Let $ I$ be an ideal of the ring $\mathbb{Z}\left[x\right]$ of all polynomials with integer coefficients such that a) the elements of $ I$ do not have a common divisor of degree greater than $ 0$, and b) $ I$ contains of a polynomial with constant term $ 1$. Prove that $ I$ contains the polynomial $ 1 + x + x^2 + ... + x^{r-1}$ for some natural number $ r$. [i]Gy. Szekeres[/i]

Let $P$ be a point inside a triangle $ABC$ with $\angle C = 90^o$ such that $AP = AC$, and let $M$ be the midpoint of $AB$ and $CH$ be the altitude. Prove that $PM$ bisects $\angle BPH$ if and only if $\angle A = 60^o$.

For $a>0$, denote by $S(a)$ the area of the part bounded by the parabolas $y=\frac 12x^2-3a$ and $y=-\frac 12x^2+2ax-a^3-a^2$. Find the maximum area of $S(a)$.

The sequence $\{a_{n}\}_{n\geq 0}$ is defined by $a_{0}=20,a_{1}=100,a_{n+2}=4a_{n+1}+5a_{n}+20(n=0,1,2,...)$. Find the smallest positive integer $h$ satisfying $1998|a_{n+h}-a_{n}\forall n=0,1,2,...$

Given a right-angled triangle $ABC$ and two perpendicular lines $x$ and $y$ passing through the vertex $A$ of its right angle. For an arbitrary point $X$ on $x$ define $y_B$ and $y_C$ as the reflections of $y$ about $XB$ and $ XC $ respectively. Let $Y$ be the common point of $y_b$ and $y_c$. Find the locus of $Y$ (when $y_b$ and $y_c$ do not coincide).

Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy \[f(x^2 + y) + f(f(x) - y) = 2f(f(x)) + 2y^2\quad\text{ for all }x, y \in \mathbb{R}.\]

Point $ E$ is selected on side $ AB$ of triangle $ ABC$ in such a way that $ AE: EB \equal{} 1: 3$ and point $ D$ is selected on side $ BC$ such that $ CD: DB \equal{} 1: 2$. The point of intersection of $ AD$ and $ CE$ is $ F$. Then $ \frac {EF}{FC} \plus{} \frac {AF}{FD}$ is: $ \textbf{(A)}\ \frac {4}{5} \qquad \textbf{(B)}\ \frac {5}{4} \qquad \textbf{(C)}\ \frac {3}{2} \qquad \textbf{(D)}\ 2 \qquad \textbf{(E)}\ \frac {5}{2}$

The sequence $\{x_n\}_n$ is defined as follows: $x_1 = 0$, and for all $n \ge 1$ $$(n + 1)^3 x_{n+1} = 2n^2 (2n + 1)x_n + 2(3n + 1).$$ Prove that $\{x_n\}_n$ contains infinitely many integer numbers.

Find all sequences $x_{1},x_{2},...,x_{n}$ of distinct positive integers such that $\frac{1}{2}=\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}$.

It is proposed to partition a set of positive integers into two disjoint subsets $ A$ and $ B$ subject to the conditions [b]i.)[/b] 1 is in $ A$ [b]ii.)[/b] no two distinct members of $ A$ have a sum of the form $ 2^k \plus{} 2, k \equal{} 0,1,2, \ldots;$ and [b]iii.)[/b] no two distinct members of B have a sum of that form. Show that this partitioning can be carried out in unique manner and determine the subsets to which 1987, 1988 and 1989 belong.

Nine numbers $a, b, c, \dots$ are arranged around a circle. All numbers of the form $a+b^c, \dots$ are prime. What is the largest possible number of different numbers among $a, b, c, \dots$?

[b]10.[/b] Let $X_1, X_2, \dots $ be independent random variables with the same distribution $P(X_i = 1) = P(X_i = -1)=\frac{1}{2}\qquad (i= 1, 2, \dots )$ Define $S_0=0, Sn=X_1 +X_2+\dots +X_n \qquad (n=1, 2, \dots$ ), $\xi (x,n) = \left | \{k : 0 \leq k \leq n, S_k= x \} \right |\qquad (x=0, \pm 1, \pm 2, \dots $), and $\alpha(n)= \left | \{ x: \xi(x,n)=a \} \right |\qquad (n=0,1,\dots$). Prove that $P(\lim \inf \alpha(n)=0) =1$ and that there is a number $0<c<\infty$ such that $P(\lim \inf \alpha(n)/\log n=c) =1$ ([b]P.24[/b]) [P. Révész]

Ana, Beto, Carlos, Diana, Elena and Fabian are in a circle, located in that order. Ana, Beto, Carlos, Diana, Elena and Fabian each have a piece of paper, where are written the real numbers $a,b,c,d,e,f$ respectively. At the end of each minute, all the people simultaneously replace the number on their paper by the sum of three numbers; the number that was at the beginning of the minute on his paper and on the papers of his two neighbors. At the end of the minute $2022, 2022$ replacements have been made and each person have in his paper it´s initial number. Find all the posible values of $abc+def$. $\textbf{Note:}$ [i]If at the beginning of the minute $N$ Ana, Beto, Carlos have the numbers $x,y,z$, respectively, then at the end of the minute $N$, Beto is going to have the number $x+y+z$[/i].

