $p_1,\ldots , p_m$ are good polynomials. Find the smallest natural number $D{}$ such that each monomial of degree $D{}$ lies in the set $\mathcal{J}$.
Among all polynomials $P(x)$ with integer coefficients for which $P(-10) = 145$ and $P(9) = 164$, compute the smallest possible value of $|P(0)|.$
The Fibonacci sequence is determined by conditions $ F_0 \equal{} 0, F1 \equal{} 1$, and $ F_k\equal{}F_{k\minus{}1}\plus{}F_{k\minus{}2}$ for all $ k \ge 2$. Let $ n$ be a positive integer and let $ P(x) \equal{} a_mx^m \plus{}. . .\plus{} a_1x\plus{} a_0$ be a polynomial that satisfies the following two conditions:
(1) $ P(F_n) \equal{} F_{n}^{2}$ ;
(2) $ P(F_k) \equal{} P(F_{k\minus{}1}) \plus{} P(F_{k\minus{}2}$ for all $ k \ge 2$.
Find the sum of the coefficients of P.
Let $A$ and $B$ be $2\times 2$ matrices with integer entries such that $A, A+B, A+2B, A+3B,$ and $A+4B$ are all invertible matrices whose inverses have integer entries. Show that $A+5B$ is invertible and that its inverse has integer entries.
Let $P$ be a quadratic (polynomial of degree two) with a positive leading coefficient and negative discriminant. Prove that there exists three quadratics $P_1, P_2, P_3$ such that:
- $P(x) = P_1(x) + P_2(x) + P_3(x)$
- $P_1, P_2, P_3$ have positive leading coefficients and zero discriminants (and hence each has a double root)
- The roots of $P_1, P_2, P_3$ are different
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.
A point in the plane with a cartesian coordinate system is called a [i]mixed point[/i] if one of its coordinates is rational and the other one is irrational. Find all polynomials with real coefficients such that their graphs do not contain any mixed point.
Show that all complex roots of the polynomial $P(z)=a_0z^n+a_1z^{n-1}+\ldots+a_{n-1}z+a_n$, where $0<a_0<\ldots<a_n$, satisfy $|z|>1$.
Consider polynomial $f(x)=ax^3+bx^2+cx+d$ where $a, b, c, d$ are nonnegative integers satisfying $ab+bc+cd+ad=20$. Find the sum of all distinct possible values of $f(1)$.
$\omega$ is a complex number such that $\omega^{2013} = 1$ and $\omega^m \neq 1$ for $m=1,2,\ldots,2012$. Find the number of ordered pairs of integers $(a,b)$ with $1 \le a, b \le 2013$ such that \[ \frac{(1 + \omega + \cdots + \omega^a)(1 + \omega + \cdots + \omega^b)}{3} \] is the root of some polynomial with integer coefficients and leading coefficient $1$. (Such complex numbers are called [i]algebraic integers[/i].)
[i]Victor Wang[/i]
Let $a \in \mathbb{R}.$ Show that for $n \geq 2$ every non-real root $z$ of polynomial $X^{n+1}-X^2+aX+1$ satisfies the condition $|z| > \frac{1}{\sqrt[n]{n}}.$
Let $n$ be a positive integer. Prove that there exist integers $a_1, a_2,..., a_n$ such that for any integer $x$, the number $(... (((x^2 + a_1)^2 + a_2)^2 + ...)^2 + a_{n-1})^2 + a_n$ is divisible by $2n - 1$.