The distances from a certain point inside a regular hexagon to three of its consecutive vertices are equal to $1, 1$ and $2$, respectively. Determine the length of this hexagon's side. (Mikhail Evdokimov)

Consider a fair coin and a fair 6-sided die. The die begins with the number 1 face up. A [i]step[/i] starts with a toss of the coin: if the coin comes out heads, we roll the die; otherwise (if the coin comes out tails), we do nothing else in this step. After 5 such steps, what is the probability that the number 1 is face up on the die?

The number $2$ is written on the board. Ana and Bruno play alternately. Ana begins. Each one, in their turn, replaces the number written by the one obtained by applying exactly one of these operations: multiply the number by $2$, multiply the number by $3$ or add $1$ to the number. The first player to get a number greater than or equal to $2011$ wins. Find which of the two players has a winning strategy and describe it.

A convex$N-$gon with equal sides is located inside a circle. Each side is extended in both directions up to the intersection with the circle so that it contains two new segments outside the polygon. Prove that one can paint some of these new $2N$ segments in red and the rest in blue so that the sum of lengths of all the red segments would be the same as for the blue ones. [i]($8$ points)[/i]

Let $p\ge 17$ be a prime. Prove that $t=3$ is the largest positive integer which satisfies the following condition: For any integers $a,b,c,d$ such that $abc$ is not divisible by $p$ and $(a+b+c)$ is divisible by $p$, there exists integers $x,y,z$ belonging to the set $\{0,1,2,\ldots , \left\lfloor \frac{p}{t} \right\rfloor - 1\}$ such that $ax+by+cz+d$ is divisible by $p$.

There are eight different symbols designed on $n\geq 2$ different T-shirts. Each shirt contains at least one symbol, and no two shirts contain all the same symbols. Suppose that for any $k$ symbols $(1\leq k\leq 7)$ the number of shirts containing at least one of the $k$ symbols is even. Determine the value of $n$.

Let $p$ be a prime and $Q(x)$ be a polynomial with integer coefficients such that $Q(0) = 0, \ Q(1) = 1$ and the remainder of $Q(n)$ is either $0$ or $1$ when divided by $p$, for every $n \in \mathbb{N}$. Prove that $Q(x)$ is of degree at least $p - 1$.

In $\triangle ABC$, consider points $D$ and $E$ on $AC$ and $AB$, respectively, and assume that they do not coincide with any of the vertices $A$, $B$, $C$. If the segments $BD$ and $CE$ intersect at $F$, consider areas $w$, $x$, $y$, $z$ of the quadrilateral $AEFD$ and the triangles $BEF$, $BFC$, $CDF$, respectively. [list=a] [*] Prove that $y^2 > xz$. [*] Determine $w$ in terms of $x$, $y$, $z$. [/list] [asy] import graph; size(10cm); real labelscalefactor = 0.5; /* changes label-to-point distance */ pen dps = linewidth(0.7) + fontsize(12); defaultpen(dps); /* default pen style */ pen dotstyle = black; /* point style */ real xmin = -2.8465032978885407, xmax = 9.445649196374966, ymin = -1.7618066305534972, ymax = 4.389732795464592; /* image dimensions */ draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282)--cycle, linewidth(0.5)); /* draw figures */ draw((3.8295013012181283,2.816337276198864)--(-0.7368327629589799,-0.5920813291311117), linewidth(0.5)); draw((-0.7368327629589799,-0.5920813291311117)--(5.672613975760373,-0.636902634996282), linewidth(0.5)); draw((5.672613975760373,-0.636902634996282)--(3.8295013012181283,2.816337276198864), linewidth(0.5)); draw((-0.7368327629589799,-0.5920813291311117)--(4.569287648059735,1.430279997142299), linewidth(0.5)); draw((5.672613975760373,-0.636902634996282)--(1.8844000180622977,1.3644681598392678), linewidth(0.5)); label("$y$",(2.74779188172294,0.23771684184669772),SE*labelscalefactor); label("$w$",(3.2941097703568736,1.8657441499758196),SE*labelscalefactor); label("$x$",(1.6660824622277512,1.0025618859342047),SE*labelscalefactor); label("$z$",(4.288408327670633,0.8168138037986672),SE*labelscalefactor); /* dots and labels */ dot((3.8295013012181283,2.816337276198864),dotstyle); label("$A$", (3.8732067323088435,2.925600853925651), NE * labelscalefactor); dot((-0.7368327629589799,-0.5920813291311117),dotstyle); label("$B$", (-1.1,-0.7565817154670613), NE * labelscalefactor); dot((5.672613975760373,-0.636902634996282),dotstyle); label("$C$", (5.763466626982254,-0.7784344310124186), NE * labelscalefactor); dot((4.569287648059735,1.430279997142299),dotstyle); label("$D$", (4.692683565259744,1.5051743434774234), NE * labelscalefactor); dot((1.8844000180622977,1.3644681598392678),dotstyle); label("$E$", (1.775346039954538,1.4942479857047448), NE * labelscalefactor); dot((2.937230516274804,0.8082418657164665),linewidth(4.pt) + dotstyle); label("$F$", (2.889834532767763,0.954), NE * labelscalefactor); clip((xmin,ymin)--(xmin,ymax)--(xmax,ymax)--(xmax,ymin)--cycle); [/asy]

