Found problems: 4275
Let $a$ and $b$ be two square-free, distinct natural numbers. Show that there exist $c>0$ such that
\[
\left | \{n\sqrt{a}\}-\{n\sqrt{b}\} \right |>\frac{c}{n^3}\]
for every positive integer $n$.
Let $a,b,c$ be three real numbers such that $1 \geq a \geq b \geq c \geq 0$. prove that if $\lambda$ is a root of the cubic equation $x^3 + ax^2 + bx + c = 0$ (real or complex), then $| \lambda | \leq 1.$
Let $\alpha$, $\beta$ and $\gamma$ be the roots of the polynomial $2023x^3-2023x^2-1$. Find $$\frac{1}{\alpha^3}+\frac{1}{\beta^3}+\frac{1}{\gamma^3}$$.
[i]Proposed by Andy Xu[/i]
Let $p(x)$ and $q(x)$ be non-constant polynomial functions with integer coeffcients. It is known that the polynomial $p(x)q(x) - 2015$ has at least $33$ different integer roots. Prove that neither $p(x)$ nor $q(x)$ can be a polynomial of degree less than three.
Three quadratic polynomials $f_1(x) = x^2+2a_1x+b_1$, $f_2(x) = x^2+2a_2x+b_2$,
$f_3(x) = x^2 + 2a_3x + b_3$ are such that $a_1a_2a_3 = b_1b_2b_3 > 1$. Prove that at
least one polynomial has two distinct roots.
Given are two polynomial $f$ and $g$ of degree $100$ with real coefficients. For each positive integer $n$ there is an integer $k$ such that
\[\frac{f(k)}{g(k)}=\frac{n+1}{n}.\]
Prove that $f$ and $g$ have a common non-constant factor.
Let $P$ be a polynomial with integer coefficients. Suppose that for $n=1,2,3,\ldots ,1998$ the number $P(n)$ is a three-digit positive integer. Prove that the polynomial $P$ has no integer roots.
A positive integer $N$ is called [i]balanced[/i], if $N=1$ or if $N$ can be written as a product of an even number of not necessarily distinct primes. Given positive integers $a$ and $b$, consider the polynomial $P$ defined by $P(x)=(x+a)(x+b)$.
(a) Prove that there exist distinct positive integers $a$ and $b$ such that all the number $P(1)$, $P(2)$,$\ldots$, $P(50)$ are balanced.
(b) Prove that if $P(n)$ is balanced for all positive integers $n$, then $a=b$.
[i]Proposed by Jorge Tipe, Peru[/i]
Given $P(x)=a_{2016}x^{2016}+a_{2015}x^{2015}+...+a_1x+a_0$
be a polynomial with real coefficients and $a_{2016} \neq 0$
satisfies
$|a_1+a_3+...+a_{2015}| > |a_0+a_2+...+a_{2016}|$
Prove that $P(x)$ has an odd number of complex roots with absolute value less than $1$ (count multiple roots also)
edited: complex roots
Find the smallest positive value of $x$ such that $x^3-9x^2+22x-16=0$.
Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?
We call a number $n$ [i]interesting [/i]if for each permutation $\sigma$ of $1,2,\ldots,n$ there exist polynomials $P_1,P_2,\ldots ,P_n$ and $\epsilon > 0$ such that:
$i)$ $P_1(0)=P_2(0)=\ldots =P_n(0)$
$ii)$ $P_1(x)>P_2(x)>\ldots >P_n(x)$ for $-\epsilon<x<0$
$iii)$ $P_{\sigma (1)} (x)>P_{\sigma (2)}(x)> \ldots >P_{\sigma (n)} (x) $ for $0<x<\epsilon$
Find all [i]interesting [/i]$n$.
[i]Proposed by Mojtaba Zare Bidaki[/i]
It is known that the number $\pi$ is transcendental, that is, it is not a root of any polynomial with integer coefficients. Using this fact, prove that the same is true for the number $\pi + \sqrt2$.
A polynomial $P(x) = 1 + x^2 + x^5 + x^{n_1} + ...+ x^{n_s} + x^{2008}$ with $n_1, ..., n_s$ are positive integers and $5 < n_1 < ... <n_s < 2008$ are given. Prove that if $P(x)$ has at least a real root, then the root is not greater than $\frac{1-\sqrt5}{2}$
Let $f$ be a function for which $f\left(\frac x3\right)=x^2+x+1$. Find the sum of all values of $z$ for which $f(3z)=7$.
