Found problems: 4275
Consider the integral $$\int_{-1}^1 x^nf(x) \; dx$$ for every $n$-th degree polynomial $f$ with integer coefficients. Let $\alpha_n$ denote the smallest positive real number that such an integral can give. Determine the limit value $$\lim_{n\to \infty} \frac{\log \alpha_n}n.$$
Let $ f(x)$ be a function defined on $ [0,\ 1]$. For $ n=1,\ 2,\ 3,\ \cdots$, a polynomial $ P_n(x)$ is defined by $ P_n(x)=\sum_{k=0}^n {}_nC{}_k f\left(\frac{k}{n}\right)x^k(1-x)^{n-k}$. Prove that $ \lim_{n\to\infty} \int_0^1 P_n(x)dx=\int_0^1 f(x)dx$.
Do there exist quadratic trinomials $P, \ \ Q, \ \ R$ such that for every integers $x$ and $y$ an integer $z$ exists satisfying $P(x)+Q(y)=R(z)?$
[i]Proposed by A. Golovanov[/i]
Find the smallest number $n$ such that there exist polynomials $f_1, f_2, \ldots , f_n$ with rational coefficients satisfying \[x^2+7 = f_1\left(x\right)^2 + f_2\left(x\right)^2 + \ldots + f_n\left(x\right)^2.\]
[i]Proposed by Mariusz Skałba, Poland[/i]
Polynomial $P(x)$ has integer coefficients. Prove, that if polynomials $P(x)$ and $P(P(P(x)))$ have common real root, they also have a common integer root.
Prove that there does not exist a polynomial $f(x)$ with integer coefficients for which $f(2008) = 0$ and $f(2010) = 1867$.
Let $n > 1$ be an integer and let $f(x) = x^n + 5 \cdot x^{n-1} + 3.$ Prove that there do not exist polynomials $g(x),h(x),$ each having integer coefficients and degree at least one, such that $f(x) = g(x) \cdot h(x).$
Find a 3rd degree polynomial whose roots are $r_a$, $r_b$ and $r_c$ where $r_a$ is the radius of the outer inscribed circle of $ABC$ with respect to $A$.
Let $s_k$ denote the sum of the $\textit{k}$th powers of the roots of the polynomial $x^3-5x^2+8x-13$. In particular, $s_0=3$, $s_1=5$, and $s_2=9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \, s_k + b \, s_{k-1} + c \, s_{k-2}$ for $k = 2$, $3$, $....$ What is $a+b+c$?
$\textbf{(A)} \; -6 \qquad \textbf{(B)} \; 0 \qquad \textbf{(C)} \; 6 \qquad \textbf{(D)} \; 10 \qquad \textbf{(E)} \; 26$
$p(x)$ is the cubic $x^3 - 3x^2 + 5x$. If $h$ is a real root of $p(x) = 1$ and $k$ is a real root of $p(x) = 5$, find $h + k$.
Define the function $\xi : \mathbb Z^2 \to \mathbb Z$ by $\xi(n,k) = 1$ when $n \le k$ and $\xi(n,k) = -1$ when $n > k$, and construct the polynomial \[ P(x_1, \dots, x_{1000}) = \prod_{n=1}^{1000} \left( \sum_{k=1}^{1000} \xi(n,k)x_k \right). \]
(a) Determine the coefficient of $x_1x_2 \dots x_{1000}$ in $P$.
(b) Show that if $x_1, x_2, \dots, x_{1000} \in \left\{ -1,1 \right\}$ then $P(x_1,x_2,\dots,x_{1000}) = 0$.
[i]Proposed by Evan Chen[/i]
Let $n$ be a natural number and let $P (t) = 1 + t + t^2 + ... + t^{2n}$.
If $x \in R$ such that $P (x)$ and $P (x^2)$ are rational numbers, prove that $x$ is rational number.
Let $ M$ be the set of all polynomials $ P(x)$ with pairwise distinct integer roots, integer coefficients and all absolut values of the coefficients less than $ 2007$. Which is the highest degree among all the polynomials of the set $ M$?
Given a positive integer $n$, suppose that $P(x,y)$ is a real polynomial such that
\[P(x,y)=\frac{1}{1+x+y} \hspace{0.5cm} \text{for all $x,y\in\{0,1,2,\dots,n\}$} \] What is the minimum degree of $P$?
[i]Proposed by Loke Zhi Kin[/i]
Let $a_0,$ $a_1,$ $\ldots,$ $a_n$ be complex numbers such that [center]$|a_nz^n+a_{n-1}z^{n-1}+\ldots+a_1z+a_0| \le 1,$ for any $z \in \mathbb{C}$ with $|z|=1.$[/center]
Prove that $|a_k| \le 1$ and $|a_0+a_1+\ldots+a_n-(n+1)a_k| \le n,$ for any $k=\overline{0,n}.$
Let $m,n$ be positive integers and $p$ be a prime number.
Show that if $\frac{7^m + p \cdot 2^n}{7^m - p \cdot 2^n}$ is an integer, then it is a prime number.
A solid in Euclidean $3$-space extends from $z = \frac{-h}{2}$ to $z = \frac{+h}{2}$ and the area of the section $z = k$ is a polynomial in $k$ of degree at most $3$. Show that the volume of the solid is $\frac{h(B + 4M + T)}{6},$ where $B$ is the area of the bottom $(z = \frac{-h}{2})$, $M$ is the area of the middle section $(z = 0),$ and $T$ is the area of the top $(z = \frac{h}{2})$. Derive the formulae for the volumes of a cone and a sphere.
Find a nonzero monic polynomial $P(x)$ with integer coefficients and minimal degree such that $P(1-\sqrt[3]2+\sqrt[3]4)=0$. (A polynomial is called $\textit{monic}$ if its leading coefficient is $1$.)
Prove that:$a+b+c+d \geq \frac{2}{3}(ab+bc+ca+ad+ac+bd)$
where $a;b;c;d$ are positive real numbers satisfying $2(ab+bc+cd+da+ac+bd)+abc+bcd+cda+dab=16$
Each of the quadratic polynomials $P(x), Q(x)$ and $P(x)+Q(x)$ with real coefficients has a repeated root. Is it guaranteed that those roots coincide?
[i]Boris Frenkin[/i]
Which polygon inscribed in a given circle has the property that the sum of the squares of the lengths of its sides is maximum?
Find all integers $a,b,m,n$, with $m>n>1$, for which the polynomial $f(X)=X^n+aX+b$ divides the polynomial $g(X)=X^m+aX+b$.
[i]Laurentiu Panaitopol[/i]
[list=1]
[*] Prove that, the sequence of remainders obtained when the Fibonacci numbers are divided by $n$ is periodic, where $n$ is a natural number.
[*] There exists no such non-constant polynomial with integer coefficients such that for every Fibonacci number $n,$ $ P(n)$ is a prime.
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Consider the matrix $ A,B\in \mathcal l{M}_3(\mathbb{C})$ with $ A=-^tA$ and $ B=^tB$. Prove that if the polinomial function defined by
\[ f(x)=\det(A+xB)\]
has a multiple root, then $ \det(A+B)=\det B$.
Let $p$ be a prime number. Find all polynomials $P$ with integer coefficients with the following properties:
$(a)$ $P(x)>x$ for all positive integers $x$.
$(b)$ The sequence defined by $p_0:=p$, $p_{n+1}:=P(p_n)$ for all positive integers $n$, satisfies the property that for all positive integers $m$ there exists some $l\geq 0$ such that $m\mid p_l$.