Find all values of the positive integer $ m$ such that there exists polynomials $ P(x),Q(x),R(x,y)$ with real coefficient satisfying the condition: For every real numbers $ a,b$ which satisfying $ a^m-b^2=0$, we always have that $ P(R(a,b))=a$ and $ Q(R(a,b))=b$.

Consider the second-degree polynomial \(P(x) = 4x^2+12x-3015\). Define the sequence of polynomials \(P_1(x)=\frac{P(x)}{2016}\) and \(P_{n+1}(x)=\frac{P(P_n(x))}{2016}\) for every integer \(n \geq 1\). [list='a'] [*]Show that exists a real number \(r\) such that \(P_n(r) < 0\) for every positive integer \(n\). [*]Find how many integers \(m\) are such that \(P_n(m)<0\) for infinite positive integers \(n\). [/list]

Prove that the polynomial $x^n+4$ can be expressed as a product of two polynomials (each with degree less than $n$) with integer coefficients, if and only if $n$ is divisible by $4$.

Find the largest real number $\lambda$ with the following property: for any positive real numbers $p,q,r,s$ there exists a complex number $z=a+bi$($a,b\in \mathbb{R})$ such that $$ |b|\ge \lambda |a| \quad \text{and} \quad (pz^3+2qz^2+2rz+s) \cdot (qz^3+2pz^2+2sz+r) =0.$$

Find all polynomials $f$ with real coefficients such that for all reals $a,b,c$ such that $ab+bc+ca = 0$ we have the following relations \[ f(a-b) + f(b-c) + f(c-a) = 2f(a+b+c). \]

Prove that $\frac{5^{125}-1}{5^{25}-1}$ is a composite number.

Determine the lowest positive integer n such that following statement is true: If polynomial with integer coefficients gets value 2 for n different integers, then it can't take value 4 for any integer.

Let $l$ be the tangent line at the point $P(s,\ t)$ on a circle $C:x^2+y^2=1$. Denote by $m$ the line passing through the point $(1,\ 0)$, parallel to $l$. Let the line $m$ intersects the circle $C$ at $P'$ other than the point $(1,\ 0)$. Note : if $m$ is the line $x=1$, then $P'$ is considered as $(1,\ 0)$. Call $T$ the operation such that the point $P'(s',\ t')$ is obtained from the point $P(s,\ t)$ on $C$. (1) Express $s',\ t'$ as the polynomials of $s$ and $t$ respectively. (2) Let $P_n$ be the point obtained by $n$ operations of $T$ for $P$. For $P\left(\frac{\sqrt{3}}{2},\ \frac{1}{2}\right)$, plot the points $P_1,\ P_2$ and $P_3$. (3) For a positive integer $n$, find the number of $P$ such that $P_n=P$.

Prove that it is possible to choose different real numbers $a_1, a_2, . . . , a_{10}$ that the equation $$(x - a_1)(x -a_2).... (x -a_{10}) = (x + a_1)(x + a_2) ...(x + a_{10})$$ will have exactly $5$ different real roots.

A bee starts flying from point $P_0$. She flies 1 inch due east to point $P_1$. For $j \ge 1$, once the bee reaches point $P_j$, she turns $30^{\circ}$ counterclockwise and then flies $j+1$ inches straight to point $P_{j+1}$. When the bee reaches $P_{2015}$ she is exactly $a\sqrt{b} + c\sqrt{d}$ inches away from $P_0$, where $a$, $b$, $c$ and $d$ are positive integers and $b$ and $d$ are not divisible by the square of any prime. What is $a+b+c+d$? $ \textbf{(A)}\ 2016 \qquad\textbf{(B)}\ 2024 \qquad\textbf{(C)}\ 2032 \qquad\textbf{(D)}\ 2040 \qquad\textbf{(E)}\ 2048$

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]

Written on a blackboard is the polynomial $x^2+x+2014$. Calvin and Hobbes take turns alternately (starting with Calvin) in the following game. At his turn, Calvin should either increase or decrease the coefficient of $x$ by $1$. And at this turn, Hobbes should either increase or decrease the constant coefficient by $1$. Calvin wins if at any point of time the polynomial on the blackboard at that instant has integer roots. Prove that Calvin has a winning stratergy.

Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

Let $\left \{ c_i \right \}_{i=0}^{\infty}$ be a sequence of non-negative real numbers with $c_{2017}>0$. A sequence of polynomials is defined as $$P_{-1}(x)=0 \ , \ P_0(x)=1 \ , \ P_{n+1}(x)=xP_n(x)+c_nP_{n-1}(x).$$ Prove that there doesn't exist any integer $n>2017$ and some real number $c$ such that $$P_{2n}(x)=P_n(x^2+c).$$ [i]Proposed by Navid Safaei[/i]

Let $P(x)$ be a polynomial with complex coefficients such that $P(0)\neq 0$. Prove that there exists a multiple of $P(x)$ with real positive coefficients if and only if $P(x)$ has no real positive root.

