Let \[ p(x) = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0, \] where the coefficients $a_i$ are integers. If $p(0)$ and $p(1)$ are both odd, show that $p(x)$ has no integral roots.

Quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$, both of which have real roots, are called friendly if for all $t \in [0,1]$ quadratic polynomial $tP(x)+(1-t)Q(x)$ also has real roots. a) Provide an example of quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ and which have real roots, that are not friendly. b) Prove that for any two quadratic polynomials $P(x)$ and $Q(x)$ with leading coefficients $1$ that have real roots, there is a quadratic polynomial $R(x)$ which has a leading coefficient $1$ and which is friendly to both $P$ and $Q$

Do there exist two polynomials with integer coefficients such that each polynomial has a coefficient with an absolute value exceeding $2015$ but all coefficients of their product have absolute values not exceeding $1$? [i]($10$ points)[/i]

(a) Let $n \geq 1$ be an integer. Prove that $X^n+Y^n+Z^n$ can be written as a polynomial with integer coefficients in the variables $\alpha=X+Y+Z$, $\beta= XY+YZ+ZX$ and $\gamma = XYZ$. (b) Let $G_n=x^n \sin(nA)+y^n \sin(nB)+z^n \sin(nC)$, where $x,y,z, A,B,C$ are real numbers such that $A+B+C$ is an integral multiple of $\pi$. Using (a) or otherwise show that if $G_1=G_2=0$, then $G_n=0$ for all positive integers $n$.

Does there exist a polynomial $P(x)$ with integral coefficients such that $P(1+\sqrt 3) = 2+\sqrt 3$ and $P(3+\sqrt 5) = 3+\sqrt 5 $? [i]Proposed by Alexander S. Golovanov, Russia[/i]

We say that a ring $A$ has property $(P)$ if any non-zero element can be written uniquely as the sum of an invertible element and a non-invertible element. a) If in $A$, $1+1=0$, prove that $A$ has property $(P)$ if and only if $A$ is a field. b) Give an example of a ring that is not a field, containing at least two elements, and having property $(P)$. [i]Dan Schwarz[/i]

Alice and Bob play a game in which they take turns removing stones from a heap that initially has $n$ stones. The number of stones removed at each turn must be one less than a prime number. The winner is the player who takes the last stone. Alice plays first. Prove that there are infinitely many such $n$ such that Bob has a winning strategy. (For example, if $n=17,$ then Alice might take $6$ leaving $11;$ then Bob might take $1$ leaving $10;$ then Alice can take the remaining stones to win.)

Let $n>1$ be an integer. Find all non-constant real polynomials $P(x)$ satisfying , for any real $x$ , the identy \[P(x)P(x^2)P(x^3)\cdots P(x^n)=P(x^{\frac{n(n+1)}{2}})\]

Initially memory of computer contained a single polynomial $x^2-1$. Every minute computer chooses any polynomial $f(x)$ from its memory and writes $f(x^2-1)$ and $f(x)^2-1$ to it, or chooses any two distinct polynomials $g(x), h(x)$ from its memory and writes polynomial $\frac{g(x) + h(x)}{2}$ to it (no polynomial is ever erased from its memory). Can it happen that after some time, memory of computer contains $P(x) = \frac{1}{1024}(x^2-1)^{2048} - 1$? [i](Proposed by Arsenii Nikolaiev)[/i]

We say that a function \( f: \mathbb{R} \to \mathbb{R} \) is morally odd if its graph is symmetric with respect to a point, that is, there exists \((x_0, y_0) \in \mathbb{R}^2\) such that if \((u, v) \in \{(x, f(x)) : x \in \mathbb{R}\}\), then \((2x_0 - u, 2y_0 - v) \in \{(x, f(x)) : x \in \mathbb{R}\}\). On the other hand, \( f \) is said to be morally even if its graph \(\{(x, f(x)) : x \in \mathbb{R}\}\) is symmetric with respect to some line (not necessarily vertical or horizontal). If \( f \) is morally even and morally odd, we say that \( f \) is parimpar. (a) Let \( S \subset \mathbb{R} \) be a bounded set and \( f: S \to \mathbb{R} \) be an arbitrary function. Prove that there exists \( g: \mathbb{R} \to \mathbb{R} \) that is parimpar such that \( g(x) = f(x) \) for all \( x \in S \). (b) Find all polynomials \( P \) with real coefficients such that the corresponding polynomial function \( P: \mathbb{R} \to \mathbb{R} \) is parimpar.

