Prove that there exist monic polynomial $f(x) $ with degree of 6 and having integer coefficients such that (1) For all integer $m$, $f(m) \ne 0$. (2) For all positive odd integer $n$, there exist positive integer $k$ such that $f(k)$ is divided by $n$.

Suppose $x, y, z, t$ are real numbers such that $\begin{cases} |x + y + z -t |\le 1 \\ |y + z + t - x|\le 1 \\ |z + t + x - y|\le 1 \\ |t + x + y - z|\le 1 \end{cases}$ Prove that $x^2 + y^2 + z^2 + t^2 \le 1$.

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

Find the function $f(x)$ that satisfies the equation $$f'(x) + x^2f(x) = 0$$ knowing that $f(1) = e$. Graph this function and calculate the tangent of the curve at the point of abscissa $1$.

Let $ax^{3}+bx^{2}+cx+d$ be a polynomial with integer coefficients $a,$ $b,$ $c,$ $d$ such that $ad$ is an odd number and $bc$ is an even number. Prove that (at least) one root of the polynomial is irrational.

Given a positive integer $ m$ and $ 0 < \delta <\pi$, construct a trigonometric polynomial $ f(x)\equal{}a_0\plus{} \sum_{n\equal{}1}^m (a_n \cos nx\plus{}b_n \sin nx)$ of degree $ m$ such that $ f(0)\equal{}1, \int_{ \delta \leq |x| \leq \pi} |f(x)|dx \leq c/m,$ and $ \max_{\minus{}\pi \leq x \leq \pi}|f'(x)| \leq c/{\delta}$, for some universal constant $ c$. [i]G. Halasz[/i]

Let $P$ be the product of the nonreal roots of $x^4-4x^3+6x^2-4x=2005$. Find $\lfloor P\rfloor$.

A sequence of real numbers $ x_0, x_1, x_2, \ldots$ is defined as follows: $ x_0 \equal{} 1989$ and for each $ n \geq 1$ \[ x_n \equal{} \minus{} \frac{1989}{n} \sum^{n\minus{}1}_{k\equal{}0} x_k.\] Calculate the value of $ \sum^{1989}_{n\equal{}0} 2^n x_n.$

[url=https://artofproblemsolving.com/community/c678145][b]Czech-Polish-Slovak Match 2018[/b][/url] [b]Austria, 24 - 27 June 2018[/b] [url=http://artofproblemsolving.com/community/c6h1667029p10595005][b]Problem 1.[/b][/url] Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$, $$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$ [i]Proposed by Walther Janous, Austria[/i] [url=http://artofproblemsolving.com/community/c6h1667030p10595011][b]Problem 2.[/b][/url] Let $ABC$ be an acute scalene triangle. Let $D$ and $E$ be points on the sides $AB$ and $AC$, respectively, such that $BD=CE$. Denote by $O_1$ and $O_2$ the circumcentres of the triangles $ABE$ and $ACD$, respectively. Prove that the circumcircles of the triangles $ABC, ADE$, and $AO_1O_2$ have a common point different from $A$. [i]Proposed by Patrik Bak, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667031p10595016][b]Problem 3.[/b][/url] There are $2018$ players sitting around a round table. At the beginning of the game we arbitrarily deal all the cards from a deck of $K$ cards to the players (some players may receive no cards). In each turn we choose a player who draws one card from each of the two neighbors. It is only allowed to choose a player whose each neighbor holds a nonzero number of cards. The game terminates when there is no such player. Determine the largest possible value of $K$ such that, no matter how we deal the cards and how we choose the players, the game always terminates after a finite number of turns. [i]Proposed by Peter Novotný, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667033p10595021][b]Problem 4.[/b][/url] Let $ABC$ be an acute triangle with the perimeter of $2s$. We are given three pairwise disjoint circles with pairwise disjoint interiors with the centers $A, B$, and $C$, respectively. Prove that there exists a circle with the radius of $s$ which contains all the three circles. [i]Proposed by Josef Tkadlec, Czechia[/i] [url=http://artofproblemsolving.com/community/c6h1667034p10595023][b]Problem 5.[/b][/url] In a $2 \times 3$ rectangle there is a polyline of length $36$, which can have self-intersections. Show that there exists a line parallel to two sides of the rectangle, which intersects the other two sides in their interior points and intersects the polyline in fewer than $10$ points. [i]Proposed by Josef Tkadlec, Czechia and Vojtech Bálint, Slovakia[/i] [url=http://artofproblemsolving.com/community/c6h1667036p10595032][b]Problem 6.[/b][/url] We say that a positive integer $n$ is [i]fantastic[/i] if there exist positive rational numbers $a$ and $b$ such that $$ n = a + \frac 1a + b + \frac 1b.$$ [b](a)[/b] Prove that there exist infinitely many prime numbers $p$ such that no multiple of $p$ is fantastic. [b](b)[/b] Prove that there exist infinitely many prime numbers $p$ such that some multiple of $p$ is fantastic. [i]Proposed by Walther Janous, Austria[/i]

