Found problems: 3597
The sum of all the roots of $ 4x^3\minus{}8x^2\minus{}63x\minus{}9\equal{}0$ is:
$ \textbf{(A)}\ 8 \qquad
\textbf{(B)}\ 2 \qquad
\textbf{(C)}\ \minus{}8 \qquad
\textbf{(D)}\ \minus{}2 \qquad
\textbf{(E)}\ 0$
Find all polynomials $P(x)$ with integer coefficients, such that for each of them there exists a positive integer $N$, such that for any positive integer $n\geq N$, number $P(n)$ is a positive integer and a divisor of $n!$.
[i]Proposed by Mykyta Kharin[/i]
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]
Find all polynomials $P(x)$ with real coefficients such that $P(2014) = 1$ and, for some integer $c$:
$xP(x-c) = (x - 2014)P(x)$
Does existes a function $f:N->N$ and for all positeve integer n
$f(f(n)+2011)=f(n)+f(f(n))$
Prove that it is impossible for polynomials $f_1(x),f_2(x),f_3(x),f_4(x)\in \mathbb{Q}[x]$ to satisfy \[f_1^2(x)+f_2^2(x)+f_3^2(x)+f_4^2(x) = x^2+7.\]
Consider polynomials $P(x)$ of degree at most $3$, each of whose coefficients is an element of $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$. How many such polynomials satisfy $P(-1) = -9$?
$\textbf{(A) } 110 \qquad \textbf{(B) } 143 \qquad \textbf{(C) } 165 \qquad \textbf{(D) } 220 \qquad \textbf{(E) } 286 $
A polynomial
\[ P(x) \equal{} c_{2004}x^{2004} \plus{} c_{2003}x^{2003} \plus{} ... \plus{} c_1x \plus{} c_0
\]has real coefficients with $ c_{2004}\not \equal{} 0$ and $ 2004$ distinct complex zeroes $ z_k \equal{} a_k \plus{} b_ki$, $ 1\leq k\leq 2004$ with $ a_k$ and $ b_k$ real, $ a_1 \equal{} b_1 \equal{} 0$, and
\[ \sum_{k \equal{} 1}^{2004}{a_k} \equal{} \sum_{k \equal{} 1}^{2004}{b_k}.
\]Which of the following quantities can be a nonzero number?
$ \textbf{(A)}\ c_0 \qquad \textbf{(B)}\ c_{2003} \qquad \textbf{(C)}\ b_2b_3...b_{2004} \qquad \textbf{(D)}\ \sum_{k \equal{} 1}^{2004}{a_k} \qquad \textbf{(E)}\ \sum_{k \equal{} 1}^{2004}{c_k}$
Given an integer $ n > 3.$ Let $ a_{1},a_{2},\cdots,a_{n}$ be real numbers satisfying $ min |a_{i} \minus{} a_{j}| \equal{} 1, 1\le i\le j\le n.$ Find the minimum value of $ \sum_{k \equal{} 1}^n|a_{k}|^3.$
How many ordered four-tuples of integers $(a,b,c,d)$ with $0 < a < b < c < d < 500$ satisfy $a + d = b + c$ and $bc - ad = 93$?
Let
\[ p(x,y) \equal{} a_0 \plus{} a_1x \plus{} a_2y \plus{} a_3x^2 \plus{} a_4xy \plus{} a_5y^2 \plus{} a_6x^3 \plus{} a_7x^2y \plus{} a_8xy^2 \plus{} a_9y^3.
\]Suppose that
\begin{align*}p(0,0) &\equal{} p(1,0) \equal{} p( \minus{} 1,0) \equal{} p(0,1) \equal{} p(0, \minus{} 1) \\&\equal{} p(1,1) \equal{} p(1, \minus{} 1) \equal{} p(2,2) \equal{} 0.\end{align*}
There is a point $ \left(\tfrac {a}{c},\tfrac {b}{c}\right)$ for which $ p\left(\tfrac {a}{c},\tfrac {b}{c}\right) \equal{} 0$ for all such polynomials, where $ a$, $ b$, and $ c$ are positive integers, $ a$ and $ c$ are relatively prime, and $ c > 1$. Find $ a \plus{} b \plus{} c$.
