Found problems: 3597
2022 Turkey Team Selection Test, 6
For a polynomial $P(x)$ with integer coefficients and a prime $p$, if there is no $n \in \mathbb{Z}$ such that $p|P(n)$, we say that polynomial $P$ [i]excludes[/i] $p$. Is there a polynomial with integer coefficients such that having degree of 5, excluding exactly one prime and not having a rational root?
2015 Dutch IMO TST, 2
Determine all polynomials P(x) with real coefficients such that
[(x + 1)P(x − 1) − (x − 1)P(x)] is a constant polynomial.
2011 Iran MO (3rd Round), 2
[b]a)[/b] Prove that for every natural numbers $n$ and $k$, we have monic polynomials of degree $n$, with integer coefficients like $A=\{P_1(x),.....,P_k(x)\}$ such that no two of them have a common factor and for every subset of $A$, the sum of elements of $A$ has all its roots real.
[b]b)[/b] Are there infinitely many monic polynomial of degree $n$ with integer coefficients like $P_1(x),P_2(x),....$ such that no two of them have a common factor and the sum of a finite number of them has all it's roots real?
[i]proposed by Mohammad Mansouri[/i]
PEN N Problems, 13
One member of an infinite arithmetic sequence in the set of natural numbers is a perfect square. Show that there are infinitely many members of this sequence having this property.
1994 Vietnam National Olympiad, 3
Do there exist polynomials $p(x), q(x), r(x)$ whose coefficients are positive integers such that $p(x) = (x^{2}-3x+3) q(x)$ and $q(x) = (\frac{x^{2}}{20}-\frac{x}{15}+\frac{1}{12}) r(x)$?
1965 AMC 12/AHSME, 7
The sum of the reciprocals of the roots of the equation $ ax^2 \plus{} bx \plus{} c \equal{} 0$ is:
$ \textbf{(A)}\ \frac {1}{a} \plus{} \frac {1}{b} \qquad \textbf{(B)}\ \minus{} \frac {c}{b} \qquad \textbf{(C)}\ \frac {b}{c} \qquad \textbf{(D)}\ \minus{} \frac {a}{b} \qquad \textbf{(E)}\ \minus{} \frac {b}{c}$
2004 India IMO Training Camp, 3
Suppose the polynomial $P(x) \equiv x^3 + ax^2 + bx +c$ has only real zeroes and let $Q(x) \equiv 5x^2 - 16x + 2004$. Assume that $P(Q(x)) = 0$ has no real roots. Prove that $P(2004) > 2004$
1993 All-Russian Olympiad, 4
If $ \{a_k\}$ is a sequence of real numbers, call the sequence $ \{a'_k\}$ defined by $ a_k' \equal{} \frac {a_k \plus{} a_{k \plus{} 1}}2$ the [i]average sequence[/i] of $ \{a_k\}$. Consider the sequences $ \{a_k\}$; $ \{a_k'\}$ - [i]average sequence[/i] of $ \{a_k\}$; $ \{a_k''\}$ - average sequence of $ \{a_k'\}$ and so on. If all these sequences consist only of integers, then $ \{a_k\}$ is called [i]Good[/i]. Prove that if $ \{x_k\}$ is a [i]good[/i] sequence, then $ \{x_k^2\}$ is also [i]good[/i].
2002 Vietnam National Olympiad, 1
Let $ a$, $ b$, $ c$ be real numbers for which the polynomial $ x^3 \plus{} ax^2 \plus{} bx \plus{} c$ has three real roots. Prove that \[ 12ab \plus{} 27c \le 6a^3 \plus{} 10\left(a^2 \minus{} 2b\right)^{\frac {3}{2}}\] When does equality occur?
2015 AMC 10, 16
If $y+4 = (x-2)^2, x+4 = (y-2)^2$, and $x \neq y$, what is the value of $x^2+y^2$?
$ \textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }20\qquad\textbf{(D) }25\qquad\textbf{(E) }\text{30} $
2006 Iran Team Selection Test, 1
We have $n$ points in the plane, no three on a line.
We call $k$ of them good if they form a convex polygon and there is no other point in the convex polygon.
Suppose that for a fixed $k$ the number of $k$ good points is $c_k$.
