Found problems: 3597
Show that the only polynomial of odd degree satisfying $p(x^2-1) = p(x)^2 - 1$ for all $x$ is $p(x) = x$
Let $ p=\sum\limits_{k=0}^n a_kX^k\in R[X] $ a polynomial such that all his roots lie in the half plane $ \{z\in C| Re(z)<0 \}. $ Prove that $ a_ka_{k+3}<a_{k+1}a_{k+2}, $ for every k=0,1,2...,n-3.
Sequences $ (x_n)$ and $ (y_n)$ are constructed as follows: $ x_0 \equal{} 365$, $ x_{n\plus{}1} \equal{} x_n\left(x^{1986} \plus{} 1\right) \plus{} 1622$, and $ y_0 \equal{} 16$, $ y_{n\plus{}1} \equal{} y_n\left(y^3 \plus{} 1\right) \minus{} 1952$, for all $ n \ge 0$. Prove that $ \left|x_n\minus{} y_k\right|\neq 0$ for any positive integers $ n$, $ k$.
Determine all real-coefficient polynomials $P(x)$ such that
\[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.
Find all functions $f: \mathbb{R} \to \mathbb{R}$ satisfying
\[f(x+f(y))-f(x)=(x+f(y))^4-x^4\]
for all $x,y \in \mathbb{R}$.
Let n be a positive integer. Find all complex numbers $x_{1}$, $x_{2}$, ..., $x_{n}$ satisfying the following system of equations:
$x_{1}+2x_{2}+...+nx_{n}=0$,
$x_{1}^{2}+2x_{2}^{2}+...+nx_{n}^{2}=0$,
...
$x_{1}^{n}+2x_{2}^{n}+...+nx_{n}^{n}=0$.
The sum of the squares of five real numbers $a_1, a_2, a_3, a_4, a_5$ equals $1$. Prove that the least of the numbers $(a_i - a_j)^2$, where $i, j = 1, 2, 3, 4,5$ and $i \neq j$, does not exceed $\frac{1}{10}.$
Let $p(x)$ be a polynomial with integer coefficients and let $n$ be an integer. Suppose that there is a positive integer $k$ for which $f^{(k)}(n) = n$, where $f^{(k)}(x)$ is the polynomial obtained as the composition of $k$ polynomials $f$. Prove that $p(p(n)) = n$.
Find all polynomials $P(x)$ of degree $1$ such that
$\underset {a\le x\le b}{max} P(x) - \underset {a\le x\le b}{min} P(x) =b-a$ , $\forall a,b\in R$ where $a < b$
Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.
Given the polynomial \[ f(x)=x^n+a_{1}x^{n-1}+a_{2}x^{n-2}+\cdots+a_{n-1}x+a_n \] with integer coefficients $a_1,a_2,\ldots,a_n$, and given also that there exist four distinct integers $a$, $b$, $c$ and $d$ such that \[ f(a)=f(b)=f(c)=f(d)=5, \] show that there is no integer $k$ such that $f(k)=8$.
Construct a set $S$ of polynomials inductively by the rules:
(i) $x\in S$;
(ii) if $f(x)\in S$, then $xf(x)\in S$ and $x+(1-x)f(x)\in S$.
Prove that there are no two distinct polynomials in $S$ whose graphs intersect within the region $\{0 < x < 1\}$.
Determine all polynomials with integral coefficients $P(x)$ such that if $a,b,c$ are the sides of a right-angled triangle, then $P(a), P(b), P(c)$ are also the sides of a right-angled triangle. (Sides of a triangle are necessarily positive. Note that it's not necessary for the order of sides to be preserved; if $c$ is the hypotenuse of the first triangle, it's not necessary that $P(c)$ is the hypotenuse of the second triangle, and similar with the others.)
Find all real polynomials $P(x)$ of degree $5$ such that $(x-1)^3| P(x)+1$ and $(x+1)^3| P(x)-1$.
Prove that there exists an infinite set of points \[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \] in the plane with the following property: For any three distinct integers $a,b,$ and $c$, points $P_a$, $P_b$, and $P_c$ are collinear if and only if $a+b+c=2014$.
The zeroes of the function $f(x)=x^2-ax+2a$ are integers. What is the sum of all possible values of $a$?
$\textbf{(A) }7\qquad\textbf{(B) }8\qquad\textbf{(C) }16\qquad\textbf{(D) }17\qquad\textbf{(E) }18$
Find all pairs $(m,n)$ of integers with $m > n > 7$ for which there exists a polynomial $p(x)$ with integer coefficients such that $p(7) = 77, p(m) = 0$, and $p(n) = 85$.
For integer $n > 1$, consider $n$ cube polynomials $P_1(x), ..., P_n(x)$ such that each polynomial has $3$ distinct real roots. Denote $S$ as the set of roots of following equation $P_1(x)P_2(x)P_3(x)... P_n(x) = 0$.
It is also known that for each $1 \le i < j \le n, P_i(x)P_j(x) = 0$ has $5$ distinct real roots.
1. Prove that if for each $a, b \in S$, there is exactly one $i \in\{1,2, 3,..., n\}$ such that $P_i(a) = P_i(b) = 0$ then $n = 7$.
2. Prove that if $n > 7$ then $|S| = 2n + 1$.
Consider a continuous function $f:[0,1]\rightarrow \mathbb{R}$ such that for any third degree polynomial function $P:[0,1]\to [0,1]$, we have
\[\int_0^1f(P(x))dx=0\]
Prove that $f(x)=0,\ (\forall)x\in [0,1]$.
[i]Mihai Piticari[/i]
Let $P_n(x)=a_0 + a_1x + \cdots + a_nx^n$, with $n \geq 2$, be a real-coefficient polynomial. Prove that if there exists $a > 0$ such that
\begin{align*}
P_n(x) = (x + a)^2 \left( \sum_{i=0}^{n-2} b_i x^i \right),
\end{align*}
where $b_i$ are positive real numbers, then there exists some $i$, with $1 \leq i \leq n-1$, such that \[a_i^2 - 4a_{i-1}a_{i+1} \leq 0.\]
Let $f (x)$ be a polynomial with integer coefficients. Prove that if $f (x)= 1979$ for four different integer values of $x$, then $f (x)$ cannot be equal to $2\times 1979$ for any integral value of $x$.
Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and
$$P(x)^2+1=(x^2+1)Q(x)^2.$$
Let $n$ be a positive integer. Find all positive integers $m$ for which there exists a polynomial $f(x) = a_{0} + \cdots + a_{n}x^{n} \in \mathbb{Z}[X]$ ($a_{n} \not= 0$) such that $\gcd(a_{0},a_{1},\cdots,a_{n},m)=1$ and $m|f(k)$ for each $k \in \mathbb{Z}$.
A polynomial $P(x)$ has unity as the coefficient of its highest power, and has the property that with natural number arguments, it can take all values of form $2^M$ , where $M$ is a natural number. Prove that the polynomial is of degree $1$.
Let $n$ be a positive integer. Show that there are positive real numbers $a_0, a_1, \dots, a_n$ such that for each choice of signs the polynomial
$$\pm a_nx^n\pm a_{n-1}x^{n-1} \pm \dots \pm a_1x \pm a_0$$
has $n$ distinct real roots.
(Proposed by Stephan Neupert, TUM, München)