Prove that if the length and breadth of a rectangle are both odd integers, then there does not exist a point $P$ inside the rectangle such that each of the distances from $P$ to the 4 corners of the rectangle is an integer.

In the figure below, how many ways are there to select two squares which do not share an edge? [asy] size(3cm); for (int t = -2; t <= 2; t=t+1) { draw( shift((t,0))*unitsquare ) ; if (t!=0) draw( shift((0,t))*unitsquare ); } [/asy] [i]Proposed by Evan Chen[/i]

Two players $A$ and $B$ take turns playing the following game: There is a pile of $2003$ stones. In his first turn, $A$ selects a divisor of $2003$ and removes this number of stones from the pile. $B$ then chooses a divisor of the number of remaining stones, and removes that number of stones from the new pile, and so on. The player who has to remove the last stone loses. Show that one of the two players has a winning strategy and describe the strategy.

In an $n \times n$ square grid, $n$ squares are marked so that every rectangle composed of exactly $n$ grid squares contains at least one marked square. Determine all possible values of $n$.

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.