Found problems: 3597
Let $ P(X) \equal{} a_{n}X^{n} \plus{} a_{n \minus{} 1}X^{n \minus{} 1} \plus{} \ldots \plus{} a_{1}X \plus{} a_{0}$ be a real polynomial of degree $ n$. Suppose $ n$ is an even number and:
a) $ a_{0} > 0$, $ a_{n} > 0$;
b) $ a_{1}^{2} \plus{} a_{2}^{2} \plus{} \ldots \plus{} a_{n \minus{} 1}^{2}\leq\frac {4\min(a_{0}^{2} , a_{n}^{2})}{n \minus{} 1}$.
Prove that $ P(x)\geq 0$ for all real values $ x$.
[i]Laurentiu Panaitopol[/i]
Find all positive integers $p$ and $q$ such that all the roots of the polynomial $(x^2 - px+q)(x^2 -qx+ p)$ are positive integers.
Find all functions $f$, of the form $f(x) = x^3 +px^2 +qx+r$ with $p$, $q$ and $r$ integers, such that $f(s) = 506$ for some integer $s$ and $f(\sqrt3) = 0$.
Let $\mathbb{Z}[x]$ denote the set of single-variable polynomials in $x$ with integer coefficients. Find all functions $\theta : \mathbb{Z}[x] \to \mathbb{Z}[x]$ (i.e. functions taking polynomials to polynomials)
such that
[list]
[*] for any polynomials $p, q \in \mathbb{Z}[x]$, $\theta(p + q) = \theta(p) + \theta(q)$;
[*] for any polynomial $p \in \mathbb{Z}[x]$, $p$ has an integer root if and only if $\theta(p)$ does.
[/list]
[i]Carl Schildkraut[/i]
Let $f(x)=\cos(\cos(\cos(\cos(\cos(\cos(\cos(\cos(x))))))))$, and suppose that the number $a$ satisfies the equation $a=\cos a$. Express $f'(a)$ as a polynomial in $a$.
If the polynomial $P(x)=ax^2+bx+c$ takes value $0$ for three different values of $x$, then prove the polynomial $P(x)$ takes value $0$ for all $x$.
Let $ P_1(x), P_2(x), \ldots, P_n(x)$ be real polynomials, i.e. they have real coefficients. Show that there exist real polynomials $ A_r(x),B_r(x) \quad (r \equal{} 1, 2, 3)$ such that
\[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_1(x) \right)^2 \plus{} \left( B_1(x) \right)^2\]
\[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_2(x) \right)^2 \plus{} x \left( B_2(x) \right)^2\]
\[ \sum^n_{s\equal{}1} \left\{ P_s(x) \right \}^2 \equiv \left( A_3(x) \right)^2 \minus{} x \left( B_3(x) \right)^2\]
Let $P$ be a polynomial such that $(x-4)P(2x) = 4(x-1)P(x)$, for every real $x$. If $P(0) \neq 0$, what is the degree of $P$?
$
\textbf{(A)}\ 0
\qquad\textbf{(B)}\ 1
\qquad\textbf{(C)}\ 2
\qquad\textbf{(D)}\ 3
\qquad\textbf{(E)}\ \text{None of the preceding}
$
Find all the functions $ f :\mathbb{R}\mapsto\mathbb{R} $ with the following property: $ \forall x$ $f(x)= f(x/2) + (x/2)f'(x)$
Suppose that $ a$ and $ b$ are real numbers such that the quadratic polynomial $ f(x)\equal{}x^2\plus{}ax\plus{}b$ has no nonnegative real roots. Prove that there exist two polynomials $ g,h$ whose coefficients are nonnegative real numbers such that: $ f(x)\equal{}\frac{g(x)}{h(x)}$ for all real numbers $ x$.
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
Find, with proof, all real numbers $x$ satisfying $x = 2\left( 2 \left( 2\left( 2\left( 2x-1 \right)-1 \right)-1 \right)-1 \right)-1$.
[i]Proposed by Evan Chen[/i]
Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as
$$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$
Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that
$$P_{2n}(x)=P_n(x^2+c).$$
[i]Proposed by Navid Safaei[/i]
Let $ k$ be a positive integer. Prove that there exist polynomials $ P_0(n),P_1(n),\dots,P_{k\minus{}1}(n)$ (which may depend on $ k$) such that for any integer $ n,$
\[ \left\lfloor\frac{n}{k}\right\rfloor^k\equal{}P_0(n)\plus{}P_1(n)\left\lfloor\frac{n}{k}\right\rfloor\plus{} \cdots\plus{}P_{k\minus{}1}(n)\left\lfloor\frac{n}{k}\right\rfloor^{k\minus{}1}.\]
($ \lfloor a\rfloor$ means the largest integer $ \le a.$)
Let $p$ be a polynomial with integer coefficients satisfying $p(16)=36,p(14)=16,p(5)=25$. Determine all possible values of $p(10)$.