Let \(a,b\) be positive integers such that \(a+1\), \(b+1\), and \(ab\) are perfect squares. Prove that $\gcd(a,b)+1$ is also a perfect square.

Joey and Chloe and their daughter Zoe all have the same birthday. Joey is 1 year older than Chloe, and Zoe is exactly 1 year old today. Today is the first of the 9 birthdays on which Chloe's age will be an integral multiple of Zoe's age. What will be the sum of the two digits of Joey's age the next time his age is a multiple of Zoe's age? $\textbf{(A) } 7 \qquad \textbf{(B) } 8 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 10 \qquad \textbf{(E) } 11 $

Let $m > n$ be positive integers. In the host country of the International Olympiad in Informatics this year, there are coins of $1$, $2$, $\ldots$, $n$ [i]alexandrias[/i], [i]lira[/i] banknotes, each worth $m$ alexandrias, and [i]pharaoh[/i] banknotes, each worth $m+n$ alexandrias. Let $A$ be a positive integer. Boris wants to exchange the amount $A$ using coins and lira, using no more than $m-1$ coins, while Vesko wants to exchange the amount $A$ using coins and pharaohs, using no more than $m$ coins. Prove that regardless of the value of $A$, the number of ways for each of them to fulfill their desire is the same.

Prove that for any real $r>0$, one can cover the circumference of a $1\times r$ rectangle with non-intersecting disks of unit radius.

Let $\omega$ be a circle with center $O$, and let $l$ be a line not intersecting $\omega$. $E$ is a point on $l$ such that $OE$ is perpendicular with $l$. Let $M$ be an arbitrary point on $M$ different from $E$. Let $A$ and $B$ be distinct points on the circle $\omega$ such that $MA$ and $MB$ are tangents to $\omega$. Let $C$ and $D$ be the foot of perpendiculars from $E$ to $MA$ and $MB$ respectively. Let $F$ be the intersection of $CD$ and $OE$. As $M$ moves, determine the locus of $F$.

For two non-zero real numbers $a, b$ , the equation, $a(x-a)^2 + b(x-b)^2 = 0$ has a unique solution. Prove that $a=\pm b$.