$\text{(A)}\ -\frac13\qquad\text{(B)}\ -\frac19 \qquad\text{(C)}\ 0 \qquad\text{(D)}\ \frac59 \qquad\text{(E)}\ \frac53$
The polynomial $f(z)=az^{2018}+bz^{2017}+cz^{2016}$ has real coefficients not exceeding $2019$, and $f(\tfrac{1+\sqrt{3}i}{2})=2015+2019\sqrt{3}i$. Find the remainder when $f(1)$ is divided by $1000$.
It is known that $ k$ and $ n$ are positive integers and \[ k \plus{} 1\leq\sqrt {\frac {n \plus{} 1}{\ln(n \plus{} 1)}}.\] Prove that there exists a polynomial $ P(x)$ of degree $ n$ with coefficients in the set $ \{0,1, \minus{} 1\}$ such that $ (x \minus{} 1)^{k}$ divides $ P(x)$.
Let $f(x)=\frac{P(x)}{Q(x)}$. Where $P(x), Q(x)$ are two non constant polynomials with no common zeros and $P(0)=P(1)=0$. Suppose $f(x)f(\frac{1}{x})=f(x)+f(\frac{1}{x})$ for all infinitely many values of $x$.
a. Show that $deg(P) <deg(Q).$
b. Show that $P'(1)=2Q'(1)- deg(Q). Q(1)$
Here $P'(x)$ denotes the derivatives of $P(x)$ as usual
Let $f$ be a polynomial with integer coefficients. Define $$a_1 = f(0)~,~a_2 = f(a_1) = f(f(0))~,$$
and $~a_n = f(a_{n-1})$ for $n \geqslant 3$.
If there exists a natural number $k \geqslant 3$ such that $a_k = 0$, then prove that either $a_1=0$ or $a_2=0$.
Given a set $S$ of integers, an allowed operation consists of the following three steps:
$\bullet$ Choose a positive integer $n$.
$\bullet$ Choose $n+1$ elements $a_0, a_1, \dots, a_n \in S$, not necessarily distinct.
$\bullet$ Add to the set $S$ all the integer roots of the polynomial $a_n x^n + a_{n-1} x^{n-1} + \dots + a_2 x^2 + a_1 x + a_0$.
Beto must choose an initial set $S$ and perform several allowed operations, so that at the end of the process $S$ contains among its elements the integers $1, 2, 3, \dots, 2023, 2024$.
Determine the smallest $k$ for which there exists an initial set $S$ with $k$ elements that allows Beto to achieve his objective.
Let $ k$ be a positive integer. Find all polynomials
\[ P(x) \equal{} a_0 \plus{} a_1 x \plus{} \cdots \plus{} a_n x^n,\]
where the $ a_i$ are real, which satisfy the equation
\[ P(P(x)) \equal{} \{ P(x) \}^k\]
Let $n$ be a natural number and $f_1$, $f_2$, ..., $f_n$ be polynomials with integers coeffcients. Show that there exists a polynomial $g(x)$ which can be factored (with at least two terms of degree at least $1$) over the integers such that $f_i(x)+g(x)$ cannot be factored (with at least two terms of degree at least $1$) over the integers for every $i$.
Let $ P(z) \equal{} z^3 \plus{} az^2 \plus{} bz \plus{} c$, where $ a$, $ b$, and $ c$ are real. There exists a complex number $ w$ such that the three roots of $ P(z)$ are $ w \plus{} 3i$, $ w \plus{} 9i$, and $ 2w \minus{} 4$, where $ i^2 \equal{} \minus{} 1$. Find $ |a \plus{} b \plus{} c|$.
Let $p$ be a polynomial with integer coefficients such that $p(15)=6$, $p(22)=1196$, and $p(35)=26$. Find an integer $n$ such that $p(n)=n+82$.
The three roots of the cubic $ 30 x^3 \minus{} 50x^2 \plus{} 22x \minus{} 1$ are distinct real numbers between $ 0$ and $ 1$. For every nonnegative integer $ n$, let $ s_n$ be the sum of the $ n$th powers of these three roots. What is the value of the infinite series
\[ s_0 \plus{} s_1 \plus{} s_2 \plus{} s_3 \plus{} \dots \, ?\]