Found problems: 3597
2010 ELMO Shortlist, 4
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$. Prove that $\displaystyle\frac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
[i]Evan O' Dorney.[/i]
2009 Turkey Team Selection Test, 1
For which $ p$ prime numbers, there is an integer root of the polynominal $ 1 \plus{} p \plus{} Q(x^1)\cdot\ Q(x^2)\ldots\ Q(x^{2p \minus{} 2})$ such that $ Q(x)$ is a polynominal with integer coefficients?
1987 India National Olympiad, 6
Prove that if coefficients of the quadratic equation $ ax^2\plus{}bx\plus{}c\equal{}0$ are odd integers, then the roots of the equation cannot be rational numbers.
2013 Balkan MO Shortlist, A5
Determine all positive integers$ n$ such that $f_n(x,y,z) = x^{2n} + y^{2n} + z^{2n} - xy - yz - zx$ divides $g_n(x,y, z) = (x - y)^{5n} + (y -z)^{5n} + (z - x)^{5n}$, as polynomials in $x, y, z$ with integer coefficients.
2012 Germany Team Selection Test, 1
Consider a polynomial $P(x) = \prod^9_{j=1}(x+d_j),$ where $d_1, d_2, \ldots d_9$ are nine distinct integers. Prove that there exists an integer $N,$ such that for all integers $x \geq N$ the number $P(x)$ is divisible by a prime number greater than 20.
[i]Proposed by Luxembourg[/i]
2012 Indonesia TST, 1
Suppose $P(x,y)$ is a homogenous non-constant polynomial with real coefficients such that $P(\sin t, \cos t) = 1$ for all real $t$. Prove that $P(x,y) = (x^2+y^2)^k$ for some positive integer $k$.
(A polynomial $A(x,y)$ with real coefficients and having a degree of $n$ is homogenous if it is the sum of $a_ix^iy^{n-i}$ for some real number $a_i$, for all integer $0 \le i \le n$.)
2014 Contests, 1
Let $f : \mathbb{Z} \rightarrow \mathbb{Z}^+$ be a function, and define $h : \mathbb{Z} \times \mathbb{Z} \rightarrow \mathbb{Z}^+$ by $h(x, y) = \gcd (f(x), f(y))$. If $h(x, y)$ is a two-variable polynomial in $x$ and $y$, prove that it must be constant.
2016 Vietnam Team Selection Test, 6
Given $16$ distinct real numbers $\alpha_1,\alpha_2,...,\alpha_{16}$. For each polynomial $P$, denote \[ V(P)=P(\alpha_1)+P(\alpha_2)+...+P(\alpha_{16}). \] Prove that there is a monic polynomial $Q$, $\deg Q=8$ satisfying:
i) $V(QP)=0$ for all polynomial $P$ has $\deg P<8$.
ii) $Q$ has $8$ real roots (including multiplicity).
2015 All-Russian Olympiad, 4
You are given $N$ such that $ n \ge 3$. We call a set of $N$ points on a plane acceptable if their abscissae are unique, and each of the points is coloured either red or blue. Let's say that a polynomial $P(x)$ divides a set of acceptable points either if there are no red dots above the graph of $P(x)$, and below, there are no blue dots, or if there are no blue dots above the graph of $P(x)$ and there are no red dots below. Keep in mind, dots of both colors can be present on the graph of $P(x)$ itself. For what least value of k is an arbitrary set of $N$ points divisible by a polynomial of degree $k$?
1985 Traian Lălescu, 1.3
Find all functions $ f:\mathbb{Q}\longrightarrow\mathbb{Q} $ with the property that
$$ f\left( p(x)\right) =p\left( f(x)\right) ,\quad\forall x\in\mathbb{Q} , $$
for all integer polynomials $ p. $
2014 Saudi Arabia Pre-TST, 2.2
Let $a_1, a_2, a_3, a_4, a_5$ be nonzero real numbers. Prove that the polynomial $$P(x)= \prod_{k=0}^{4} a_{k+1}x^4 + a_{k+2}x^3 + a_{k+3}x^2 + a_{k+4}x + a_{k+5}$$, where $a_{5+i} = a_i$ for $i = 1,2, 3,4$, has a root with negative real part.
1984 USAMO, 5
$P(x)$ is a polynomial of degree $3n$ such that
\begin{eqnarray*}
P(0) = P(3) = \cdots &=& P(3n) = 2, \\
P(1) = P(4) = \cdots &=& P(3n-2) = 1, \\
P(2) = P(5) = \cdots &=& P(3n-1) = 0, \quad\text{ and }\\
&& P(3n+1) = 730.\end{eqnarray*}
Determine $n$.