Let $A=(a_{ij})\in \mathcal{M}_p(\mathbb{C})$ such that $a_{12}=a_{23}=\ldots=a_{p-1,p}=1$ and $a_{ij}=0$ for any other entry.
a)Prove that $A^{p-1}\neq O_p$ and $A^p=O_p$.
b)If $X\in \mathcal{M}_{p}(\mathbb{C})$ and $AX=XA$, prove that there exist $a_1,a_2,\ldots,a_p\in \mathbb{C}$ such that:
\[X=\left( \begin{array}{ccccc} a_1 & a_2 & a_3 & \ldots & a_p \\ 0 & a_1 & a_2 & \ldots & a_{p-1} \\ 0 & 0 & a_1 & \ldots & a_{p-2} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ 0 & 0 & 0 & \ldots & a_1 \end{array} \right)\]
c)If there exist $B,C\in \mathcal{M}_p(\mathbb{C})$ such that $(I_p+A)^n=B^n+C^n,\ (\forall)n\in \mathbb{N}^*$, prove that $B=O_p$ or $C=O_p$.
Let $\{u_n \}$ be the sequence defined by its first two terms $u_0, u_1$ and the recursion formula
\[u_{n+2 }= u_n - u_{n+1}.\]
[b](a)[/b] Show that $u_n$ can be written in the form $u_n = \alpha a^n + \beta b^n$, where $a, b, \alpha, \beta$ are constants independent of $n$ that have to be determined.
[b](b)[/b] If $S_n = u_0 + u_1 + \cdots + u_n$, prove that $S_n + u_{n-1}$ is a constant independent of $n.$ Determine this constant.
Let $ a$, $ b$, $ c$, $ d$, $ e$, $ f$ be positive integers and let $ S = a+b+c+d+e+f$.
Suppose that the number $ S$ divides $ abc+def$ and $ ab+bc+ca-de-ef-df$. Prove that $ S$ is composite.
َA natural number $n$ is given. Let $f(x,y)$ be a polynomial of degree less than $n$ such that for any positive integers $x,y\leq n, x+y \leq n+1$ the equality $f(x,y)=\frac{x}{y}$ holds. Find $f(0,0)$.
When \[x^5, \quad x+\frac{1}{x}\quad \text{and}\quad 1+\frac{2}{x} + \frac{3}{x^2}\] are multiplied, the product is a polynomial of degree
$ \textbf{(A)}\ 2\qquad\textbf{(B)}\ 3\qquad\textbf{(C)}\ 6\qquad\textbf{(D)}\ 7\qquad\textbf{(E)}\ 8 $
$ n $ is a natural number, and $ S $ is the sum of all the solutions of the equations
$$ x^2+a_k\cdot x+a_k=0,\quad a_k\in\mathbb{R} ,\quad k\in\{ 1,2,...,n\} . $$
Show that if $ |S|>2n\left( \sqrt[n]{n} -1\right) , $ then at least one of the equations has real solutions.
Determine all functions $ f: \mathbb{N}_0 \rightarrow \mathbb{N}_0$ such that $ f(f(n))\plus{}f(n)\equal{}2n\plus{}6$ for all $ n \in \mathbb{N}_0$.
Let $m,n\geq 2$ be integers. Let $f(x_1,\dots, x_n)$ be a polynomial with real coefficients such that $$f(x_1,\dots, x_n)=\left\lfloor \frac{x_1+\dots + x_n}{m} \right\rfloor\text{ for every } x_1,\dots, x_n\in \{0,1,\dots, m-1\}.$$ Prove that the total degree of $f$ is at least $n$.
Let $A_1A_2A_3A_4$ be a non-cyclic quadrilateral. Let $O_1$ and $r_1$ be the circumcentre and the circumradius of the triangle $A_2A_3A_4$. Define $O_2,O_3,O_4$ and $r_2,r_3,r_4$ in a similar way. Prove that
\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]
[i]Proposed by Alexey Gladkich, Israel[/i]
Let $Q(x)=a_0+a_1x+\dots+a_nx^n$ be a polynomial with integer coefficients, and $0\le a_i<3$ for all $0\le i\le n$.
Given that $Q(\sqrt{3})=20+17\sqrt{3}$, compute $Q(2)$.