Let $P(x)$ be a polynomial with integer coefficients and roots $1997$ and $2010$. Suppose further that $|P(2005)|<10$. Determine what integer values $P(2005)$ can get.

Consider a quadratic polynomial $ax^2+bx+c$ with real coefficients satisfying $a\ge 2$, $b\ge 2$, $c\ge 2$. Adam and Boris play the following game. They alternately take turns with Adam first. On Adam’s turn, he can choose one of the polynomial’s coefficients and replace it with the sum of the other two coefficients. On Boris’s turn, he can choose one of the polynomial’s coefficients and replace it with the product of the other two coefficients. The winner is the player who first produces a polynomial with two distinct real roots. Depending on the values of $a$, $b$ and $c$, determine who has a winning strategy.

Factor the polynomial $(a+b+c)^3- a^3 -b^3 -c^3$

Let $P(x)$ be polynomial with integer coefficients. Prove, that if for any natural $k$ holds equality: $ \underbrace{P(P(...P(0)...))}_{n -times}=0$ then $P(0)=0$ or $P(P(0))=0$

Let $P(x)$ denote the polynomial \[3\sum_{k=0}^{9}x^k + 2\sum_{k=10}^{1209}x^k + \sum_{k=1210}^{146409}x^k.\]Find the smallest positive integer $n$ for which there exist polynomials $f,g$ with integer coefficients satisfying $x^n - 1 = (x^{16} + 1)P(x) f(x) + 11\cdot g(x)$. [i]Victor Wang.[/i]

Let $n$ be a natural number. Find all real numbers $x$ satisfying the equation $$\sum^n_{k=1}\frac{kx^k}{1+x^{2k}}=\frac{n(n+1)}4.$$

Let $p,q,r\in R $ with $pqr=1$. Prove that $$\left(\frac{1}{1-p}\right)^2+\left(\frac{1}{1-q}\right)^2+\left(\frac{1}{1-r}\right)^2\ge 1$$ [i](Real Matrik)[/i]

What is the smallest possible value of $x^2+2y^2-x-2y-xy$?

Let $ x,y$ are real numbers such that $x^2+2cosy=1$. Find the ranges of $x-cosy$.

Given [i]Fibonacci[/i] sequence $(F_n),$ and a positive integer $m$, denote $k(m)$ by the smallest positive integer satisfying $F_{n+k(m)}\equiv F_n(\bmod m),$ for all natural numbers $n$, $s$ is a positive integer. Prove that: a) ${F_{{{3.2}^{s - 1}}}} \equiv 0(\bmod {2^s})$ and ${F_{{{3.2}^{s - 1}} + 1}} \equiv 1(\bmod {2^s}).$ b) $k({2^s}) = {3.2^{s - 1}}.$

Given a positive integer $n$, we say that a polynomial $P$ with real coefficients is $n$-pretty if the equation $P(\lfloor x \rfloor)=\lfloor P(x) \rfloor$ has exactly $n$ real solutions. Show that for each positive integer $n$ [list=a] [*] there exists an n-pretty polynomial; [*] any $n$-pretty polynomial has a degree of at least $\tfrac{2n+1}{3}$. [/list] ([i]Remark.[/i] For a real number $x$, we denote by $\lfloor x \rfloor$ the largest integer smaller than or equal to $x$.)

$4$ men stand at the entrance of a dark tunnel. Man $A$ needs $10$ minutes to pass through the tunnel, man $B$ needs $5$ minutes, man $C$ needs $2$ minutes and man $D$ needs $1$ minute. There is only one torch, that may be used from anyone that passes through the tunnel. Additionaly, at most $2$ men can pass through at the same time using the existing torch. Determine the smallest possible time the four men need to reach the exit of the tunnel.

Compute $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1}}}}...}$

Suppose $a,\,b,$ and $c$ are three complex numbers with product $1$. Assume that none of $a,\,b,$ and $c$ are real or have absolute value $1$. Define \begin{tabular}{c c c} $p=(a+b+c)+\left(\dfrac 1a+\dfrac 1b+\dfrac 1c\right)$ & \text{and} & $q=\dfrac ab+\dfrac bc+\dfrac ca$. \end{tabular} Given that both $p$ and $q$ are real numbers, find all possible values of the ordered pair $(p,q)$. [i]David Altizio[/i]

A positive integer is called a [b]double number[/b] if its decimal representation consists of a block of digits, not commencing with 0, followed immediately by an identical block. So, for instance, 360360 is a double number, but 36036 is not. Show that there are infinitely many double numbers which are perfect squares.

Find all function $ f: \mathbb{R}^{+} \to \mathbb{R}^{+}$ such that for every three real positive number $x,y,z$ : $$ x+f(y) , f(f(y)) + z , f(f(z))+f(x) $$ are length of three sides of a triangle and for every postive number $p$ , there is a triangle with these sides and perimeter $p$. [i]Proposed by Amirhossein Zolfaghari [/i]

Determine all polynomials $P(x)$ with real coefficients such that \[(x+1)P(x-1)-(x-1)P(x)\] is a constant polynomial.

Determine all functions $f: \mathbb{R} \mapsto \mathbb{R}$ satisfying \[f\left(\sqrt{2} \cdot x\right) + f\left(4 + 3 \cdot \sqrt{2} \cdot x \right) = 2 \cdot f\left(\left(2 + \sqrt{2}\right) \cdot x\right)\] for all $x$.

For which values of the real parameter $r$ the equation $r^2 x^2+2rx+4=28r^2$ has two distinct integer roots?

Prove that for every natural number $ n$ there exists a polynomial $ p(x)$ with integer coefficients such that$ p(1),p(2),...,p(n)$ are distinct powers of $ 2$ .

Does the Cauchy problem $u|_{y=x^2}=1$, $(\nabla u)^2=1$ have a smooth solution in the domain $y\ge x^2$? In the domain $y\le x^2$?

[b]p1.[/b] There are $16$ students in a class. Each month the teacher divides the class into two groups. What is the minimum number of months that must pass for any two students to be in different groups in at least one of the months? [b]p2.[/b] Find all functions $f(x)$ defined for all real $x$ that satisfy the equation $2f(x) + f(1 - x) = x^2$. [b]p3.[/b] Arrange the digits from $1$ to $9$ in a row (each digit only once) so that every two consecutive digits form a two-digit number that is divisible by $7$ or $13$. [b]p4.[/b] Prove that $\cos 1^o$ is irrational. [b]p5.[/b] Consider $2n$ distinct positive Integers $a_1,a_2,...,a_{2n}$ not exceeding $n^2$ ($n>2$). Prove that some three of the differences $a_i- a_j$ are equal . PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Find all polynomials P(x) such that P(x)+P(1/x)=x+1/x

Prove that any real solution of $x^3+px+q=0$, where $p,q$ are real numbers, satisfies the inequality $4qx\le p^2$.

Find all positive integers $m$ such that $$\{\sqrt{m}\} = \{\sqrt{m+ 2011}\}.$$