Prove that the polynom $ X^3-aX-a+1 $ has three integer roots, for an infinite number of integers $ a. $ [i]Liviu Parsan[/i]

a) $P(x),R(x)$ are polynomials with rational coefficients and $P(x)$ is not the zero polynomial. Prove that there exist a non-zero polynomial $Q(x)\in\mathbb Q[x]$ that \[P(x)\mid Q(R(x)).\] b) $P,R$ are polynomial with integer coefficients and $P$ is monic. Prove that there exist a monic polynomial $Q(x)\in\mathbb Z[x]$ that \[P(x)\mid Q(R(x)).\]

Find all polynomials $p(x)$ with real coeffcients such that \[p(a + b - 2c) + p(b + c - 2a) + p(c + a - 2b) = 3p(a - b) + 3p(b - c) + 3p(c - a)\] for all $a, b, c\in\mathbb{R}$. [i](2nd Benelux Mathematical Olympiad 2010, Problem 2)[/i]

Find a polynomial $Q(x)$ such that $(2x^2 - 6x + 5)Q(x)$ is a polynomial with all positive coefficients.

Let $ a_{1}, a_{2}...a_{n}$ be non-negative reals, not all zero. Show that that (a) The polynomial $ p(x) \equal{} x^{n} \minus{} a_{1}x^{n \minus{} 1} \plus{} ... \minus{} a_{n \minus{} 1}x \minus{} a_{n}$ has preceisely 1 positive real root $ R$. (b) let $ A \equal{} \sum_{i \equal{} 1}^n a_{i}$ and $ B \equal{} \sum_{i \equal{} 1}^n ia_{i}$. Show that $ A^{A} \leq R^{B}$.

Prove that if the polynomial $ f $ which is not identical to zero satisfies for every real $ x $ the equality $$ f(x)f(x + 3) = f(x^2 + x + 3), $$then it has no real roots .

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 $p$.

Suppose $P(x)$ is a cubic polynomial with integer coefficients such $P(\sqrt{5})=5$ and $P(\sqrt[3]{5})=5\sqrt[3]{5}$.

