Two circles $\omega_1,\omega_2$ intersect each other at points $A,B$. Let $PQ$ be a common tangent line of these two circles with $P \in \omega_1$ and $Q \in \omega_2$. An arbitrary point $X$ lies on $\omega_1$. Line $AX$ intersects $ \omega_2$ for the second time at $Y$ . Point $Y'\ne Y$ lies on $\omega_2$ such that $QY = QY'$. Line $Y'B$ intersects $ \omega_1$ for the second time at $X'$. Prove that $PX = PX'$. Proposed by Morteza Saghafian

For all $n>1$, let $f(n)$ be the biggest divisor of $n$ except itself. Does there exists a positive integer $k$ such that the equality $n-f(n)=k$ has exactly $2023$ solutions?

Let $ABCD$ be a cyclic quadrilateral with $\angle BAD < \angle ADC$. Let $M$ be the midpoint of the arc $CD$ not containing $A$. Suppose there is a point $P$ inside $ABCD$ such that $\angle ADB = \angle CPD$ and $\angle ADP = \angle PCB$. Prove that lines $AD, PM$, and $BC$ are concurrent.

Ranu starts with one standard die on a table. At each step, she rolls all the dice on the table: if all of them show a 6 on top, then she places one more die on the table; otherwise, she does nothing more on this step. After 2013 such steps, let $D$ be the number of dice on the table. What is the expected value (average value) of $6^D$?

Let $a, b, c, d, e, f$ are nonzero digits such that the natural numbers $\overline{abc}, \overline{def}$ and $\overline{abcdef }$ are squares. a) Prove that $\overline{abcdef}$ can be represented in two different ways as a sum of three squares of natural numbers. b) Give an example of such a number.

A group of mathematicians is attending a conference. We say that a mathematician is $k-$[i]content[/i] if he is in a room with at least $k$ people he admires or if he is admired by at least $k$ other people in the room. It is known that when all participants are in a same room then they are all at least $3k + 1$-content. Prove that you can assign everyone into one of $2$ rooms in a way that everyone is at least $k$-content in his room and neither room is empty. [i]Admiration is not necessarily mutual and no one admires himself.[/i] [i]Matija Bucić[/i]

Let $ S$ be the smallest subset of the integers with the property that $ 0\in S$ and for any $ x\in S$, we have $ 3x\in S$ and $ 3x \plus{} 1\in S$. Determine the number of non-negative integers in $ S$ less than $ 2008$.

Find $x$ such that trigonometric \[\frac{\sin 3x \cos (60^\circ -4x)+1}{\sin(60^\circ - 7x) - \cos(30^\circ + x) + m}=0\] where $m$ is a fixed real number.

Do there exist a convex quadrilateral and a point $P$ inside it such that the sum of distances from $P$ to the vertices of the quadrilateral is greater than its perimeter? (A.Akopyan)

Given a cube with a side of $4$. Is it possible to completely cover $3$ of its faces, which have a common vertex, with sixteen rectangular paper strips measuring $1 \times3$?

An open chain was made from $54$ identical single cardboard squares, connecting them hingedly at the vertices. Any square (except for the extreme ones) is connected to its neighbors by two opposite vertices. Is it possible to completely cover a $3\times 3 \times3$ surface with this chain of squares?

In an isosceles triangle $ABC$ ($AB=AC$) on the side $BC$, point $M$ is marked so that the segment $CM$ is equal to the altitude of the triangle drawn on this side, and on the side $AB$, point $K$ is marked so that the angle $\angle KMC$ is right. Find the angle $\angle ACK$.

Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.

How many ways are there to color every integer either red or blue such that $n$ and $n + 7$ are the same color for all integers $n,$ and there does not exist an integer $k$ such that $k, k + 1,$ and $2k$ are all the same color?

A particular country has seven distinct cities, conveniently named $C_1,C_2,\dots,C_7.$ Between each pair of cities, a direction is chosen, and a one-way road is constructed in that direction connecting the two cities. After the construction is complete, it is found that any city is reachable from any other city, that is, for distinct $1 \leq i, j \leq 7,$ there is a path of one-way roads leading from $C_i$ to $C_j.$ Compute the number of ways the roads could have been configured. Pictured on the following page are the possible configurations possible in a country with three cities, if every city is reachable from every other city. [Insert Diagram] [i]Proposed by Ezra Erives[/i]

$\omega$ is circumcircle of triangle $ABC$ and $BB_1, CC_1$ are bisectors of ABC. $I$ is center incirle . $B_1 C1$ and $\omega$ intersects at $M$ and $N$ . Find the ratio of circumradius of $ABC$ to circumradius $MIN$.

The lengths of the sides of a convex decagon are no greater than 1. Prove that for each inner point $M$ of the decagon there is at least one vertex $A$, for which $MA\leq \frac{\sqrt{5}+1}{2}$.

