Found problems: 3597
2023 Belarusian National Olympiad, 11.6
Let $a$ be some integer. Prove that the polynomial $x^4(x-a)^4+1$ can not be a product of two non-constant polynomials with integer coefficients
1992 IMO Longlists, 51
Let $ f, g$ and $ a$ be polynomials with real coefficients, $ f$ and $ g$ in one variable and $ a$ in two variables. Suppose
\[ f(x) \minus{} f(y) \equal{} a(x, y)(g(x) \minus{} g(y)) \forall x,y \in \mathbb{R}\]
Prove that there exists a polynomial $ h$ with $ f(x) \equal{} h(g(x)) \text{ } \forall x \in \mathbb{R}.$
1996 IMO Shortlist, 6
Let $ n$ be an even positive integer. Prove that there exists a positive inter $ k$ such that
\[ k \equal{} f(x) \cdot (x\plus{}1)^n \plus{} g(x) \cdot (x^n \plus{} 1)\]
for some polynomials $ f(x), g(x)$ having integer coefficients. If $ k_0$ denotes the least such $ k,$ determine $ k_0$ as a function of $ n,$ i.e. show that $ k_0 \equal{} 2^q$ where $ q$ is the odd integer determined by $ n \equal{} q \cdot 2^r, r \in \mathbb{N}.$
Note: This is variant A6' of the three variants given for this problem.
2024 IFYM, Sozopol, 6
Let $P(x)$ be a polynomial in one variable with integer coefficients. Prove that the number of pairs $(m,n)$ of positive integers such that $2^n + P(n) = m!$, is finite.
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 $i + j + k \geq n$.
1995 Kurschak Competition, 2
Consider a polynomial in $n$ variables with real coefficients. We know that if every variable is $\pm1$, the value of the polynomial is positive, or negative if the number of $-1$'s is even, or odd, respectively. Prove that the degree of this polynomial is at least $n$.
2003 Balkan MO, 1
Can one find 4004 positive integers such that the sum of any 2003 of them is not divisible by 2003?
2015 JBMO Shortlist, A2
If $x^3-3\sqrt3 x^2 +9x - 3\sqrt3 -64=0$ find the value of $x^6-8x^5+13x^4-5x^3+49x^2-137x+2015$ .
2008 China Team Selection Test, 3
Let $ z_{1},z_{2},z_{3}$ be three complex numbers of moduli less than or equal to $ 1$. $ w_{1},w_{2}$ are two roots of the equation $ (z \minus{} z_{1})(z \minus{} z_{2}) \plus{} (z \minus{} z_{2})(z \minus{} z_{3}) \plus{} (z \minus{} z_{3})(z \minus{} z_{1}) \equal{} 0$. Prove that, for $ j \equal{} 1,2,3$, $\min\{|z_{j} \minus{} w_{1}|,|z_{j} \minus{} w_{2}|\}\leq 1$ holds.
2014 Hanoi Open Mathematics Competitions, 12
Find a polynomial $Q(x)$ such that $(2x^2 - 6x + 5)Q(x)$ is a polynomial with all positive coefficients.
2022 Indonesia TST, N
Let $n$ be a natural number, with the prime factorisation
\[ n = p_1^{e_1} p_2^{e_2} \cdots p_r^{e_r} \] where $p_1, \ldots, p_r$ are distinct primes, and $e_i$ is a natural number. Define
\[ rad(n) = p_1p_2 \cdots p_r \] to be the product of all distinct prime factors of $n$. Determine all polynomials $P(x)$ with rational coefficients such that there exists infinitely many naturals $n$ satisfying $P(n) = rad(n)$.
2023 Spain Mathematical Olympiad, 4
Let $x_1\leq x_2\leq x_3\leq x_4$ be real numbers. Prove that there exist polynomials of degree two $P(x)$ and $Q(x)$ with real coefficients such that $x_1$, $x_2$, $x_3$ and $x_4$ are the roots of $P(Q(x))$ if and only if $x_1+x_4=x_2+x_3$.
2011 India IMO Training Camp, 2
Let the real numbers $a,b,c,d$ satisfy the relations $a+b+c+d=6$ and $a^2+b^2+c^2+d^2=12.$ Prove that
\[36 \leq 4 \left(a^3+b^3+c^3+d^3\right) - \left(a^4+b^4+c^4+d^4 \right) \leq 48.\]
[i]Proposed by Nazar Serdyuk, Ukraine[/i]
2007 Silk Road, 4
The set of polynomials $f_1, f_2, \ldots, f_n$ with real coefficients is called [i]special [/i], if for any different $i,j,k \in \{ 1,2, \ldots, n\}$ polynomial $\dfrac{2}{3}f_i + f_j + f_k$ has no real roots, but for any different $p,q,r,s \in \{ 1,2, \ldots, n\}$ of a polynomial $f_p + f_q + f_r + f_s$ there is a real root.
a) Give an example of a [i]special [/i] set of four polynomials whose sum is not a zero polynomial.
b) Is there a [i]special [/i] set of five polynomials?
