Found problems: 3597
The Fibonacci sequence $F_n$ is defined by $F_1=F_2=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq3$.
Let $m,n\geq1$ be integers. Find the minimal degree $d$ for which there exists a polynomial $f(x)=a_dx^d+a_{d-1}x^{d-1}+\dots+a_1x+a_0$, which satisfies $f(k)=F_{m+k}$ for all $k=0,1,...,n$.
Let $\alpha$ and $\beta$ be the roots of $x^{2} - qx + 1$, where $q$ is a rational number larger than $2$. Let $s_1 = \alpha + \beta$, $t_1 = 1$, and for all integers $n \geq 2$:
$s_n = \alpha^n + \beta^n$
$t_n = s_{n-1} + 2s_{n-2} + \cdot \cdot \cdot + (n - 1)s_{1} + n$
Prove that, for all odd integers $n$, $t_n$ is the square of a rational number.
For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]
For every positive integer $n\ge 3$, let $\phi_n$ be the set of all positive integers less than and coprime to $n$. Consider the polynomial:
$$P_n(x)=\sum_{k\in\phi_n} {x^{k-1}}.$$
a. Prove that $P_n(x)=(x^{r_n}+1)Q_n(x)$ for some positive integer $r_n$ and polynomial $Q_n(x)\in\mathbb{Z}[x]$ (not necessary non-constant polynomial).
b. Find all $n$ such that $P_n(x)$ is irreducible over $\mathbb{Z}[x]$.
Let $f(x,y)$ be a polynomial with real coefficients in the real variables $x$ and $y$ defined over the entire $xy$-plane. What are the possibilities for the range of $f(x,y)?$
Find all polynomials $P$ such that $P(x)+3P(x+2)=3P(x+1)+P(x+3)$ for all real numbers $x$.
Fix a positive integer $a > 1$. Consider triples $(f(x), g(x), h(x))$ of polynomials with integer coefficients, such that
1. $f$ is a monic polynomial with $\deg f \ge 1$.
2. There exists a positive integer $N$ such that $g(x)>0$ for $x \ge N$ and for all positive integers $n \ge N$, we have $f(n) \mid a^{g(n)} + h(n)$.
Find all such possible triples.
[i]Proposed by Mainak Ghosh and Rijul Saini[/i]
If the polynomial $ P\left(x\right)$ satisfies $ 2P\left(x\right) \equal{} P\left(x \plus{} 3\right) \plus{} P\left(x \minus{} 3\right)$ for every real number $ x$, degree of $ P\left(x\right)$ will be at most
$\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 2 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ \text{None}$
a) We have a geometric sequence of $3$ terms. If the sum of these terms is $26$ , and their sum of squares is $364$ , find the terms of the sequence.
b) Suppose that $a,b,c,u,v,w$ are positive real numbers , and each of $a,b,c$ and $u,v,w$ are geometric sequences. Suppose also that $a+u,b+v,c+w$ are an arithmetic sequence. Prove that $a=b=c$ and $u=v=w$
c) Let $a,b,c,d$ be real numbers (not all zero), and let $f(x,y,z)$ be the polynomial in three variables defined by$$f(x,y,z) = axyz + b(xy + yz + zx) + c(x+y+z) + d$$.Prove that $f(x,y,z)$ is reducible if and only if $a,b,c,d$ is a geometric sequence.
Let $n\ge 2$ be an even integer and $a,b$ real numbers such that $b^n=3a+1$. Show that the polynomial $P(X)=(X^2+X+1)^n-X^n-a$ is divisible by $Q(X)=X^3+X^2+X+b$ if and only if $b=1$.
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
For numbers $a,b \in \mathbb{R}$ we consider the sets:
$$A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\}$$
Find all $a,b > 1$ for which there exists two real , non-constant polynomials $P,Q$ with positive leading coefficients st for each $r \in \mathbb{R}$:
$$ P(r) \in A \iff Q(r) \in B$$
Set $P$ as a polynomial function by $p_n(x)=\sum_{k=0}^{n-1} x^k$.
a) Prove that for $m,n\in N$, when dividing $p_n(x)$ by $p_m(x)$, the remainder is $$p_i(x),\forall i=0,1,...,m-1.$$
b) Find all the positive integers $i,j,k$ that make $$p_i(x)+p_j(x^2)+p_k(x^4)=p_{100}(x).$$
[i](square1zoa)[/i]
Let a be a non-zero integer, and $n \ge 3$ another integer. Prove that the following polynomial is irreducible in the ring of integer polynomials (i.e. it cannot be written as a product of two non-constant integer polynomials):
$$f(x) = x^n + ax^{n-1} + ax^{n-2} +... + ax -1$$
Prove that if $ a $ and $ b $ are real numbers and the polynomial $ ax^3 - ax^2 + 9bx - b $ has three positive roots, then they are equal.
Let $f(x)$ be a quadratic polynomial with two real roots in the interval $[-1,1]$. Prove that if the maximum value of $|f(x)|$ in the interval $[-1,1]$ is equal to $1$, then the maximum value of $|f'(x)|$ in the interval $[-1,1]$ is not less than $1$.
We are given a polynomial $f(x)=x^6-11x^4+36x^2-36$. Prove that for an arbitrary prime number $p$, $f(x)\equiv 0\pmod{p}$ has a solution.
Let $ f(x) \equal{} x^8 \plus{} 4x^6 \plus{} 2x^4 \plus{} 28x^2 \plus{} 1.$ Let $ p > 3$ be a prime and suppose there exists an integer $ z$ such that $ p$ divides $ f(z).$ Prove that there exist integers $ z_1, z_2, \ldots, z_8$ such that if \[ g(x) \equal{} (x \minus{} z_1)(x \minus{} z_2) \cdot \ldots \cdot (x \minus{} z_8),\] then all coefficients of $ f(x) \minus{} g(x)$ are divisible by $ p.$
Prove that there is no nonconstant polynomial $f(x)$ with integral coefficients such that $f(n)$ is prime for all $n \in \mathbb{N}$.
Find a necessary and sufficient condition on the natural number $ n$ for the equation
\[ x^n \plus{} (2 \plus{} x)^n \plus{} (2 \minus{} x)^n \equal{} 0
\]
to have a integral root.
Find all the polynomials with real coefficients which satisfy $ (x^2-6x+8)P(x)=(x^2+2x)P(x-2)$ for all $x\in \mathbb{R}$.
Let $n > 1$ and $p(x)=x^n+a_{n-1}x^{n-1} +...+a_0$ be a polynomial with $n$ real roots (counted
with multiplicity). Let the polynomial $q$ be defined by
$$q(x) = \prod_{j=1}^{2015} p(x + j)$$.
We know that $p(2015) = 2015$. Prove that $q$ has at least $1970$ different roots $r_1, ..., r_{1970}$
such that $|r_j| < 2015$ for all $ j = 1, ..., 1970$.
Find all polynomials $p(x)$ with integer coefficients such that for each positive integer $n$, the number $2^n - 1$ is divisible by $p(n)$.
Let $P(x)=x^3+ax^2+b$ and $Q(x)=x^3+bx+a$, where $a$ and $b$ are nonzero real numbers. Suppose that the roots of the equation $P(x)=0$ are the reciprocals of the roots of the equation $Q(x)=0$. Prove that $a$ and $b$ are integers. Find the greatest common divisor of $P(2013!+1)$ and $Q(2013!+1)$.
Do there exist natural numbers $a, b$ and $c$ such that $a^2+b^2+c^2$ is divisible by $2013(ab+bc+ca)$?
[i]Proposed by Mahan Malihi[/i]