Let $ABC$ be a triangle, and let $\ell_{1}$ and $\ell_{2}$ be two parallel lines. For $i=1,2$, let $\ell_{i}$ meet the lines $BC$, $CA$, and $AB$ at $X_{i}$, $Y_{i}$, and $Z_{i}$, respectively. Suppose that the line through $X_{i}$ perpendicular to $BC$, the line through $Y_{i}$ perpendicular to $CA$, and finally the line through $Z_{i}$ perpendicular to $AB$, determine a non-degenerate triangle $\Delta_{i}$.
Show that the circumcircles of $\Delta_{1}$ and $\Delta_{2}$ are tangent to each other.
In an acute-angled triangle $A B C$, point $H$ is the foot of the altitude from $A$. Let $P$ be a moving point such that the bisectors $k$ and $\ell$ of angles $P B C$ and $P C B$, respectively, intersect each other on the line segment $A H$. Let $k$ and $A C$ meet at $E$, let $\ell$ and $A B$ meet at $F$, and let $E F$ and $A H$ meet at $Q$. Prove that, as $P$ varies, the line $P Q$ passes through a fixed point.
Let $ABC$ and $A'B'C'$ be two triangles having the same circumcircle $\omega$, and the same orthocentre $H$. Let $\Omega$ be the circumcircle of the triangle determined by the lines $AA'$, $BB'$, and $CC'$. Prove that $H$, the centre of $\omega$, and the centre of $\Omega$ are collinear.
(Denmark)
Let $A A^{\prime} B C C^{\prime} B^{\prime}$ be a convex cyclic hexagon such that $A C$ is tangent to the incircle of the triangle $A^{\prime} B^{\prime} C^{\prime}$, and $A^{\prime} C^{\prime}$ is tangent to the incircle of the triangle $A B C$. Let the lines $A B$ and $A^{\prime} B^{\prime}$ meet at $X$ and let the lines $B C$ and $B^{\prime} C^{\prime}$ meet at $Y$.
Prove that if $X B Y B^{\prime}$ is a convex quadrilateral, then it has an incircle.
For each $1 \leqslant i \leqslant 9$ and $T \in \mathbb{N}$, define $d_{i}(T)$ to be the total number of times the digit $i$ appears when all the multiples of $1829$ between $1$ and $T$ inclusive are written out in base $10$.
Show that there are infinitely many $T \in \mathbb{N}$ such that there are precisely two distinct values among $d_{1}(T), d_{2}(T), \ldots, d_{9}(T)$.
Let $k$ be a positive integer. A sequence of integers $\langle a_i \rangle_{i=1}^\infty$ is called $k$-pop if the following holds: for every $n \in \mathbb{N}$, $a_n$ is equal to the number of distinct elements in the set $\{a_1, \dots, a_{n+k}\}$. Determine, as a function of $k$, how many $k$-pop sequences there are.
Let $r \gt 0$ be a real number. All the interior points of the disc $D(r)$ of radius $r$ are colored with one of two colors, red or blue.
1. If $r \gt \frac{\pi}{\sqrt{3}}$, show that we can find two points $A$ and $B$ in the interior of the disc such that the distance $AB = \pi$ and $A$ and $B$ have the same color.
2. Does the conclusion in (a) hold if $r \gt \frac{\pi}{2}$?
Let $N \ge 3$ be an integer. In the country of Sibyl, there are $N^2$ towns arranged as the vertices of an $N \times N$ grid, with each pair of towns corresponding to an adjacent pair of vertices on the grid connected by a road. Several automated drones are each given the instruction to traverse a rectangular path starting and ending at the same town, following the roads of the country. It turned out that each road was traversed at least once by some drone. Determine the minimum number of drones that must be operating.
Let $A$, $B$ be two points on a plane and $M$ be the midpoint of $AB$. We firstly choose a point $P$ on the segment $AB$, other than $A$, $B$, $M$. At step $i$ we choose a red point $P_i$ then choose one of $A$ and $B$, call it $X_i$, and reflect $P_i$ with respect to $X_i$ to get $Q_i$, then color the midpoint of $Q_iX_i$ red. Is it possible that after a few steps we color the $M$ by red?
We call a natural number $n$ *nice* if it has no zero digit in its decimal representation and it has a digit $b$ in this representation such that the resulting number would be a divisor of $n$, after its removal. Prove that there are only finitely many *nice* numbers.
Let $ABCD$ be a quadrilateral and $\omega$ is a circle inscribed in it. The circle $\omega$ is tangent to $BC$ and $AD$ at $E$ and $F$, respectively and $DE$ meets $\omega$ at $X$, for the second time. If $AB$, $CD$ are tangent to the circumcircle of $DXF$, prove that $AFXC$ is cyclic.
Given $1400$ real numbers, prove that among them there are at least three numbers $x$, $y$, $z$ such that
$$
\left| \frac{(x-y)(y-z)(z-x)}{1+x^4+y^4+z^4} \right| \lt \frac{9}{1000} .
$$
Is it possible to write $1400$ natural numbers (not necessarily distinct) around a circle such that $2021$ is used at least once and each number is the sum of the greatest common divisor of the two previous numbers and the greatest common divisor of the two next numbers? For example, if $a, b, c, d, e$ are five consecutive numbers on the circle then $c = \gcd(a, b) + \gcd(d, e)$.
*Morteza* has $100$ sets. At each step *Mahdi* can choose two distinct sets from them and Morteza tells him the intersection and union of those two sets. Find the least possible number of steps that *Mahdi* can determine all of $100$ sets.
For a positive integer $n$, let $\tau(n)$ and $\sigma(n)$ be the number of positive divisors of $n$ and the sum of positive divisors of $n$, respectively. Let $a$ and $b$ be positive integers such that $\sigma(a^n)$ divides $\sigma(b^n)$ for all $n \in \mathbb{N}$. Prove that each prime factor of $\tau(a)$ divides $\tau(b)$.
Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$, respectively. Point $X$ is the reflection of $D$ with respect to $B$. Suppose that the line $DE$ is tangent to the $A$-excircle at $Z$. Let the circumcircle of triangle $XZE$ intersect $\omega$ at $K$, for the second time. Prove that the intersection of $BK$ and $AZ$ lies on $\omega$.
Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$. Let $M$ and $N$ be midpoints of the sides $AD$ and $BC$, respectively. Suppose that $\omega_1$, $\omega_2$ and $\omega_3$ are circumcircles of triangles $ADP$, $BCP$ and $OMN$, respectively. Let $E$ and $F$ be the intersection point of $\omega_1$ and $\omega_3$ which is not on the arc $APD$ of $\omega_1$, and the intersection point of $\omega_2$ and $\omega_3$ which is not on the arc $BPC$ of $\omega_2$, respectively. Prove that $OE = OF$.
Let $m$, $n$ and $a_1, a_2, \dots, a_m$ be arbitrary positive integers. Ali and Mohammad play the following game: At each step, Ali chooses $b_1, b_2, \dots, b_m \in \mathbb{N}$, then Mohammad chooses a positive integer $s$ and obtains a new sequence $\{c_i = a_i + b_{i+s}\}_{i=1}^m$, where
$$
b_{m+1} = b_1,\ b_{m+2} = b_2,\ \dots,\ b_{m+s} = b_s.
$$
Through a finite number of steps, Ali intends to make all the numbers divisible by $n$. Find all positive integers $m$ and $n$ such that Ali, independent of the initial values $a_1, a_2, \dots, a_m$, has a winning strategy.