Found problems: 3597
Let $P(x)=ax^3+bx^2+cx+d$ be a polynomial with real coefficients such that \[\min\{d,b+d\}> \max\{|{c}|,|{a+c}|\}\]
Prove that $P(x)$ do not have a real root in $[-1,1]$.
For some constant $k$ the polynomial $p(x) = 3x^2 + kx + 117$ has the property that $p(1) = p(10)$. Evaluate $p(20)$.
Find all real solutions to the equation
$$(x+1)^{2001}+(x+1)^{2000}(x-2)+(x+1)^{1999}(x-2)^2+...+(x+1)^2(x-2)^{1999}+(x+1)^{2000}(x-2)+(x+1)^{2001}=0$$
A game is played in three moves. The first player picks any real number, then the second player makes it the coefficient of a cubic, except that the coefficient of $x^3$ is already fixed at $1$. Can the first player make his choices so that the final cubic has three distinct integer roots?
Suppose that $k,m,n$ are positive integers with $k \le n$. Prove that:
\[\sum_{r=0}^m \dfrac{k \binom{m}{r} \binom{n}{k}}{(r+k) \binom{m+n}{r+k}} = 1\]
Consider the set $ M = \{1,2, \ldots ,2008\}$. Paint every number in the set $ M$ with one of the three colors blue, yellow, red such that each color is utilized to paint at least one number. Define two sets:
$ S_1=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have the same color and }2008 | (x + y + z)\}$;
$ S_2=\{(x,y,z)\in M^3\ \mid\ x,y,z\text{ have three pairwisely different colors and }2008 | (x + y + z)\}$.
Prove that $ 2|S_1| > |S_2|$ (where $ |X|$ denotes the number of elements in a set $ X$).
Let $P(x)$ be a polynomial with integer coefficients. Prove that the polynomial $Q(x) = P(x^4)P(x^3)P(x^2)P(x)+1$ has no integer roots.
Find all polynomials $p(x)$ with real coe ffcients such that
\[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\]
for all $a, b, c\in\mathbb{R}$.
[i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]
$p(x)$ is a polynomial of degree $n$ with every coefficient $0 $ or $\pm1$, and $p(x)$ is divisible by $ (x - 1)^k$ for some integer $ k > 0$. $q$ is a prime such that $\frac{q}{\ln q}< \frac{k}{\ln n+1}$. Show that the complex $q$-th roots of unity must be roots of $ p(x). $
Find all real non-zero polynomials satisfying $P(x)^3+3P(x)^2=P(x^{3})-3P(-x)$ for all $x\in\mathbb{R}$.
Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]
Fix two positive integers $a,k\ge2$, and let $f\in\mathbb{Z}[x]$ be a nonconstant polynomial. Suppose that for all sufficiently large positive integers $n$, there exists a rational number $x$ satisfying $f(x)=f(a^n)^k$. Prove that there exists a polynomial $g\in\mathbb{Q}[x]$ such that $f(g(x))=f(x)^k$ for all real $x$.
[i]Victor Wang.[/i]
On the graph of a polynomial with integer coefficients, two points are chosen with integer coordinates. Prove that if the distance between them is an integer, then the segment that connects them is parallel to the horizontal axis.
Find all positive integers $n$ such that there exist a permutation $\sigma$ on the set $\{1,2,3, \ldots, n\}$ for which
\[\sqrt{\sigma(1)+\sqrt{\sigma(2)+\sqrt{\ldots+\sqrt{\sigma(n-1)+\sqrt{\sigma(n)}}}}}\]
is a rational number.
Find all $f:\mathbb{R}\to\mathbb{R}$ such that for all $x,y\in\mathbb{R}$, $f(x)+f(y) = f(x+y)$ and $f(x^{2013}) = f(x)^{2013}$.
[i]Proposed by Calvin Deng[/i]
Let $n$ be a natural number and $P_1, P_2, ... , P_n$ are polynomials with integer coefficients, each of degree at least $2$. Let $S$ be the set of all natural numbers $N$ for which there exists a natural number $a$ and an index $1 \le i \le n$ such that $P_i(a) = N$. Prove, that there are infinitely many primes that do not belong to $S$.
The function $f$ has the property that, for each real number $x,$ \[ f(x)+f(x-1) = x^2. \] If $f(19)=94,$ what is the remainder when $f(94)$ is divided by 1000?
Let $P (x)$ be a polynomial with integer coefficients and $P (n) =m$ for some integers $n, m$ ($m \ne 10$). Prove that $P (n + km)$ is divisible by $m$ for any integer $k$.
Let $P(x)$ be a non-zero polynomial with real coefficient so that $P(0)=0$.Prove that for any positive real number $M$ there exist a positive integer $d$ so that for any monic polynomial $Q(x)$ with degree at least $d$ the number of integers $k$ so that $|P(Q(k))| \le M$ is at most equal to the degree of $Q$.
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$.
Let $J\subsetneq I\subseteq \mathbb R$ be non-empty open intervals, and let $f_1, f_2,\ldots$ be real polynomials satisfying the following conditions:
[list]
[*] $f_i(x)\ge 0$ for all $i\ge 1$ and $x\in I$,
[*] $\sum\limits_{i=1}^\infty f_i(x)$ is finite for all $x\in I$,
[*] $\sum\limits_{i=1}^\infty f_i(x)=1$ for all $x\in J$.
[/list]
Do these conditions imply that $\sum\limits_{i=1}^\infty f_i(x)=1$ also for all $x\in I$?
[i]Proposed by András Imolay, Budapest[/i]
Let $A,B\in \mathcal{M}_n(\mathbb{C})$ such that $AB=BA$ and $\det B\neq 0$.
a) If $|\det(A+zB)|=1$ for any $z\in \mathbb{C}$ such that $|z|=1$, then $A^n=O_n$.
b) Is the question from a) still true if $AB\neq BA$ ?
Prove that $1 + x + x^2 + x^3 + ...+ x^{2011} \ge 0$ for every $x \ge - 1$ .
[list=1]
[*] Let $G$ be a $(4, 4)$ unoriented graph, 2-regulate, containing a cycle with the length 3. Find the characteristic polynomial $P_G (\lambda)$ , its spectrum $Spec (G)$ and draw the graph $G$.
[*] Let $G'$ be another 2-regulate graph, having its characteristic polynomial $P_{G'} (\lambda) = \lambda^4 - 4\lambda^2 + \alpha, \alpha \in \mathbb{R}$. Find the spectrum $Spec(G')$ and draw the graph $G'$.
[*] Are the graphs $G$ and $G'$ cospectral or isomorphic?
[/list]
Show that there is no polynomial $P(x)$ of degree $998$ with real coefficients which satisfies $P(x^2 + 1) = P(x)^2 - 1$ for all $x$.