Found problems: 3597
2021 European Mathematical Cup, 4
Find all positive integers $d$ for which there exist polynomials $P(x)$ and $Q(x)$ with real coefficients such that degree of $P$ equals $d$ and
$$P(x)^2+1=(x^2+1)Q(x)^2.$$
2002 AMC 10, 11
Let $P(x)=kx^3+2k^2x^2+k^3$. Find the sum of all real numbers $k$ for which $x-2$ is a factor of $P(x)$.
$\textbf{(A) }-8\qquad\textbf{(B) }-4\qquad\textbf{(C) }0\qquad\textbf{(D) }5\qquad\textbf{(E) }8$
2013 IMO Shortlist, A6
Let $m \neq 0 $ be an integer. Find all polynomials $P(x) $ with real coefficients such that
\[ (x^3 - mx^2 +1 ) P(x+1) + (x^3+mx^2+1) P(x-1) =2(x^3 - mx +1 ) P(x) \]
for all real number $x$.
1954 AMC 12/AHSME, 41
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$
1984 Vietnam National Olympiad, 2
Given two real numbers $a, b$ with $a \neq 0$, find all polynomials $P(x)$ which satisfy
\[xP(x - a) = (x - b)P(x).\]
2021 Taiwan TST Round 1, A
Let $\mathcal{A}$ denote the set of all polynomials in three variables $x, y, z$ with integer coefficients. Let $\mathcal{B}$ denote the subset of $\mathcal{A}$ formed by all polynomials which can be expressed as
\begin{align*}
(x + y + z)P(x, y, z) + (xy + yz + zx)Q(x, y, z) + xyzR(x, y, z)
\end{align*}
with $P, Q, R \in \mathcal{A}$. Find the smallest non-negative integer $n$ such that $x^i y^j z^k \in \mathcal{B}$ for all non-negative integers $i, j, k$ satisfying $i + j + k \geq n$.
1996 Israel National Olympiad, 2
Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$
2013 CentroAmerican, 3
Determine all pairs of non-constant polynomials $p(x)$ and $q(x)$, each with leading coefficient $1$, degree $n$, and $n$ roots which are non-negative integers, that satisfy $p(x)-q(x)=1$.
2014 ELMO Shortlist, 11
Let $p$ be a prime satisfying $p^2\mid 2^{p-1}-1$, and let $n$ be a positive integer. Define
\[ f(x) = \frac{(x-1)^{p^n}-(x^{p^n}-1)}{p(x-1)}. \]
Find the largest positive integer $N$ such that there exist polynomials $g(x)$, $h(x)$ with integer coefficients and an integer $r$ satisfying $f(x) = (x-r)^N g(x) + p \cdot h(x)$.
[i]Proposed by Victor Wang[/i]
2009 USA Team Selection Test, 8
Fix a prime number $ p > 5$. Let $ a,b,c$ be integers no two of which have their difference divisible by $ p$. Let $ i,j,k$ be nonnegative integers such that $ i \plus{} j \plus{} k$ is divisible by $ p \minus{} 1$. Suppose that for all integers $ x$, the quantity
\[ (x \minus{} a)(x \minus{} b)(x \minus{} c)[(x \minus{} a)^i(x \minus{} b)^j(x \minus{} c)^k \minus{} 1]\]
is divisible by $ p$. Prove that each of $ i,j,k$ must be divisible by $ p \minus{} 1$.
[i]Kiran Kedlaya and Peter Shor.[/i]
2012 Pre - Vietnam Mathematical Olympiad, 2
Let $(a_n)$ defined by: $a_0=1, \; a_1=p, \; a_2=p(p-1)$, $a_{n+3}=pa_{n+2}-pa_{n+1}+a_n, \; \forall n \in \mathbb{N}$. Knowing that
(i) $a_n>0, \; \forall n \in \mathbb{N}$.
(ii) $a_ma_n>a_{m+1}a_{n-1}, \; \forall m \ge n \ge 0$.
Prove that $|p-1| \ge 2$.
2022 Belarusian National Olympiad, 10.7
Find all positive integers $a$ for which there exists a polynomial $p(x)$ with integer coefficients such that
$p(\sqrt{2}+1)=2-\sqrt{2}$ and $p(\sqrt{2}+2)=a$
2018 Iran Team Selection Test, 3
$n>1$ and distinct positive integers $a_1,a_2,\ldots,a_{n+1}$ are given. Does there exist a polynomial $p(x)\in\Bbb{Z}[x]$ of degree $\le n$ that satisfies the following conditions?
a. $\forall_{1\le i < j\le n+1}: \gcd(p(a_i),p(a_j))>1 $
b. $\forall_{1\le i < j < k\le n+1}: \gcd(p(a_i),p(a_j),p(a_k))=1 $
[i]Proposed by Mojtaba Zare[/i]
1992 All Soviet Union Mathematical Olympiad, 576
If you have an algorithm for finding all the real zeros of any cubic polynomial, how do you find the real solutions to $x = p(y), y = p(x)$, where $p$ is a cubic polynomial?