[b]p1.[/b] Compute the degree of the least common multiple of the polynomials $x - 1$, $x^2 - 1$, $x^3 - 1$,$...$, $x^{10} -1$. [b]p2.[/b] A line in the $xy$ plane is called wholesome if its equation is $y = mx+b$ where $m$ is rational and $b$ is an integer. Given a point with integer coordinates $(x,y)$ on a wholesome line $\ell$, let $r$ be the remainder when $x$ is divided by $7$, and let $s$ be the remainder when y is divided by $7$. The pair $(r, s)$ is called an [i]ingredient[/i] of the line $\ell$. The (unordered) set of all possible ingredients of a wholesome line $\ell$ is called the [i]recipe [/i] of $\ell$. Compute the number of possible recipes of wholesome lines. [b]p3.[/b] Let $\tau (n)$ be the number of distinct positive divisors of $n$. Compute $\sum_{d|15015} \tau (d)$, that is, the sum of $\tau (d)$ for all $d$ such that $d$ divides $15015$. [b]p4.[/b] Suppose $2202010_b - 2202010_3 = 71813265_{10}$. Compute $b$. ($n_b$ denotes the number $n$ written in base $b$.) [b]p5.[/b] Let $x = (3 -\sqrt5)/2$. Compute the exact value of $x^8 + 1/x^8$. [b]p6.[/b] Compute the largest integer that has the same number of digits when written in base $5$ and when written in base $7$. Express your answer in base $10$. [b]p7.[/b] Three circles with integer radii $a$, $b$, $c$ are mutually externally tangent, with $a \le b \le c$ and $a < 10$. The centers of the three circles form a right triangle. Compute the number of possible ordered triples $(a, b, c)$. [b]p8.[/b] Six friends are playing informal games of soccer. For each game, they split themselves up into two teams of three. They want to arrange the teams so that, at the end of the day, each pair of players has played at least one game on the same team. Compute the smallest number of games they need to play in order to achieve this. [b]p9.[/b] Let $A$ and $B$ be points in the plane such that $AB = 30$. A circle with integer radius passes through $A$ and $B$. A point $C$ is constructed on the circle such that $AC$ is a diameter of the circle. Compute all possible radii of the circle such that $BC$ is a positive integer. [b]p10.[/b] Each square of a $3\times 3$ grid can be colored black or white. Two colorings are the same if you can rotate or reflect one to get the other. Compute the total number of unique colorings. [b]p11.[/b] Compute all positive integers $n$ such that the sum of all positive integers that are less than $n$ and relatively prime to $n$ is equal to $2n$. [b]p12.[/b] The distance between a point and a line is defined to be the smallest possible distance between the point and any point on the line. Triangle $ABC$ has $AB = 10$, $BC = 21$, and $CA = 17$. Let $P$ be a point inside the triangle. Let $x$ be the distance between $P$ and $\overleftrightarrow{BC}$, let $y$ be the distance between $P$ and $\overleftrightarrow{CA}$, and let $z$ be the distance between $P$ and $\overleftrightarrow{AB}$. Compute the largest possible value of the product $xyz$. [b]p13.[/b] Alice, Bob, David, and Eve are sitting in a row on a couch and are passing back and forth a bag of chips. Whenever Bob gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac13$ , and he passes it on in the same direction with probability $\frac23$ . Whenever David gets the bag of chips, he passes the bag back to the person who gave it to him with probability $\frac14$ , and he passes it on with probability $\frac34$ . Currently, Alice has the bag of chips, and she is about to pass it to Bob when Cathy sits between Bob and David. Whenever Cathy gets the bag of chips, she passes the bag back to the person who gave it to her with probability $p$, and passes it on with probability $1-p$. Alice realizes that because Cathy joined them on the couch, the probability that Alice gets the bag of chips back before Eve gets it has doubled. Compute $p$. [b]p14.[/b] Circle $O$ is in the plane. Circles $A$, $B$, and $C$ are congruent, and are each internally tangent to circle $O$ and externally tangent to each other. Circle $X$ is internally tangent to circle $O$ and externally tangent to circles $A$ and $B$. Circle $X$ has radius $1$. Compute the radius of circle $O$. [img]https://cdn.artofproblemsolving.com/attachments/f/d/8ddab540dca0051f660c840c0432f9aa5fe6b0.png[/img] [b]p15.[/b] Compute the number of primes $p$ less than 100 such that $p$ divides $n^2 +n+1$ for some integer $n$. PS. You had better use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let real coefficient polynomial $f(x) = x^n + a_1 \cdot x^{n-1} + \ldots + a_n$ has real roots $b_1, b_2, \ldots, b_n$, $n \geq 2,$ prove that $\forall x \geq max\{b_1, b_2, \ldots, b_n\}$, we have \[f(x+1) \geq \frac{2 \cdot n^2}{\frac{1}{x-b_1} + \frac{1}{x-b_2} + \ldots + \frac{1}{x-b_n}}.\]

Consider the polynomial \[P(x)=\prod_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots(x^{1024}+1024).\] The coefficient of $x^{2012}$ is equal to $2^a$. What is $a$? $ \textbf{(A)}\ 5\qquad\textbf{(B)}\ 6\qquad\textbf{(C)}\ 7\qquad\textbf{(D)}\ 10\qquad\textbf{(E)}\ 24 $

Find all real parameters $a$ for which all the roots of the equation $$x^6+3x^5+(6-a)x^4+(7-2a)x^3+(6-a)x^2+3x+1$$are real.

Find all integer-coefficient polynomials $f(x)$ satisfying the following conditions for every integer $n \geqslant 2$: [list] [*] $f(n) > 0$. [*] $f(n)$ divides $n^{f(n)} - 1$. [/list]

Find all polynomials $f\in Z[X],$ such that for each odd prime $p$ $$f(p)|(p-3)!+\frac{p+1}{2}.$$

Find all complex polynomial $ P(x)$ such that for any three integers $ a,b,c$ satisfying $ a \plus{} b \plus{} c\not \equal{} 0, \frac{P(a) \plus{} P(b) \plus{} P(c)}{a \plus{} b \plus{} c}$ is an integer.

[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.