Does there exist a function $f: \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0}$ such that \[ f(ab) = f(a)b + af(b) \] for all $a,b \in \mathbb{Z}_{\geq 0}$ and $f(p) > p^p$ for every prime number $p$? [i] (Here, $\mathbb{Z}_{\geq 0}$ denotes the set of non-negative integers.)[/i]

Find all pairs of naturals $a,b$ with $a <b$, such that the sum of the naturals greater than $a$ and less than $ b$ equals $1998$.

Positive real numbers $a$,$b$,$c$ are given such that $abc=1$.Prove that $$2(a+b+c)+\frac{9}{(ab+bc+ca)^2}\geq7.$$

Compute $$\sum^{\infty}_{n=1} \left( \frac{1}{n^2 + 3n} - \frac{1}{n^2 + 3n + 2}\right)$$

Given the equation \[ 4x^3 \plus{} 4x^2y \minus{} 15xy^2 \minus{} 18y^3 \minus{} 12x^2 \plus{} 6xy \plus{} 36y^2 \plus{} 5x \minus{} 10y \equal{} 0,\] find all positive integer solutions.

Given real numbers $a,c,d$ show that there exists at most one function $f:\mathbb{R}\rightarrow\mathbb{R}$ which satisfies: \[f(ax+c)+d\le x\le f(x+d)+c\quad\text{for any}\ x\in\mathbb{R}\]

Given non-zero reals $ a$, $ b$, find all functions $ f: \mathbb{R} \longmapsto \mathbb{R}$, such that for every $ x, y \in \mathbb{R}$, $ y \neq 0$, $ f(2x) \equal{} af(x) \plus{} bx$ and $ \displaystyle f(x)f(y) \equal{} f(xy) \plus{} f \left( \frac {x}{y} \right)$.

Let $a$ be a real number such that $\frac{1}{a}=a-[a]$. Show that $a$ is irrational. Clarification: The brackets indicate the integer part of the number they enclose.

Find all the (x,y) integer ,if $y^2+2y=x^4+20x^3+104x^2+40x+2003$

A [i]palindrome[/i], such as $ 83438$, is a number that remains the same when its digits are reversed. The numbers $ x$ and $ x \plus{} 32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of x? $ \textbf{(A)}\ 20\qquad \textbf{(B)}\ 21\qquad \textbf{(C)}\ 22\qquad \textbf{(D)}\ 23\qquad \textbf{(E)}\ 24$

Find all positive integers $k$ for which there exists a nonlinear function $f:\mathbb{Z} \rightarrow\mathbb{Z}$ such that the equation $$f(a)+f(b)+f(c)=\frac{f(a-b)+f(b-c)+f(c-a)}{k}$$ holds for any integers $a,b,c$ satisfying $a+b+c=0$ (not necessarily distinct). [i]Evan Chen[/i]

Two functions are defined by equations: $f(a) = a^2 + 3a + 2$ and $g(b, c) = b^2 - b + 3c^2 + 3c$. Prove that for any positive integer $a$ there exist positive integers $b$ and $c$ such that $f(a) = g(b, c)$.

What is the number $x =\sqrt{4+\sqrt7}-\sqrt{4-\sqrt7}-\sqrt2$, positive, negative or zero?

Let $N$ be a positive integer. Consider the sequence $a_1, a_2, ..., a_N$ of positive integers, none of which is a multiple of $2^{N+1}$. For $n \ge N +1$, the number $a_n$ is defined as follows: choose $k$ to be the number among $1, 2, ..., n - 1$ for which the remainder obtained when $a_k$ is divided by $2^n$ is the smallest, and define $a_n = 2a_k$ (if there are more than one such $k$, choose the largest such $k$). Prove that there exist $M$ for which $a_n = a_M$ holds for every $n \ge M$.

Let $P(X)=a_{2n+1}X^{2n+1}+a_{2n}X^{2n}+...+a_1X+a_0$ be a polynomial with all positive coefficients. Prove that there exists a permutation $(b_{2n+1},b_{2n},...,b_1,b_0)$ of numbers $(a_{2n+1},a_{2n},...,a_1,a_0)$ such that the polynomial $Q(X)=b_{2n+1}X^{2n+1}+b_{2n}X^{2n}+...+b_1X+b_0$ has exactly one real root.

If $\sum_{k=1}^{N} \frac{2k+1}{(k^2+k)^2}= 0.9999$ then determine the value of $N$.