There are 94 safes and 94 keys. Each key can open only one safe, and each safe can be opened by only one key. We place randomly one key into each safe. 92 safes are then randomly chosen, and then locked. What is the probability that we can open all the safes with the two keys in the two remaining safes? (Once a safe is opened, the key inside the safe can be used to open another safe.)

Let $ABCD$ be a rectangle of centre $O$, such that $\angle DAC=60^{\circ}$. The angle bisector of $\angle DAC$ meets $DC$ at $S$. Lines $OS$ and $AD$ meet at $L$, and lines $BL$ and $AC$ meet at $M$. Prove that lines $SM$ and $CL$ are parallel.

Let $ABC$ be an acute triangle. Let $D$ be a point on side $AB$ and $E$ be a point on side $AC$ such that lines $BC$ and $DE$ are parallel. Let $X$ be an interior point of $BCED$. Suppose rays $DX$ and $EX$ meet side $BC$ at points $P$ and $Q$, respectively, such that both $P$ and $Q$ lie between $B$ and $C$. Suppose that the circumcircles of triangles $BQX$ and $CPX$ intersect at a point $Y \neq X$. Prove that the points $A, X$, and $Y$ are collinear.

Let $ n\ge2$ be a positive integer and denote by $ S_n$ the set of all permutations of the set $ \{1,2,\ldots,n\}$. For $ \sigma\in S_n$ define $ l(\sigma)$ to be $ \displaystyle\min_{1\le i\le n\minus{}1}\left|\sigma(i\plus{}1)\minus{}\sigma(i)\right|$. Determine $ \displaystyle\max_{\sigma\in S_n}l(\sigma)$.

Let $M,N$ and $P$ be the midpoints of sides $BC,CA$ and $AB$ respectively, of the acute triangle $ABC.$ Let $A',B'$ and $C'$ be the antipodes of $A,B$ and $C$ in the circumcircle of triangle $ABC.$ On the open segments $MA',NB'$ and $PC'$ we consider points $X,Y$ and $Z$ respectively such that \[\frac{MX}{XA'}=\frac{NY}{YB'}=\frac{PZ}{ZC'}.\][list=a] [*]Prove that the lines $AX,BY,$ and $CZ$ are concurrent at some point $S.$ [*]Prove that $OS<OG$ where $O$ is the circumcenter and $G$ is the centroid of triangle $ABC.$ [/list]

Let $F(n)$ denote the smallest positive integer greater than $n$ whose sum of digits is equal to the sum of the digits of $n$. For example, $F(2019) = 2028$. Compute $F(1) + F(2) + \dots + F(1000).$ [i]Proposed by Sean Li[/i]

Regular hexagon $ABCDEF$ has side length $2$. Let $G$ be the midpoint of $\overline{AB}$, and let $H$ be the midpoint of $\overline{DE}$. What is the perimeter of $GCHF$? $ \textbf{(A)}\ 4\sqrt3 \qquad \textbf{(B)}\ 8 \qquad \textbf{(C)}\ 4\sqrt5 \qquad \textbf{(D)}\ 4\sqrt7 \qquad \textbf{(E)}\ 12$

As illustrated below, we can dissect every triangle $ABC$ into four pieces so that piece 1 is a triangle similar to the original triangle, while the other three pieces can be assembled into a triangle also similar to the original triangle. Determine the ratios of the sizes of the three triangles and verify that the construction works. [asy] import olympiad;size(350);defaultpen(linewidth(0.7)+fontsize(10)); path p=origin--(13,0)--(9,8)--cycle; path p2=rotate(180)*p, p3=shift(-26,0)*scale(2)*p, p4=shift(-27,-24)*scale(3)*p, p1=shift(-53,-24)*scale(4)*p; pair A=(-53,-24), B=(-8,16), C=(12,-24), D=(-17,8), E=(-1,-24), F=origin, G=(-13,0), H=(-9,-8); label("1", centroid(A,D,E)); label("2", centroid(F,G,H)); label("3", (-10,6)); label("4", (0,-15)); draw(p2^^p3^^p4); filldraw(p1, white, black); pair point = centroid(F,G,H); label("$\mathbf{A}$", A, dir(point--A)); label("$\mathbf{B}$", B, dir(point--B)); label("$\mathbf{C}$", C, dir(point--C)); label("$\mathbf{D}$", D, dir(point--D)); label("$\mathbf{E}$", E, dir(point--E)); label("$\mathbf{F}$", F, dir(point--F)); label("$\mathbf{G}$", G, dir(point--G)); label("$\mathbf{H}$", H, dir(point--H)); real x=90; draw(shift(x)*p1); label("1", centroid(shift(x)*A,shift(x)*D,shift(x)*E)); draw(shift(130,0)*p4); draw(shift(130,0)*shift(-27,-24)*p); draw(shift(130,0)*shift(-1,-24)*p3); label("2", shift(130,0)*shift(-27,-24)*centroid(F,(9,8),(13,0))); label("3", shift(130,0)*shift(-1,-24)*(-10,6)); label("4", shift(130,0)*(0,-15)); label("Piece 2 rotated $180^\circ$", (130,10));[/asy]