2014 Brazil National Olympiad, 4
The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as
\[P_0(x)=x,\quad P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1),\quad n\ge 1.\]
Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$.
2006 China Team Selection Test, 3
Let $a_{i}$ and $b_{i}$ ($i=1,2, \cdots, n$) be rational numbers such that for any real number $x$ there is:
\[x^{2}+x+4=\sum_{i=1}^{n}(a_{i}x+b)^{2}\]
Find the least possible value of $n$.
2008 China Team Selection Test, 2
Prove that for all $ n\geq 2,$ there exists $ n$-degree polynomial $ f(x) \equal{} x^n \plus{} a_{1}x^{n \minus{} 1} \plus{} \cdots \plus{} a_{n}$ such that
(1) $ a_{1},a_{2},\cdots, a_{n}$ all are unequal to $ 0$;
(2) $ f(x)$ can't be factorized into the product of two polynomials having integer coefficients and positive degrees;
(3) for any integers $ x, |f(x)|$ isn't prime numbers.
2007 Harvard-MIT Mathematics Tournament, 6
Consider the polynomial $P(x)=x^3+x^2-x+2$. Determine all real numbers $r$ for which there exists a complex number $z$ not in the reals such that $P(z)=r$.
2017 China Team Selection Test, 4
Show that there exists a degree $58$ monic polynomial
$$P(x) = x^{58} + a_1x^{57} + \cdots + a_{58}$$
such that $P(x)$ has exactly $29$ positive real roots and $29$ negative real roots and that $\log_{2017} |a_i|$ is a positive integer for all $1 \leq i \leq 58$.
2019 Serbia National Math Olympiad, 2
For the sequence of real numbers $a_1,a_2,\dots ,a_k$ we say it is [i]invested[/i] on the interval $[b,c]$ if there exists numbers $x_0,x_1,\dots ,x_k$ in the interval $[b,c]$ such that $|x_i-x_{i-1}|=a_i$ for $i=1,2,3,\dots k$ .
A sequence is [i]normed[/i] if all its members are not greater than $1$ . For a given natural $n$ , prove :
a)Every [i]normed[/i] sequence of length $2n+1$ is [i]invested[/i] in the interval $\left[ 0, 2-\frac{1}{2^n} \right ]$.
b) there exists [i]normed[/i] sequence of length $4n+3$ wich is not [i]invested[/i] on $\left[ 0, 2-\frac{1}{2^n} \right ]$.
2020 Silk Road, 3
A polynomial $ Q (x) = k_n x ^ n + k_ {n-1} x ^ {n-1} + \ldots + k_1 x + k_0 $ with real coefficients is called [i]powerful[/i] if the equality $ | k_0 | = | k_1 | + | k_2 | + \ldots + | k_ {n-1} | + | k_n | $, and [i]non-increasing[/i] , if $ k_0 \geq k_1 \geq \ldots \geq k_ {n-1} \geq k_n $.
Let for the polynomial $ P (x) = a_d x ^ d + a_ {d-1} x ^ {d-1} + \ldots + a_1 x + a_0 $ with nonzero real coefficients, where $ a_d> 0 $, the polynomial $ P (x) (x-1) ^ t (x + 1) ^ s $ is [i]powerful[/i] for some non-negative integers $ s $ and $ t $ ($ s + t> 0 $). Prove that at least one of the polynomials $ P (x) $ and $ (- 1) ^ d P (-x) $ is [i]nonincreasing[/i].
2016 Iran MO (3rd Round), 1
Let $F$ be a subset of the set of positive integers with at least two elements and $P(x)$ be a polynomial with integer coefficients such that for any two distinct elements of $F$ like $a$ and $b$, the following two conditions hold
[list]
[*] $a+b \in F$, and
[*] $\gcd(P(a),P(b))=1$.
[/list]
Prove that $P(x)$ is a constant polynomial.
1988 All Soviet Union Mathematical Olympiad, 485
The sequence of integers an is given by $a_0 = 0, a_n = p(a_n-1)$, where $p(x)$ is a polynomial whose coefficients are all positive integers. Show that for any two positive integers $m, k$ with greatest common divisor $d$, the greatest common divisor of $a_m$ and $a_k$ is $a_d$.
Kvant 2022, M2714
Let $f{}$ and $g{}$ be polynomials with integers coefficients. The leading coefficient of $g{}$ is equal to 1. It is known that for infinitely many natural numbers $n{}$ the number $f(n)$ is divisible by $g(n)$ . Prove that $f(n)$ is divisible by $g(n)$ for all positive integers $n{}$ such that $g(n)\neq 0$.
[i]From the folklore[/i]
1991 AIME Problems, 1
Find $x^2+y^2$ if $x$ and $y$ are positive integers such that \[xy+x+y = 71\qquad\text{and}\qquad x^2y+xy^2 = 880.\]