$2024$ ones and $2024$ twos are arranged in a circle in some order. Is it always possible to divide the circle into [b]a)[/b] two (contiguous) parts with equal sums? [b]b)[/b] three (contiguous) parts with equal sums? [i]Proposed by Fedir Yudin[/i]

Construct a function $ f:[0,1]\longrightarrow\mathbb{R} $ that is primitivable, bounded, and doesn't touch its bounds. [i]Dorian Popa[/i]

Triangle $ABC$ has circumcircle $\Omega$ and circumcenter $O$. A circle $\Gamma$ with center $A$ intersects the segment $BC$ at points $D$ and $E$, such that $B$, $D$, $E$, and $C$ are all different and lie on line $BC$ in this order. Let $F$ and $G$ be the points of intersection of $\Gamma$ and $\Omega$, such that $A$, $F$, $B$, $C$, and $G$ lie on $\Omega$ in this order. Let $K$ be the second point of intersection of the circumcircle of triangle $BDF$ and the segment $AB$. Let $L$ be the second point of intersection of the circumcircle of triangle $CGE$ and the segment $CA$. Suppose that the lines $FK$ and $GL$ are different and intersect at the point $X$. Prove that $X$ lies on the line $AO$. [i]Proposed by Greece[/i]

Let $300$ students participate to the Olympiad. Between each $3$ participants there is a pair that are not friends. Hamza enumerates participants in some order and denotes by $x_i$ the number of friends of $i$-th participant. It occurs that $\{x_1,x_2,...,x_{299},x_{300}\} = \{1, 2,..., N - 1,N\}$ Find the biggest possible value for $N$.

On a $999\times 999$ board a [i]limp rook[/i] can move in the following way: From any square it can move to any of its adjacent squares, i.e. a square having a common side with it, and every move must be a turn, i.e. the directions of any two consecutive moves must be perpendicular. A [i]non-intersecting route[/i] of the limp rook consists of a sequence of pairwise different squares that the limp rook can visit in that order by an admissible sequence of moves. Such a non-intersecting route is called [i]cyclic[/i], if the limp rook can, after reaching the last square of the route, move directly to the first square of the route and start over. How many squares does the longest possible cyclic, non-intersecting route of a limp rook visit? [i]Proposed by Nikolay Beluhov, Bulgaria[/i]

In the space $n \geq 3$ points are given. Every pair of points determines some distance. Suppose all distances are different. Connect every point with the nearest point. Prove that it is impossible to obtain (closed) polygonal line in such a way.

Let $AB$ be a segment of length $26$, and let points $C$ and $D$ be located on $AB$ such that $AC=1$ and $AD=8$. Let $E$ and $F$ be points on one of the semicircles with diameter $AB$ for which $EC$ and $FD$ are perpendicular to $AB$. Find $EF$. $\textbf{(A) }5\qquad\textbf{(B) }5\sqrt2\qquad\textbf{(C) }7\qquad\textbf{(D) }7\sqrt2\qquad\textbf{(E) }12$

Let $ABC$ be an equilateral triangle with each side of length 1. Let $X$ be a point chosen uniformly at random on side $\overline{AB}$. Let $Y$ be a point chosen uniformly at random on side $\overline{AC}$. (Points $X$ and $Y$ are chosen independently.) Let $p$ be the probability that the distance $XY$ is at most $\dfrac{1}{\sqrt[4]{3}}\,$. What is the value of $900p$, rounded to the nearest integer?

One afternoon at the park there were twice as many dogs as there were people, and there were twice as many people as there were snakes. The sum of the number of eyes plus the number of legs on all of these dogs, people, and snakes was $510$. Find the number of dogs that were at the park.

Consider the function $ f(x) \equal{} \frac {2x^3 \minus{} 3}{3x^2 \minus{} 1}$. $ 1.$ Prove that there is a continuous function $ g(x)$ on $ \mathbb{R}$ satisfying $ f(g(x)) \equal{} x$ and $ g(x) > x$ for all real $ x$. $ 2.$ Show that there exists a real number $ a > 1$ such that the sequence $ \{a_n\}$, $ n \equal{} 1, 2, \ldots$, defined as follows $ a_0 \equal{} a$, $ a_{n \plus{} 1} \equal{} f(a_n)$, $ \forall n\in\mathbb{N}$ is periodic with the smallest period $ 1995$.

We say that a set $A \subset \mathbb Z$ is irregular if, for any different elements $x, y \in A$, there is no element of the form $x + k(y -x)$ different from $x$ and $y$ (where $k$ is an integer). Is there an infinite irregular set?