1994 Balkan MO, 2
Let $n$ be an integer. Prove that the polynomial $f(x)$ has at most one zero, where \[ f(x) = x^4 - 1994 x^3 + (1993+n)x^2 - 11x + n . \]
[i]Greece[/i]
2001 Nordic, 3
Determine the number of real roots of the equation
${x^8 -x^7 + 2x^6- 2x^5 + 3x^4 - 3x^3 + 4x^2 - 4x + \frac{5}{2}= 0}$
2007 All-Russian Olympiad, 2
Given polynomial $P(x) = a_{0}x^{n}+a_{1}x^{n-1}+\dots+a_{n-1}x+a_{n}$. Put $m=\min \{ a_{0}, a_{0}+a_{1}, \dots, a_{0}+a_{1}+\dots+a_{n}\}$. Prove that $P(x) \ge mx^{n}$ for $x \ge 1$.
[i]A. Khrabrov [/i]
2020 CIIM, 4
For each polynomial $P(x)$ with real coefficients, define
$P_0=P(0)$ and $P_j(x)=x^j\cdot P^{(j)}(x)$
where $P^{(j)}$ denotes the $j$-th derivative of $P$ for $j\geq 1$.
Prove that there exists one unique sequence of real numbers $b_0, b_1, b_2, \dots$ such that for each polynomial $P(x)$ with real coefficients and for each $x$ real, we have
$P(x)=b_0P_0+\sum_{k\geq 1}b_kP_k(x)=b_0P_0+b_1P_1(x)+b_2P_2(x)+\dots$
2018 PUMaC Live Round, 7.1
Find the number of nonzero terms of the polynomial $P(x)$ if $$x^{2018}+x^{2017}+x^{2016}+x^{999}+1=(x^4+x^3+x^2+x+1)P(x).$$
2020 Taiwan TST Round 2, 4
Alice and Bob are stuck in quarantine, so they decide to play a game. Bob will write down a polynomial $f(x)$ with the following properties:
(a) for any integer $n$, $f(n)$ is an integer;
(b) the degree of $f(x)$ is less than $187$.
Alice knows that $f(x)$ satisfies (a) and (b), but she does not know $f(x)$. In every turn, Alice picks a number $k$ from the set $\{1,2,\ldots,187\}$, and Bob will tell Alice the value of $f(k)$. Find the smallest positive integer $N$ so that Alice always knows for sure the parity of $f(0)$ within $N$ turns.
[i]Proposed by YaWNeeT[/i]
1999 AMC 12/AHSME, 12
What is the maximum number of points of intersection of the graphs of two different fourth degree polynomial functions $ y \equal{} p(x)$ and $ y \equal{} q(x)$, each with leading coefficient $ 1$?
$ \textbf{(A)}\ 1 \qquad \textbf{(B)}\ 2 \qquad \textbf{(C)}\ 3 \qquad \textbf{(D)}\ 4 \qquad \textbf{(E)}\ 8$
2007 AIME Problems, 14
Let $f(x)$ be a polynomial with real coefficients such that $f(0) = 1,$ $f(2)+f(3)=125,$ and for all $x$, $f(x)f(2x^{2})=f(2x^{3}+x).$ Find $f(5).$
2024 Bulgarian Autumn Math Competition, 10.3
Find all polynomials $P$ with integer coefficients, for which there exists a number $N$, such that for every natural number $n \geq N$, all prime divisors of $n+2^{\lfloor \sqrt{n} \rfloor}$ are also divisors of $P(n)$.
2012 Graduate School Of Mathematical Sciences, The Master Course, Kyoto University, A1
Find the smallest positive integer value of $N$ such that field $K=\mathbb{Q}(\sqrt{N},\ \sqrt{i+1})$, where $i=\sqrt{-1}$, is Galois extension on $\mathbb{Q}$, then find the Galois group $Gal(K/\mathbb{Q}).$
2003 National Olympiad First Round, 24
If $3a=1+\sqrt 2$, what is the largest integer not exceeding $9a^4-6a^3+8a^2-6a+9$?
$
\textbf{(A)}\ 8
\qquad\textbf{(B)}\ 9
\qquad\textbf{(C)}\ 10
\qquad\textbf{(D)}\ 12
\qquad\textbf{(E)}\ \text{None of the preceding}
$