Let $n,k$ be given natural numbers. Find the smallest possible cardinality of a set $A$ with the following property: There exist subsets $A_1,A_2,\ldots,A_n$ of $A$ such that the union of any $k$ of them is $A$, but the union of any $k-1$ of them is never $A$.

We are given a cyclic quadrilateral $ABCD$ with a point $E$ on the diagonal $AC$ such that $AD=AE$ and $CB=CE$. Let $M$ be the center of the circumcircle $k$ of the triangle $BDE$. The circle $k$ intersects the line $AC$ in the points $E$ and $F$. Prove that the lines $FM$, $AD$ and $BC$ meet at one point. [i](4th Middle European Mathematical Olympiad, Individual Competition, Problem 3)[/i]

A [i]string of length $n$[/i] is a sequence of $n$ characters from a specified set. For example, $BCAAB$ is a string of length 5 with characters from the set $\{A,B,C\}$. A [i]substring[/i] of a given string is a string of characters that occur consecutively and in order in the given string. For example, the string $CA$ is a substring of $BCAAB$ but $BA$ is not a substring of $BCAAB$. [list=a][*]List all strings of length 4 with characters from the set $\{A,B,C\}$ in which both the strings $AB$ and $BA$ occur as substrings. (For example, the string $ABAC$ should appear in your list.) [*]Determine the number of strings of length 7 with characters from the set $\{A,B,C\}$ in which $CC$ occures as a substring. [*]Let $f(n)$ be the number of strings of length $n$ with characters from the set $\{A,B,C\}$ such that [list][*]$CC$ occurs as a substring, and[*]if either $AB$ or $BA$ occurs as a substring then there is an occurrence of the substring $CC$ to its left.[/list] (for example, when $n\;=\;6$, the strings $CCAABC$ and $ACCBBB$ and $CCABCC$ satisfy the requirements, but the strings $BACCAB$ and $ACBBAB$ and $ACBCAC$ do not). Prove that $f(2097)$ is a multiple of $97$.[/list]

Let $P(x)$ be given a polynomial of degree 4, having 4 positive roots. Prove that the equation \[(1-4 \cdot x) \cdot \frac{P(x)}{x^2} + (x^2 + 4 \cdot x - 1) \cdot \frac{P'(x)}{x^2} - P''(x) = 0\] has also 4 positive roots.

Given a right triangle $ABC$ ($\angle A <45^o$,$ \angle C = 90^o$), on the sides $AC$ and $AB$ which are selected points $D,E$ respectively, such that $BD = AD$ and $CB = CE$. Let the segments $BD$ and $CE$ intersect at the point $O$. Prove that $\angle DOE = 90^o$.

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

Let $P$ be a point in the interior of quadrilateral $ABCD$ such that the circumcircles of triangles $PDA, PAB,$ and $PBC$ are pairwise distinct but congruent. Let the lines $AD$ and $BC$ meet at $X$. If $O$ is the circumcenter of triangle $XCD$, prove that $OP \perp AB$.

Let $f:\mathbb R\to\mathbb R$ be an infinitely differentiable function. Assume that for every $x\in\mathbb R$ there is an $n\in\mathbb N$ (depending on $x$) such that $$f^{(n)}(x)=0.$$Prove that $f$ is a polynomial.

Gary purchased a large beverage, but drank only $m/n$ of this beverage, where $m$ and $n$ are relatively prime positive integers. If Gary had purchased only half as much and drunk twice as much, he would have wasted only $\frac{2}{9}$ as much beverage. Find $m+n$.

Let $a,b,c$ be integer numbers such that $(a+b+c) \mid (a^{2}+b^{2}+c^{2})$. Show that there exist infinitely many positive integers $n$ such that $(a+b+c) \mid (a^{n}+b^{n}+c^{n})$. [i]Laurentiu Panaitopol[/i]

The squares in a $7\times7$ grid are colored one of two colors: green and purple. The coloring has the property that no green square is directly above or to the right of a purple square. Find the total number of ways this can be done.

Let $a, b, c, d$ be distinct non-zero real numbers satisfying the following two conditions: $ac = bd$ and $\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a}= 4$. Determine the largest possible value of the expression $\frac{a}{c}+\frac{c}{a}+\frac{b}{d}+\frac{d}{b}$.

