There is a $2n\times 2n$ rectangular grid and a chair in each cell of the grid. Now, there are $2n^2$ pairs of couple are going to take seats. Define the distance of a pair of couple to be the sum of column difference and row difference between them. For example, if a pair of couple seating at $(3,3)$ and $(2,5)$ respectively, then the distance between them is $|3-2|+|3-5|=3$. Moreover, define the total distance to be the sum of the distance in each pair. Find the maximal total distance among all possibilities.

Find all tuples of positive integers $(a,b,c)$ such that $$a^b+b^c+c^a=a^c+b^a+c^b$$

For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ Proposed by Warut Suksompong, Thailand

Given a $ \triangle ABC $ and three points $ D, E, F $ such that $ DB = DC, $ $ EC = EA, $ $ FA = FB, $ $ \measuredangle BDC = \measuredangle CEA = \measuredangle AFB. $ Let $ \Omega_D $ be the circle with center $ D $ passing through $ B, C $ and similarly for $ \Omega_E, \Omega_F. $ Prove that the radical center of $ \Omega_D, \Omega_E, \Omega_F $ lies on the Euler line of $ \triangle DEF. $ Proposed by Telv Cohl

Determine all surjective functions $ f: \mathbb{Z} \to \mathbb{Z} $ such that $$ f\left(xyz+xf\left(y\right)+yf\left(z\right)+zf\left(x\right)\right)=f\left(x\right)f\left(y\right)f\left(z\right) $$for all $ x,y,z $ in $ \mathbb{Z} $

Let $ABC$ be a triangle such that $BC>AB$, $L$ be the internal angle bisector of $\angle ABC$. Let $P,Q$ be the feet from $A,C$ to $L$, respectively. Suppose $M,N$ are the midpoints of $\overline{AC}$ and $\overline{BC}$, respectively. Let $O$ be the circumcenter of triangle $PQM$, and the circumcircle intersects $AC$ at point $H$. Prove that $O,M,N,H$ are concyclic.

Given a circle and four points $B,C,X,Y$ on it. Assume $A$ is the midpoint of $BC$, and $Z$ is the midpoint of $XY$. Let $L_1,L_2$ be lines perpendicular to $BC$ and pass through $B,C$ respectively. Let the line pass through $X$ and perpendicular to $AX$ intersects $L_1,L_2$ at $X_1,X_2$ respectively. Similarly, let the line pass through $Y$ and perpendicular to $AY$ intersects $L_1,L_2$ at $Y_1,Y_2$ respectively. Assume $X_1Y_2$ intersects $X_2Y_1$ at $P$. Prove that $\angle AZP=90^o.$ Proposed by William Chao

Let $a,b,c,d$ be positive real numbers satisfying $a+b+c+d=4$. Prove that $$\frac{a^2}{b}+\frac{b^2}{c}+\frac{c^2}{d}+\frac{d^2}{a}\geq 4+(a-d)^2$$

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. Proposed by Dorlir Ahmeti, Albania

Find all integer $c\in\{0,1,...,2016\}$ such that the number of $f:\mathbb{Z}\rightarrow\{0,1,...,2016\}$ which satisfy the following condition is minimal: (1) $f$ has periodic $2017$ (2) $f(f(x)+f(y)+1)-f(f(x)+f(y))\equiv c\pmod{2017}$ Proposed by William Chao

Let $n \geq 3$ be a positive integer. Find the maximum number of diagonals in a regular $n$-gon one can select, so that any two of them do not intersect in the interior or they are perpendicular to each other.

Let $I$ be the incentre of a non-equilateral triangle $ABC$, $I_A$ be the $A$-excentre, $I'_A$ be the reflection of $I_A$ in $BC$, and $l_A$ be the reflection of line $AI'_A$ in $AI$. Define points $I_B$, $I'_B$ and line $l_B$ analogously. Let $P$ be the intersection point of $l_A$ and $l_B$. Prove that $P$ lies on line $OI$ where $O$ is the circumcentre of triangle $ABC$. Let one of the tangents from $P$ to the incircle of triangle $ABC$ meet the circumcircle at points $X$ and $Y$. Show that $\angle XIY = 120^{\circ}$.