Let $k,n\ge 1$ be relatively prime integers. All positive integers not greater than $k+n$ are written in some order on the blackboard. We can swap two numbers that differ by $k$ or $n$ as many times as we want. Prove that it is possible to obtain the order $1,2,\dots,k+n-1, k+n$.

For a positive number $n$, we write $d (n)$ for the number of positive divisors of $n$. Determine all positive integers $k$ for which exist positive integers $a$ and $b$ with the property $k = d (a) = d (b) = d (2a + 3b)$.

Let $X$ be the intersection of the diagonals $AC$ and $BD$ of convex quadrilateral $ABCD$. Let $P$ be the intersection of lines $AB$ and $CD$, and let $Q$ be the intersection of lines $PX$ and $AD$. Suppose that $\angle ABX = \angle XCD = 90^o$. Prove that $QP$ is the angle bisector of $\angle BQC$.

Let ( M , g ) be a Riemannian manifold. Extend the metric tensor g to the set of tangents TM with the following specification: if $a,b\in T_v TM \, (v\in T_p M)$, then $$\tilde g_v (a, b): = g_p (\dot {\alpha} (0), \dot {\beta} (0) ) + g_p (D _ {\alpha} X(0) , D_{\beta} Y(0) )$$ where $\alpha, \beta$ are curves in M such that $\alpha(0) = \beta(0) = p$. X and Y are vector fields along $\alpha,\beta$ respectively, with the condition $\dot X (0) = a,\dot Y(0) = b$. $D _{\alpha}$ and $D _{\beta}$ are the operators of the covariant derivative along the corresponding curves according to the Levi-Civita connection. Is the eigenfunction from the Riemannian manifold (M,g) to the Riemannian manifold $(TM, \tilde g)$ harmonic?

In a text editor program, initially there is a footprint symbol (L) that we want to multiply. Unfortunately, our computer has been the victim of a hacker attack, and only two functions are working: Copy and Paste, each costing 1 Dürer dollar to use. Using the Copy function, we can select one or more consecutive symbols from the existing ones, and the computer memorizes their number. When using the Paste function, the computer adds as many new footprint symbols to the sequence as were selected in the last Copy. If no Copy has been done yet, Paste cannot be used. Let $D(n)$ denote the minimum number of Dürer dollars required to obtain exactly $n$ footprint symbols. Prove that for any positive integer $k$, there exists a positive integer $N$ such that \[D(N)=D(N+1)+1=D(N+2)=D(N+3)+1=D(N+4)=\ldots=D(N+2k-1)+1=D(N+2k).\] [i]Based on a problem of the Dürer Competition[/i]

Let $n \geq 2$ be a natural number. Find a way to assign natural numbers to the vertices of a regular $2n$-gon such that the following conditions are satisfied: (1) only digits $1$ and $2$ are used; (2) each number consists of exactly $n$ digits; (3) different numbers are assigned to different vertices; (4) the numbers assigned to two neighboring vertices differ at exactly one digit.

In quadrilateral $ABCD$, $BC=8$, $CD=12$, $AD=10$, and $m\angle A= m\angle B = 60^\circ$. Given that $AB=p + \sqrt{q}$, where $p$ and $q$ are positive integers, find $p+q$.

In the circle $\Omega$ the hexagon $ABCDEF$ is inscribed. It is known that the point $D{}$ divides the arc $BC$ in half, and the triangles $ABC$ and $DEF$ have a common inscribed circle. The line $BC$ intersects segments $DF$ and $DE$ at points $X$ and $Y$ and the line $EF$ intersects segments $AB$ and $AC$ at points $Z$ and $T$ respectively. Prove that the points $X, Y, T$ and $Z$ lie on the same circle. [i]Proposed by D. Brodsky[/i]

Divide a regular hexagon into $24$ identical small equilateral triangles as shown. To each of the $19$ points that are vertices of at least one of the equilateral triangles, a different number is assigned. Find the maximum possible number of small equilateral triangles with their vertices having numbers in ascending order going clockwise. [asy] size(5.5cm); int n=6; pair[] V= sequence(new pair(int i){return dir(360*i/n);}, n); V.cyclic=true; for(int i=0;i<n;++i){ draw(V[i]--V[i+1],black+0.7bp); draw(V[i]--V[i+3],black+0.7bp); draw(midpoint(V[i]--V[i+1])--midpoint(V[i+2]--V[i+3]),black+0.7bp); } [/asy]

What is the positive difference between the sum of the first $5$ positive even integers and the first $5$ positive odd integers? $\textbf{(A) }2\qquad\textbf{(B) }3\qquad\textbf{(C) }4\qquad\textbf{(D) }5\qquad\textbf{(E) }6$

The numbers $1,2,3,\dots,1000$ are written on the board. Patya and Vassya are playing a game. They take turn alternatively erasing a number from the board. Patya begins. If after a turn all numbers (maybe one) on the board be divisible by a natural number greater than $1$ the player who last played loses. If after some number of steps the only remaining number on the board be $1$ then they call it a draw. Determine the result of the game if they both play their best.