Let $n$ be a positive integer. For all $i, j \in \{1,2,...,n\}$ define $a_{j,i} = 1$ if $j = i$ and $a_{j,i} = 0$ otherwise. Also, for $i = n+1,...,2n$ and $j = 1,...,n$ define $a_{j,i} = -\frac{1}{n}$. Prove that for any permutation $p$ of the set $\{1,2,...,2n\}$ the following inequality holds: $\sum_{j=1}^{n}\left|\sum_{k=1}^{n} a_{j,p}(k)\right| \ge \frac{n}{2}$

Let $$x_0 = 5\,\, ,\, \,\,x_{n+1} = x_n +\frac{1}{x_n}.$$ Prove that $45 < x_{1000} < 45.1$.

The sides and diagonals of a regular $n$-gon $R$ are colored in $k$ colors so that: (i) For each color $a$ and any two vertices $A$,$B$ of $R$ , the segment $AB$ is of color $a$ or there is a vertex $C$ such that $AC$ and $BC$ are of color $a$. (ii) The sides of any triangle with vertices at vertices of $R$ are colored in at most two colors. Prove that $k\leq 2$.

Let $ABC$ be an acute triangle with circumcircle $\Omega$. Let $B_0$ be the midpoint of $AC$ and let $C_0$ be the midpoint of $AB$. Let $D$ be the foot of the altitude from $A$ and let $G$ be the centroid of the triangle $ABC$. Let $\omega$ be a circle through $B_0$ and $C_0$ that is tangent to the circle $\Omega$ at a point $X\not= A$. Prove that the points $D,G$ and $X$ are collinear. [i]Proposed by Ismail Isaev and Mikhail Isaev, Russia[/i]

Azambuja writes a rational number $q$ on a blackboard. One operation is to delete $q$ and replace it by $q+1$; or by $q-1$; or by $\frac{q-1}{2q-1}$ if $q \neq \frac{1}{2}$. The final goal of Azambuja is to write the number $\frac{1}{2018}$ after performing a finite number of operations. [b]a)[/b] Show that if the initial number written is $0$, then Azambuja cannot reach his goal. [b]b)[/b] Find all initial numbers for which Azambuja can achieve his goal.

Find the smallest number of the form $1...1$ in its decimal expression which is divisible by $\underbrace{\hbox{3...3}}_{\hbox{100}}$,.

Let $x_1, x_2, \ldots$ be a sequence of $0,1$, such that it satisfies the following three conditions: 1) $x_2=x_{100}=1$, $x_i=0$ for $1 \leq i \leq 100$ and $i \neq 2,100$; 2) $x_{2n-1}=x_{n-50}+1, x_{2n}=x_{n-50}$ for $51 \leq n \leq 100$; 3) $x_{2n}=x_{n-50}, x_{2n-1}=x_{n-50}+x_{n-100}$ for $n>100$. Show that the sequence is periodic.

Consider all ordered pairs $(m, n)$ of positive integers satisfying $59 m - 68 n = mn$. Find the sum of all the possible values of $n$ in these ordered pairs.

One day while the Kubik family attends one of Michael's baseball games, Tony gets bored and walks to the creek a few yards behind the baseball field. One of Tony's classmates Mitchell sees Tony and goes to join him. While playing around the creek, the two boys find an ordinary six-sided die buried in sediment. Mitchell washes it off in the water and challenges Tony to a contest. Each of the boys rolls the die exactly once. Mitchell's roll is $3$ higher than Tony's. "Let's play once more," says Tony. Let $a/b$ be the probability that the difference between the outcomes of the two dice is again exactly $3$ (regardless of which of the boys rolls higher), where $a$ and $b$ are relatively prime positive integers. Find $a+b$.

$2015$ points are given in a plane such that from any five points we can choose two points with distance less than $1$ unit. Prove that $504$ of the given points lie on a unit disc.

A rook is randomly placed on an otherwise empty $8 \times 8$ chessboard. Owen makes moves with the rook by randomly choosing $1$ of the $14$ possible moves. Find the expected value of the number of moves it takes Owen to move the rook to the top left square. Note that a rook can move any number of squares either in the horizontal or vertical direction each move.

Let $n\ge 3$ be a positive integer. Find the largest real number $t_n$ as a function of $n$ such that the inequality \[\max\left(|a_1+a_2|, |a_2+a_3|, \dots ,|a_{n-1}+a_{n}| , |a_n+a_1|\right) \ge t_n \cdot \max(|a_1|,|a_2|, \dots ,|a_n|)\] holds for all real numbers $a_1, a_2, \dots , a_n$ . [i]Proposed by Rohan Goyal and Rijul Saini[/i]

Determine all real-valued functions $f$ on the set of real numbers satisfying \[2f(x)=f(x+y)+f(x+2y)\] for all real numbers $x$ and all non-negative real numbers $y$.