Let $ABC$ be a triangle. Circle $\Gamma$ passes through point $A$, meets segments $AB$ and $AC$ again at $D$ and $E$ respectively, and intersects segment $BC$ at $F$ and $G$ such that $F$ lies between $B$ and $G$. The tangent to circle $(BDF)$ at $F$ and the tangent to circle $(CEG)$ at $G$ meet at $T$. Suppose that points $A$ and $T$ are distinct. Prove that line $AT$ is parallel to $BC$.

Let $x, y, z$ be positive real numbers such that $x^2+y^2+z^2=3$. Prove that $$\frac{x+1}{z+x+1}+\frac{y+1}{x+y+1}+\frac{z+1}{y+z+1}\geq\frac{(xy+yz+zx+\sqrt{xyz})^2}{(x+y)(y+z)(z+x)}.$$

Find all pairs of positive integers $(m, n)$ satisfying the equation $$m!+n!=m^n+1.$$

A $1\times 2019$ board is filled with numbers $1, 2, \dots, 2019$ in an increasing order. In each step, three consecutive tiles are selected, then one of the following operations is performed: $\text{(i)}$ the number in the middle is increased by $2$ and its neighbors are decreased by $1$, or $\text{(ii)}$ the number in the middle is decreased by $2$ and its neighbors are increased by $1$. After several such operations, the board again contains all the numbers $1, 2,\dots, 2019$. Prove that each number is in its original position.

Let $\{a_n\}$ be a sequence of positive integers such that $a_{n+1} = a_n^2+1$ for all $n \geq 1$. Prove that there is no positive integer $N$ such that $$\prod_{k=1}^N(a_k^2+a_k+1)$$is a perfect square.

Prove that the unit square can be tiled with rectangles (not necessarily of the same size) similar to a rectangle of size $1\times(3+\sqrt[3]{3})$.

Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ satisfying $$f(\max \left\{ x, y \right\} + \min \left\{ f(x), f(y) \right\}) = x+y $$for all $x,y \in \mathbb{R}$.

For any positive integer $m \geq 2$, let $p(m)$ be the smallest prime dividing $m$ and $P(m)$ be the largest prime dividing $m$. Let $C$ be a positive integer. Define sequences $\{a_n\}$ and $\{b_n\}$ by $a_0 = b_0 = C$ and, for each positive integer $k$ such that $a_{k-1}\geq 2$, $$a_k=a_{k-1}-\frac{a_{k-1}}{p(a_{k-1})};$$and, for each positive integer $k$ such that $b_{k-1}\geq 2$, $$b_k=b_{k-1}-\frac{b_{k-1}}{P(b_{k-1})}$$It is easy to see that both $\{a_n\}$ and $\{b_n\}$ are finite sequences which terminate when they reach the number $1$. Prove that the numbers of terms in the two sequences are always equal.

Let $ABC$ be an acute triangle and $\Gamma$ be its circumcircle. Line $\ell$ is tangent to $\Gamma$ at $A$ and let $D$ and $E$ be distinct points on $\ell$ such that $AD = AE$. Suppose that $B$ and $D$ lie on the same side of line $AC$. The circumcircle $\Omega_1$ of $\vartriangle ABD$ meets $AC$ again at $F$. The circumcircle $\Omega_2$ of $\vartriangle ACE$ meets $AB$ again at $G$. The common chord of $\Omega_1$ and $\Omega_2$ meets $\Gamma$ again at $H$. Let $K$ be the reflection of $H$ across line $BC$ and let $L$ be the intersection of $BF$ and $CG$. Prove that $A, K$ and $L$ are collinear.

Does there exist a set $S$ of positive integers satisfying the following conditions? $\text{(i)}$ $S$ contains $2020$ distinct elements; $\text{(ii)}$ the number of distinct primes in the set $\{\gcd(a, b) : a, b \in S, a \neq b\}$ is exactly $2019$; and $\text{(iii)}$ for any subset $A$ of $S$ containing at least two elements, $\sum\limits_{a,b\in A; a<b} ab$ is not a prime power.

Let $P$ be an interior point of a circle $\Gamma$ centered at $O$ where $P \ne O$. Let $A$ and $B$ be distinct points on $\Gamma$. Lines $AP$ and $BP$ meet $\Gamma$ again at $C$ and $D$, respectively. Let $S$ be any interior point on line segment $PC$. The circumcircle of $\vartriangle ABS$ intersects line segment $PD$ at $T$. The line through $S$ perpendicular to $AC$ intersects $\Gamma$ at $U$ and $V$ . The line through $T$ perpendicular to $BD$ intersects $\Gamma$ at $X$ and $Y$ . Let $M$ and $N$ be the midpoints of $UV$ and $XY$ , respectively. Let $AM$ and $BN$ meet at $Q$. Suppose that $AB$ is not parallel to $CD$. Show that $P, Q$, and $O$ are collinear if and only if $S$ is the midpoint of $PC$.

A nonempty set $S$ is called Bally if for every $m\in S$, there are fewer than $\frac{1}{2}m$ elements of $S$ which are less than $m$. Determine the number of Bally subsets of $\{1, 2, . . . , 2020\}$.