Let $n$ be a positive even integer, and let $c_1, c_2, \dots, c_{n-1}$ be real numbers satisfying \[ \sum_{i=1}^{n-1} \left\lvert c_i-1 \right\rvert < 1. \] Prove that \[ 2x^n - c_{n-1}x^{n-1} + c_{n-2}x^{n-2} - \dots - c_1x^1 + 2 \] has no real roots.

For a positive integer $n$, a cubic polynomial $p(x)$ is said to be [i]$n$-good[/i] if there exist $n$ distinct integers $a_1, a_2, \ldots, a_n$ such that all the roots of the polynomial $p(x) + a_i = 0$ are integers for $1 \le i \le n$. Given a positive integer $n$ prove that there exists an $n$-good cubic polynomial.

Let $P(x)$ be a polynomial with rational coefficients. Prove that there exists a positive integer $n$ such that the polynomial $Q(x)$ defined by \[Q(x)= P(x+n)-P(x)\] has integer coefficients.

Is there any polynomial $P(x)$ with degree $n$ such that $ \underbrace{P(...(P(x))...)}_{m\,\, times \,\, P}$ has all roots from $1,2,..., mn$ ?

Let $n$ be a positive integer. Find all polynomials $P$ with real coefficients such that $$P(x^2+x-n^2)=P(x)^2+P(x)$$ for all real numbers $x$.

In the expression $(x + 100) (x + 99) ... (x-99) (x-100)$, the brackets were expanded and similar terms were given. The expression $x^{201} + ...+ ax^2 + bx + c$ turned out. Find the numbers $a$ and $c$.

Let $p(x)$ be a polynomial of degree 4 with leading coefficients 1. Suppose $p(1)=1$, $p(2)=2$, $p(3)=3$, $p(4)=4$. Then $p(5)=$ [list=1] [*] 5 [*] $\frac{25}{6}$ [*] 29 [*] 35 [/list]

Let $A\in\mathbb{R}^{n\times n}$ such that $3A^3=A^2+A+I$. Show that the sequence $A^k$ converges to an idempotent matrix. (idempotent: $B^2=B$)

The graph of $ y \equal{} x^6 \minus{} 10x^5 \plus{} 29x^4 \minus{} 4x^3 \plus{} ax^2$ lies above the line $ y \equal{} bx \plus{} c$ except at three values of $ x$, where the graph and the line intersect. What is the largest of those values? $ \textbf{(A)}\ 4 \qquad \textbf{(B)}\ 5 \qquad \textbf{(C)}\ 6 \qquad \textbf{(D)}\ 7 \qquad \textbf{(E)}\ 8$

$p(x_1, x_2, ... , x_n)$ is a polynomial with integer coefficients. For each positive integer $r, k(r)$ is the number of $n$-tuples $(a_1, a_2,... , a_n)$ such that $0 \le a_i \le r-1 $ and $p(a_1, a_2, ... , a_n)$ is prime to $r$. Show that if $u$ and $v$ are coprime then $k(u\cdot v) = k(u)\cdot k(v)$, and if p is prime then $k(p^s) = p^{n(s-1)} k(p)$.

An ordered pair $(x, y)$ of integers is a primitive point if the greatest common divisor of $x$ and $y$ is $1$. Given a finite set $S$ of primitive points, prove that there exist a positive integer $n$ and integers $a_0, a_1, \ldots , a_n$ such that, for each $(x, y)$ in $S$, we have: $$a_0x^n + a_1x^{n-1} y + a_2x^{n-2}y^2 + \cdots + a_{n-1}xy^{n-1} + a_ny^n = 1.$$ [i]Proposed by John Berman, United States[/i]

$p$ is a polynomial with integer coefficients and for every natural $n$ we have $p(n)>n$. $x_k $ is a sequence that: $x_1=1, x_{i+1}=p(x_i)$ for every $N$ one of $x_i$ is divisible by $N.$ Prove that $p(x)=x+1$

Denote $f(x) = x^4 + 2x^3 - 2x^2 - 4x+4$. Prove that there are infinitely many primes $p$ that satisfies the following. For all positive integers $m$, $f(m)$ is not a multiple of $p$.

Find the smallest possible value of a real number $c$ such that for any $2012$-degree monic polynomial \[P(x)=x^{2012}+a_{2011}x^{2011}+\ldots+a_1x+a_0\] with real coefficients, we can obtain a new polynomial $Q(x)$ by multiplying some of its coefficients by $-1$ such that every root $z$ of $Q(x)$ satisfies the inequality \[ \left\lvert \operatorname{Im} z \right\rvert \le c \left\lvert \operatorname{Re} z \right\rvert. \]

If the polynomial $P(x)=ax^2+bx+c$ takes value $0$ for three different values of $x$, then prove the polynomial $P(x)$ takes value $0$ for all $x$.