Day 1
The sequence ${a_1, a_2, ..., a_{2019}}$ satisfies the following condition. $a_1=1, a_{n+1}=2019a_{n}+1$ Now let $x_1, x_2, ..., x_{2019}$ real numbers such that $x_1=a_{2019}, x_{2019}=a_1$ (The others are arbitary.) Prove that $\sum_{k=1}^{2018} (x_{k+1}-2019x_k-1)^2 \ge \sum_{k=1}^{2018} (a_{2019-k}-2019a_{2020-k}-1)^2$
Triangle $ABC$ is an scalene triangle. Let $I$ the incenter, $\Omega$ the circumcircle, $E$ the $A$-excenter of triangle $ABC$. Let $\Gamma$ the circle centered at $E$ and passes $A$. $\Gamma$ and $\Omega$ intersect at point $D(\neq A)$, and the perpendicular line of $BC$ which passes $A$ meets $\Gamma$ at point $K(\neq A)$. $L$ is the perpendicular foot from $I$ to $AC$. Now if $AE$ and $DK$ intersects at $F$, prove that $BE\cdot CI=2\cdot CF\cdot CL$.
Suppose that positive integers $m,n,k$ satisfy the equations $$m^2+1=2n^2, 2m^2+1=11k^2.$$Find the residue when $n$ is divided by $17$.
Let $(x_1, y_1, z_1), (x_2, y_2, z_2), \cdots, (x_{19}, y_{19}, z_{19})$ be integers. Prove that there exist pairwise distinct subscripts $i, j, k$ such that $x_i+x_j+x_k$, $y_i+y_j+y_k$, $z_i+z_j+z_k$ are all multiples of $3$.
Day 2
Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$
In acute triangle $ABC$, $AB>AC$. Let $I$ the incenter, $\Omega$ the circumcircle of triangle $ABC$, and $D$ the foot of perpendicular from $A$ to $BC$. $AI$ intersects $\Omega$ at point $M(\neq A)$, and the line which passes $M$ and perpendicular to $AM$ intersects $AD$ at point $E$. Now let $F$ the foot of perpendicular from $I$ to $AD$. Prove that $ID\cdot AM=IE\cdot AF$.
For prime $p\equiv 1\pmod{7} $, prove that there exists some positive integer $m$ such that $m^3+m^2-2m-1$ is a multiple of $p$.
There are two countries $A$ and $B$, where each countries have $n(\ge 2)$ airports. There are some two-way flights among airports of $A$ and $B$, so that each airport has exactly $3$ flights. There might be multiple flights among two airports; and there are no flights among airports of the same country. A travel agency wants to plan an exotic traveling course which travels through all $2n$ airports exactly once, and returns to the initial airport. If $N$ denotes the number of all exotic traveling courses, then prove that $\frac{N}{4n}$ is an even integer. (Here, note that two exotic traveling courses are different if their starting place are different.)