Let $ABCD$ be a cyclic quadrilateral inscribed in a circle $\Omega$ such that $AB<CD$. Suppose that $AC$ meets $BD$ at $E$, $AD$ meets $BC$ at $F$, and $\Omega$ meets $(FAE)$, $(FBE)$ at $X$, $Y$, respectively. Prove that if $XY$ is diameter of $\Omega$, then $XY$ is perpendicular to $EF$.

Let $n\ge 2$ be a positive integer. There are $n$ real coefficient polynomials $P_1(x),P_2(x),\cdots ,P_n(x)$ which is not all the same, and their leading coefficients are positive. Prove that $$\deg(P_1^n+P_2^n+\cdots +P_n^n-nP_1P_2\cdots P_n)\ge (n-2)\max_{1\le i\le n}(\deg P_i)$$and find when the equality holds.

Let $n\ge 3$ be a positive integer. Amy wrote all the integers from $1$ to $n^2$ on the $n\times n$ grid, so that each cell contains exactly one number. For $i=1,2,\cdots ,n^2-1$, the cell containing $i$ shares a common side with the cell containing $i+1$. Each turn, Bred can choose one cell, and check what number is written. Bred wants to know where $1$ is written by less than $3n$ turns. Determine whether $n$ such that Bred can always achieve his goal is infinite.

There are $2022$ students in winter school. Two arbitrary students are friend or enemy each other. Each turn, we choose a student $S$, make friends of $S$ enemies, and make enemies of $S$ friends. This continues until it satisfies the final condition. Final Condition : For any partition of students into two non-empty groups $A$, $B$, there exist two students $a$, $b$ such that $a\in A$, $b\in B$, and $a$, $b$ are friend each other. Determine the minimum value of $n$ such that regardless of the initial condition, we can satisfy the final condition with no more than $n$ turns.

Let $ABDC$ be a cyclic quadrilateral inscribed in a circle $\Omega$. $AD$ meets $BC$ at $P$, and $\Omega$ meets lines passing $A$ and parallel to $DB$, $DC$ at $E$, $F$, respectively. $X$ is a point on $\Omega$ such that $PA=PX$. Prove that the lines $BE$, $CF$, and $DX$ are concurrent.

Determine all positive integers $(x_1,x_2,x_3,y_1,y_2,y_3)$ such that $y_1+ny_2^n+n^2y_3^{2n}$ divides $x_1+nx_2^n+n^2x_3^{2n}$ for all positive integer $n$.

Prove that equation $y^2=x^3+7$ doesn't have any solution on integers.

Let $ABC$ be an acute triangle such that $AB<AC$. Let $\Omega$ be its circumcircle, $O$ be its circumcenter, and $l$ be the internal angle bisector of $\angle BAC$. Suppose that the tangents to $\Omega$ at $B$ and $C$ intersect at $X$. Let $\omega$ be a circle whose center is $X$ and passes $B$, and $Y$ be the intersection of $l$ and $\omega$ which is chosen inside $\triangle ABC$. Let $D,E$ be the projections of $Y$ onto $AB,AC$, respectively. $OY$ meets $BC$ at $Z$. $ZD,ZE$ meet $l$ at $P,Q$, respectively. Prove that $BQ$ and $CP$ are parallel.

Let $n\ge 2$ be a positive integer. $S$ is a set of $2n$ airports. For two arbitrary airports $A,B$, if there is an airway from $A$ to $B$, then there is an airway from $B$ to $A$. Suppose that $S$ has only one independent set of $n$ airports. Let the independent set $X$. Prove that there exists an airport $P\in S\setminus X$ which satisfies following condition. Condition : For two arbitrary distinct airports $A,B\in S\setminus \{P\}$, if there exists a path connecting $A$ and $B$, then there exists a path connecting $A$ and $B$ which does not pass $P$.

For a finite set $A$ of positive integers and its subset $B$, call $B$ a half subset of $A$ when it satisfies the equation $\sum_{a\in A}a=2\sum_{b\in B}b$. For example, if $A=\{1,2,3\}$, then $\{1,2\}$ and $\{3\}$ are half subset of $A$. Determine all positive integers $n$ such that there exists a finite set $A$ which has exactly $n$ half subsets.

Determine all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)-9f(y))=(x+3y)^2f(x-3y)$$for all $x,y\in \mathbb{R}$.

Let $ABC$ be an acute triangle with incenter $I$ and circumcircle $\Omega$. The line passing $I$ and perpendicular to $AI$ meets $AB, AC$ at $D, E$, respectively. $A$-excircle of $\triangle{ABC}$ meets $BC$ at $T$. $AT$ meets $\Omega$ at $P$. The line passing $P$ and parallel to $BC$ meets $\Omega$ at $Q$. The intersection of $QI$ and $AT$ is $K$. Prove that $Q,D,K,E$ are concyclic.