There are positive integers $b$ and $c$ such that the polynomial $2x^2 + bx + c$ has two real roots which differ by $30.$ Find the least possible value of $b + c.$

a) Let $a\in \mathbb{R}$ and $f \colon \mathbb{R} \to \mathbb{R}$ be a continuous function for which there exists an antiderivative $F \colon \mathbb{R} \to \mathbb{R} $, such that $F(x)+a\cdot f(x) \geq 0$, for any $x \in \mathbb{R}$, and$ \lim_{|x| \to \infty} \frac{F(x)}{e^{|\alpha \cdot x|}}=0$ holds for any $\alpha \in \mathbb{R}^*$. Prove that $F(x) \geq 0$ for all $x \in \mathbb{R}$. b) Let $n\geq 2$ be a positive integer, $g \in \mathbb{R}[X]$, $g = X^n + a_1X^{n-1}+ \dots + a_{n-1}X+a_n$ be a polynomial with all of its roots being real, and $f \colon \mathbb{R} \to \mathbb{R}$ a polynomial function such that $f(x)+a_1\cdot f'(x)+a_2\cdot f^{(2)}(x)+\dots+a_n\cdot f^{(n)}(x) \geq 0$ for any $x \in \mathbb{R}$. Prove that $f(x) \geq 0$ for all $x \in \mathbb{R}$.

[b]two variable ploynomial[/b] $P(x,y)$ is a two variable polynomial with real coefficients. degree of a monomial means sum of the powers of $x$ and $y$ in it. we denote by $Q(x,y)$ sum of monomials with the most degree in $P(x,y)$. (for example if $P(x,y)=3x^4y-2x^2y^3+5xy^2+x-5$ then $Q(x,y)=3x^4y-2x^2y^3$.) suppose that there are real numbers $x_1$,$y_1$,$x_2$ and $y_2$ such that $Q(x_1,y_1)>0$ , $Q(x_2,y_2)<0$ prove that the set $\{(x,y)|P(x,y)=0\}$ is not bounded. (we call a set $S$ of plane bounded if there exist positive number $M$ such that the distance of elements of $S$ from the origin is less than $M$.) time allowed for this question was 1 hour.

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.

Let $ q_{0}, q_{1}, \cdots$ be a sequence of integers such that a) for any $ m > n$, $ m \minus{} n$ is a factor of $ q_{m} \minus{} q_{n}$, b) item $ |q_n| \le n^{10}$ for all integers $ n \ge 0$. Show that there exists a polynomial $ Q(x)$ satisfying $ q_{n} \equal{} Q(n)$ for all $ n$.

A polynomial $p(x)$ with integer coefficients satisfies the equality $$p(\sqrt{2}+\sqrt{3})=\sqrt{2}-\sqrt{3}$$ a) Find all possible values of $p(\sqrt{2}-\sqrt{3})$ b) Find an example of any polynomial $p(x)$ which satisfies the condition.

Let $A=\{a_1,a_2,\cdots,a_{2010}\}$ and $B=\{b_1,b_2,\cdots,b_{2010}\}$ be two sets of complex numbers. Suppose \[\sum_{1\leq i<j\leq 2010} (a_i+a_j)^k=\sum_{1\leq i<j\leq 2010}(b_i+b_j)^k\] holds for every $k=1,2,\cdots, 2010$. Prove that $A=B$.

Find all polynomials $p(x)$ such that for all reals $a, b$ and $c$, with $a+b+c=0$, satisfies $$p(a^3)+p(b^3)+p(c^3)=3p(abc)$$

Let $f$ be a polynomial function with integer coefficients and $p$ be a prime number. Suppose there are at least four distinct integers satisfying $f(x) = p$. Show that $f$ does not have integer zeros.

For any polynomial $P(x)=a_0+a_1x+\ldots+a_kx^k$ with integer coefficients, the number of odd coefficients is denoted by $o(P)$. For $i-0,1,2,\ldots$ let $Q_i(x)=(1+x)^i$. Prove that if $i_1,i_2,\ldots,i_n$ are integers satisfying $0\le i_1<i_2<\ldots<i_n$, then: \[ o(Q_{i_{1}}+Q_{i_{2}}+\ldots+Q_{i_{n}})\ge o(Q_{i_{1}}). \]