The circle with radius $4$ passing through $A$ and $B$ is tangent to the circle with radius $3$ passing through $A$ and $C$, where $|AB|=2$. If the line $BC$ is tangent to the second circle, what is $|BC|$? $ \textbf{(A)}\ 7 \qquad\textbf{(B)}\ 2 + \dfrac{\sqrt{43}}2 \qquad\textbf{(C)}\ \dfrac 52 \qquad\textbf{(D)}\ 4 + \sqrt 9 \qquad\textbf{(E)}\ \sqrt 7 $

One day when Wendy is riding her horse Vanessa, they get to a field where some tourists are following Martin (the tour guide) on some horses. Martin and some of the workers at the stables are each leading extra horses, so there are more horses than people. Martin's dog Berry runs around near the trail as well. Wendy counts a total of $28$ heads belonging to the people, horses, and dog. She counts a total of $92$ legs belonging to everyone, and notes that nobody is missing any legs. Upon returning home Wendy gives Alexis a little problem solving practice, "I saw $28$ heads and $92$ legs belonging to people, horses, and dogs. Assuming two legs per person and four for the other animals, how many people did I see?" Alexis scribbles out some algebra and answers correctly. What is her answer?

In a company of $6$ sousliks each souslik has $4$ friends. Is it always possible to divide this company into two groups of $3$ sousliks such that in both groups all sousliks are friends?

Find all integers $n$, $n \ge 1$, such that $n \cdot 2^{n+1}+1$ is a perfect square.

A necklace contains 2024 pearls, each one of them having one of the following colours: black, green and yellow. Each moment, we will switch each one of all pearls simultaneously to a new one following the following rules: i) If its two neighbours are of the same colour, then it'll be switched to that same colour. ii) If its two neighbours are of different colours, then it'll be switched to the third colour. a) Does there exist any necklace that can be transformed into a necklace that consists of only yellow pearls if initially half of the pearls are black and the other half is green? b) Does there exist a necklace that can be transformed into a necklace that consists of only yellow pearls if initially 998 pearls are black and the rest 1026 pearls are green?

$11$ theatrical groups participated in a festival. Each day, some of the groups were scheduled to perform while the remaining groups joined the general audience. At the conclusion of the festival, each group had seen, during its days off, at least $1$ performance of every other group. At least how many days did the festival last?

Let triangle $\triangle{ABC}$ have a right angle at $C$, and let $M$ be the midpoint of the hypotenuse $AB$. Choose a point $D$ on line $BC$ so that angle $\angle{CDM}$ measures $30$ degrees. Prove that the segments $AC$ and $MD$ have equal lengths.

A finite collection of squares has total area $4$. Show that they can be arranged to cover a square of side $1$.

[b]O[/b]n a blackboard a stranger writes the values of $s_7(n)^2$ for $n=0,1,...,7^{20}-1$, where $s_7(n)$ denotes the sum of digits of $n$ in base $7$. Compute the average value of all the numbers on the board.

Let $\Sigma$ be a finite set. For $x,y \in \Sigma^{\ast}$, define \[x\preceq y\] if $x$ is a sub-string ([b]not necessarily contiguous[/b]) of $y$. For example, $ac \preceq abc$. We call a set $S\subseteq \Sigma^{\ast}$ [b][u]good[/u][/b] if $\forall x,y \in \Sigma^{\ast}$, $$ x\preceq y, \; y \in S \; \; \; \Rightarrow \; x\in S .$$ Prove or disprove: Every good set is regular.

Each pair of opposite sides of a convex hexagon has the following property: the distance between their midpoints is equal to $\dfrac{\sqrt{3}}{2}$ times the sum of their lengths. Prove that all the angles of the hexagon are equal.

The arithmetic, geometric and harmonic mean of two distinct positive integers are different numbers. Find the smallest possible value for the arithmetic mean.

The Incredible Hulk can double the distance he jumps with each succeeding jump. If his first jump is 1 meter, the second jump is 2 meters, the third jump is 4 meters, and so on, then on which jump will he first be able to jump more than 1 kilometer? $\textbf{(A)}\ 9^\text{th} \qquad \textbf{(B)}\ 10^\text{th} \qquad \textbf{(C)}\ 11^\text{th} \qquad \textbf{(D)}\ 12^\text{th} \qquad \textbf{(E)}\ 13^\text{th}$

Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\tfrac12$ is $\tfrac{a-b\pi}c$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. What is $a+b+c$? $\textbf{(A) }59\qquad\textbf{(B) }60\qquad\textbf{(C) }61\qquad\textbf{(D) }62\qquad\textbf{(E) }63$