Let $p$ and $q$ be integers. Show that there exists an interval $I$ of length $1/q$ and a polynomial $P$ with integral coefficients such that
\[ \left|P(x)-\frac pq \right| < \frac{1}{q^2}\]for all $x \in I.$
Let $p = \overline{abcd}$ be a $4$-digit prime number. Prove that the equation $ax^3+bx^2+cx+d=0$ has no rational roots.
Consider a polynomial with coefficients of real numbers $ \phi(x)\equal{}ax^3\plus{}bx^2\plus{}cx\plus{}d$ with three positive real roots. Assume that $ \phi(0)<0$, prove that \[ 2b^3\plus{}9a^2d\minus{}7abc \le 0.\]
[i]Hery Susanto, Malang[/i]
Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.
A triplet of polynomials $u,v,w \in \mathbb{R}[x,y,z]$ is called [i]smart[/i] if there exists polynomials $P,Q,R\in \mathbb{R}[x,y,z]$ such that the following polynomial identity holds :$$u^{2019}P +v^{2019 }Q+w^{2019} R=2019$$
a) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y+z$$ [i]smart[/i]?
b) Is the triplet of polynomials $$u=x+2y+3 , \;\;\;\; v=y+z+2, \;\;\;\;\;w=x+y-z$$ [i]smart[/i]?
[i]Proposed by Arturas Dubickas (Vilnius University).
[/i]
Let $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0$ be a monic polynomial of degree $n>2$, with real coefficients and all its roots real and different from zero. Prove that for all $k=0,1,2,\cdots,n-2$, at least one of the coefficients $a_k,a_{k+1}$ is different from zero.
Find all polynomials $f$ such that $f$ has non-negative integer coefficients, $f(1)=7$ and $f(2)=2017$.
Let be two polynoms $ P,Q\in\mathbb{C} [X] $ with degree at least $ 1, $ and such that $ P $ has only simple roots. Prove that the following affirmations are equivalent:
$ \text{(i)} P\circ Q $ is divisible by $ P. $
$ \text{(ii)} $ The evaluation of $ Q $ at any root of $ P $ is a root of $ P. $
[i]Marcel Țena[/i]
Find all integer $n$ such that $n-5$ divide $n^2+n-27$.
Let $P$ be a polynomial with integer coefficients. Assume that there exists a positive integer $n$ with $P(n^2)=2022$. Prove that there cannot be a positive rational number $r$ with $P(r^2)=2024$.
Let $p > 3$ be a prime number, and let $F_p$ denote the (fi nite) set of residue classes modulo $p$.
Let $S_d$ denote the set of $2$-variable polynomials $P(x, y)$ with coefficients in $F_p$, total degree $\le d$, and satisfying $P(x, y) = P(y,- x -y)$. Show that $$|S_d| = p^{\lceil (d+1)(d+2)/6 \rceil}$$.
[i]The total degree of a $2$-variable polynomial $P(x, y)$ is the largest value of $i + j$ among monomials $x^iy^j$
[/i] appearing in $P$.
Given an integer $n\geq 2,$ let $a_{n},b_{n},c_{n}$ be integer numbers such that \[
\left( \sqrt[3]{2}-1\right) ^{n}=a_{n}+b_{n}\sqrt[3]{2}+c_{n}\sqrt[3]{4}.
\] Prove that $c_{n}\equiv 1\pmod{3} $ if and only if $n\equiv 2\pmod{3}.$
Let $Q$ be a quadriatic polynomial having two different real zeros. Prove that there is a non-constant monic polynomial $P$ such that all coefficients of the polynomial $Q(P(x))$ except the leading one are (by absolute value) less than $0.001$.
Determine all real polynomials $p$ such that $p(x+p(x))=x^2p(x)$ for all $x$.