2013 Romania Team Selection Test, 3
Given an integer $n\geq 2$, determine all non-constant polynomials $f$ with complex coefficients satisfying the condition
\[1+f(X^n+1)=f(X)^n.\]
2009 Kyrgyzstan National Olympiad, 3
For function $ f: \mathbb{R} \to \mathbb{R}$ given that $ f(x^2 +x +3) +2 \cdot f(x^2 - 3x + 5) = 6x^2 - 10x +17$, calculate $ f(2009)$.
1998 Mediterranean Mathematics Olympiad, 2
Prove that the polynomial $z^{2n} + z^n + 1\ (n \in \mathbb{N})$ is divisible by the polynomial $z^2 + z + 1$ if and only if $n$ is not a multiple of $3$.
2013 Turkey Junior National Olympiad, 1
Let $x, y, z$ be real numbers satisfying $x+y+z=0$ and $x^2+y^2+z^2=6$. Find the maximum value of
\[ |(x-y)(y-z)(z-x) | \]
1978 Putnam, B3
The sequence $(Q_{n}(x))$ of polynomials is defined by
$$Q_{1}(x)=1+x ,\; Q_{2}(x)=1+2x,$$
and for $m \geq 1 $ by
$$Q_{2m+1}(x)= Q_{2m}(x) +(m+1)x Q_{2m-1}(x),$$
$$Q_{2m+2}(x)= Q_{2m+1}(x) +(m+1)x Q_{2m}(x).$$
Let $x_n$ be the largest real root of $Q_{n}(x).$ Prove that $(x_n )$ is an increasing sequence and that $\lim_{n\to \infty} x_n =0.$
2010 Indonesia TST, 1
Let $ f$ be a polynomial with integer coefficients. Assume that there exists integers $ a$ and $ b$ such that $ f(a)\equal{}41$ and $ f(b)\equal{}49$. Prove that there exists an integer $ c$ such that $ 2009$ divides $ f(c)$.
[i]Nanang Susyanto, Jogjakarta[/i]
1976 IMO, 2
Let $P_{1}(x)=x^{2}-2$ and $P_{j}(x)=P_{1}(P_{j-1}(x))$ for j$=2,\ldots$ Prove that for any positive integer n the roots of the equation $P_{n}(x)=x$ are all real and distinct.
2015 Korea Junior Math Olympiad, 7
For a polynomial $f(x)$ with integer coefficients and degree no less than $1$, prove that there are infinitely many primes $p$ which satisfies the following.
There exists an integer $n$ such that $f(n) \not= 0$ and $|f(n)|$ is a multiple of $p$.
2006 Iran MO (3rd Round), 3
Find all real $x,y,z$ that \[\left\{\begin{array}{c}x+y+zx=\frac12\\ \\ y+z+xy=\frac12\\ \\ z+x+yz=\frac12\end{array}\right.\]
2013 Iran MO (3rd Round), 2
Suppose that $a,b$ are two odd positive integers such that $2ab+1 \mid a^2 + b^2 + 1$. Prove that $a=b$.
(15 points)
2010 Contests, 4
Determine whether there exists a polynomial $f(x_1, x_2)$ with two variables, with integer coefficients, and two points $A=(a_1, a_2)$ and $B=(b_1, b_2)$ in the plane, satisfying the following conditions:
(i) $A$ is an integer point (i.e $a_1$ and $a_2$ are integers);
(ii) $|a_1-b_1|+|a_2-b_2|=2010$;
(iii) $f(n_1, n_2)>f(a_1, a_2)$ for all integer points $(n_1, n_2)$ in the plane other than $A$;
(iv) $f(x_1, x_2)>f(b_1, b_2)$ for all integer points $(x_1, x_2)$ in the plane other than $B$.
[i]Massimo Gobbino, Italy[/i]
2023 Azerbaijan IZhO TST, 2
P(x) is polynomial such that, polynomial P(P(x)) is strictly monotone in all real number line. Prove that polynomial P(x) is also strictly monotone in all real number line.
2016 Estonia Team Selection Test, 6
A circle is divided into arcs of equal size by $n$ points ($n \ge 1$). For any positive integer $x$, let $P_n(x)$ denote the number of possibilities for colouring all those points, using colours from $x$ given colours, so that any rotation of the colouring by $ i \cdot \frac{360^o}{n}$ , where i is a positive integer less than $n$, gives a colouring that differs from the original in at least one point. Prove that the function $P_n(x)$ is a polynomial with respect to $x$.