Compute the number of $7$-digit positive integers that start $\textit{or}$ end (or both) with a digit that is a (nonzero) composite number.

Given any set $S$ of positive integers, show that at least one of the following two assertions holds: (1) There exist distinct finite subsets $F$ and $G$ of $S$ such that $\sum_{x\in F}1/x=\sum_{x\in G}1/x$; (2) There exists a positive rational number $r<1$ such that $\sum_{x\in F}1/x\neq r$ for all finite subsets $F$ of $S$.

Prove that if $a,b,c$ are positive numbers with $abc=1$, then \[\frac{a}{b} +\frac{b}{c} + \frac{c}{a} \ge a + b + c. \]

Let $n$ be a positive integer greater than or equal to $6$, and suppose that $a_1, a_2, ...,a_n$ are real numbers such that the sums $a_i + a_j$ for $1 \le i<j\le n$, taken in some order, form consecutive terms of an arithmetic progression $A$, $A + d$, $...$ ,$A + (k-1)d$, where $k = n(n-1)/2$. What are the possible values of $d$?

Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ that satisfy the following conditions: a. $x+f(y+f(x))=y+f(x+f(y)) \quad \forall x,y \in \mathbb{R}$ b. The set $I=\left\{\frac{f(x)-f(y)}{x-y}\mid x,y\in \mathbb{R},x\neq y \right\}$ is an interval. [i]Proposed by Navid Safaei[/i]

Show that if $t_1 , t_2, t_3, t_4, t_5$ are real numbers, then $$\sum_{j=1}^{5} (1-t_j )\exp \left( \sum_{k=1}^{j} t_k \right) \leq e^{e^{e^{e}}}.$$

Natural number $N$ is given. Let $p_N$ - biggest prime, that $ \leq N$. On every move we replace $N$ by $N-p_N$. We repeat this until we get $0$ or $1$. Prove that exists such number $N$, that we need exactly $1000$ turns to make $0$

A list with $2007$ positive integers is written on a board, such that the arithmetic mean of all the numbers is $12$. Then, seven consecutive numbers are erased from the board. The arithmetic mean of the remaining numbers is $11.915$. The seven erased numbers have this property: the sixth number is half of the seventh, the fifth number is half of the sixth, and so on. Determine the $7$ erased numbers.

Find all values of $a \in \mathbb{R}$ for which the polynomial \begin{align*} f(x)=x^4-2x^3 + \left(5-6a^2 \right)x^2 + \left(2a^2-4 \right)x + \left(a^2 -2 \right)^2 \end{align*} has exactly three real roots.

Let $\omega$ be the circumcircle of a triangle $ABC$ ($AB>AC$), $E$ be the midpoint of the arc $AC$ which does not contain point $B,$ аnd $F$ the midpoint of the arc $AB$ which does not contain point $C.$ Lines $AF$ and $BE$ meet at point $P,$ line $CF$ and $AE$ meet at point $R,$ and the tangent to $\omega$ at point $A$ meets line $BC$ at point $Q.$ Prove that points $P,Q,R$ are collinear. (M. Kurskiy)

Marla has a large white cube that has an edge of 10 feet. She also has enough green paint to cover 300 square feet. Marla uses all the paint to create a white square centered on each face, surrounded by a green border. What is the area of one of the white squares, in square feet? $\textbf{(A)}\hspace{.05in}5\sqrt2 \qquad \textbf{(B)}\hspace{.05in}10 \qquad \textbf{(C)}\hspace{.05in}10\sqrt2 \qquad \textbf{(D)}\hspace{.05in}50 \qquad \textbf{(E)}\hspace{.05in}50\sqrt2 $

Ezzie is walking around the perimeter of a regular hexagon. Each vertex of the hexagon has an instruction telling him to move clockwise or counterclockwise around the hexagon. However, when he leaves a vertex the instruction switches from clockwise to counterclockwise on that vertex, or vice versa. We say that a configuration $C$ of Ezzie’s position and the instructions on the vertices is [I]irrepeatable[/I] if, when starting from configuration $C,$ configuration $C$ only appears finitely many more times. Find the number of irrepeatable configurations.