Show that the following sum is independent of the structure of points and only depends on $n$ :
\[ \sum_{i=3}^n (-1)^i c_i \]
KoMaL A Problems 2020/2021, A. 801
For which values of positive integer $m$ is it possible to find polynomials $P, Q\in\mathbb{C} [x]$, with degrees at least two, such that \[x(x+1)\cdots(x+m-1)=P(Q(x)).\][i]Proposed by Navid Safaei, Tehran[/i]
2000 Turkey MO (2nd round), 2
Let define $P_{n}(x)=x^{n-1}+x^{n-2}+x^{n-3}+ \dots +x+1$ for every positive integer $n$. Prove that for every positive integer $a$ one can find a positive integer $n$ and polynomials $R(x)$ and $Q(x)$ with integer coefficients such that \[P_{n}(x)= [1+ax+x^{2}R(x)] Q(x).\]
2002 Tournament Of Towns, 3
Show that if the last digit of the number $x^2+xy+y^2$ is $0$ (where $x,y\in\mathbb{N}$ ) then last two digits are zero.
2007 Indonesia TST, 3
Let $a, b, c$ be positive reals such that $a + b + c = 1$ and $P(x) = 3^{2005}x^{2007 }- 3^{2005}x^{2006} - x^2$.
Prove that $P(a) + P(b) + P(c) \le -1$.
2012 Indonesia TST, 1
Given a positive integer $n$.
(a) If $P$ is a polynomial of degree $n$ where $P(x) \in \mathbb{Z}$ for every $x \in \mathbb{Z}$, prove that for every $a,b \in \mathbb{Z}$ where $P(a) \neq P(b)$,
\[\text{lcm}(1, 2, \ldots, n) \ge \left| \dfrac{a-b}{P(a) - P(b)} \right|\]
(b) Find one $P$ (for each $n$) such that the equality case above is achieved for some $a,b \in \mathbb{Z}$.
2011 Brazil Team Selection Test, 1
Let $P_1$, $P_2$ and $P_3$ be polynomials of degree two with positive coefficient leader and real roots . Prove that if each pair of polynomials has a common root , then the polynomial $P_1 + P_2 + P_3$ has also real roots.
2010 ISI B.Math Entrance Exam, 7
We are given $a,b,c \in \mathbb{R}$ and a polynomial $f(x)=x^3+ax^2+bx+c$ such that all roots (real or complex) of $f(x)$ have same absolute value. Show that $a=0$ iff $b=0$.
1971 IMO Longlists, 16
Knowing that the system
\[x + y + z = 3,\]\[x^3 + y^3 + z^3 = 15,\]\[x^4 + y^4 + z^4 = 35,\]
has a real solution $x, y, z$ for which $x^2 + y^2 + z^2 < 10$, find the value of $x^5 + y^5 + z^5$ for that solution.
2015 Tournament of Towns, 3
Each coefficient of a polynomial is an integer with absolute value not exceeding $2015$.
Prove that every positive root of this polynomial exceeds $\frac{1}{2016}$.
[i]($6$ points)[/i]
1987 Romania Team Selection Test, 11
Let $P(X,Y)=X^2+2aXY+Y^2$ be a real polynomial where $|a|\geq 1$. For a given positive integer $n$, $n\geq 2$ consider the system of equations: \[ P(x_1,x_2) = P(x_2,x_3) = \ldots = P(x_{n-1},x_n) = P(x_n,x_1) = 0 . \] We call two solutions $(x_1,x_2,\ldots,x_n)$ and $(y_1,y_2,\ldots,y_n)$ of the system to be equivalent if there exists a real number $\lambda \neq 0$, $x_1=\lambda y_1$, $\ldots$, $x_n= \lambda y_n$. How many nonequivalent solutions does the system have?
[i]Mircea Becheanu[/i]
2008 China National Olympiad, 3
Find all triples $(p,q,n)$ that satisfy
\[q^{n+2} \equiv 3^{n+2} (\mod p^n) ,\quad p^{n+2} \equiv 3^{n+2} (\mod q^n)\]
where $p,q$ are odd primes and $n$ is an positive integer.
2002 All-Russian Olympiad Regional Round, 9.2
A monic quadratic polynomial $f$ with integer coefficients attains prime values at three consecutive integer points.show that it attains a prime value at some other integer point as well.
2004 Kurschak Competition, 2
Find the smallest positive integer $n\neq 2004$ for which there exists a polynomial $f\in\mathbb{Z}[x]$ such that the equation $f(x)=2004$ has at least one, and the equation $f(x)=n$ has at least $2004$ different integer solutions.
2015 Canadian Mathematical Olympiad Qualification, 2
A polynomial $f(x)$ with integer coefficients is said to be [i]tri-divisible[/i] if $3$ divides $f(k)$ for any integer $k$. Determine necessary and sufficient conditions for a polynomial to be tri-divisible.