Alice and Bob are play a game in a $(2n)*(2n)$ chess boared.Alice starts from the top right point moving 1 unit in every turn.Bob starts from the down left square and does the same.The goal of Alice is to make a closed shape ending in its start position and cannot reach any point that was reached before by any of players .if a players cannot move the other player keeps moving.what is the maximum are of the shape that the first player can build with every strategy of second player.

There are 8 points in the plane.we write down the area of each triangle having all vertices amoung these points(totally 56 numbers).Let them be $a_1,a_2,\dots a_{56}$.Prove that there is a choice of plus or minus such that: $$\pm a_1 \pm a_2 \dots \pm a_{56}=0$$

Find the smallest positive integer $n$ such that we can write numbers $1,2,\dots ,n$ in a 18*18 board such that: i)each number appears at least once ii)In each row or column,there are no two numbers having difference 0 or 1

Let $n$ be a positive integer.Consider all $2^n$ binary strings of length $n$.We say two of these strings are neighbors if they differ in exactly 1 digit.We have colored $m$ strings.In each moment,we can color any uncolored string which is neighbor with at least 2 colored strings.After some time,all the strings are colored.Find the minimum possible value of $m$.

Incircle of triangle $ABC$ is tangent to sides $BC,CA,AB$ at $D,E,F$,respectively.Points $P,Q$ are inside angle $BAC$ such that $FP=FB,FP||AC$ and $EQ=EC,EQ||AB$.Prove that $P,Q,D$ are collinear.

Two intersecting circles $\omega_1$ and $\omega_2$ are given.Lines $AB,CD$ are common tangents of $\omega_1,\omega_2$($A,C \in \omega_1 ,B,D \in \omega_2$) Let $M$ be the midpoint of $AB$.Tangents through $M$ to $\omega_1$ and $\omega_2$(other than $AB$) intersect $CD$ at $X,Y$.Let $I$ be the incenter of $MXY$.Prove that $IC=ID$.

$H$ is the orthocenter of acude triangle $ABC$.Let $\omega$ be the circumcircle of $BHC$ with center $O'$.$\Omega$ is the nine-point circle of $ABC$.$X$ is an arbitrary point on arc $BHC$ of $\omega$ and $AX$ intersects $\Omega$ at $Y$.$P$ is a point on $\Omega$ such that $PX=PY$.Prove that $O'PX=90$.

for acute triangle $\triangle ABC$ with orthocenter $H$, and $E,F$ the feet of altitudes for $B,C$, we have $P$ on $EF$ such as that $HO \perp HP$. $Q$ is on segment $AH$ so $HM \perp PQ$. prove $QA=3QH$

$n\ge 2 $ is an integer.Prove that the number of natural numbers $m$ so that $0 \le m \le n^2-1,x^n+y^n \equiv m (mod n^2)$ has no solutions is at least $\binom{n}{2}$

Prove that for every prime number $p$ there exist infinity many natural numbers $n$ so that they satisfy: $2^{2^{2^{ \dots ^{2^n}}}} \equiv n^{2^{2^{\dots ^{2}}}} (mod p)$ Where in both sides $2$ appeared $1397$ times

Find all functions $f:\mathbb{N}\to \mathbb{N}$ so that for every natural numbers $m,n$ :$f(n)+2mn+f(m)$ is a perfect square.

Prove that for any natural numbers$a,b$ there exist infinity many prime numbers $p$ so that $Ord_p(a)=Ord_p(b)$(Proving that there exist infinity prime numbers $p$ so that $Ord_p(a) \ge Ord_p(b)$ will get a partial mark)

For positive real numbers$a,b,c$such that $ab+ac+bc=1$ prove that: $\prod\limits_{cyc} (\sqrt{bc}+\frac{1}{2a+\sqrt{bc}}) \ge 8abc$

Find all functions $f: \mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that: $f(x^3+xf(xy))=f(xy)+x^2f(x+y) \forall x,y \in \mathbb{R}^{\ge 0}$

A)Let $x,y$ be two complex numbers on the unit circle so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{5 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-x|+|z-y| \ge |zx-y|$ B)Let $x,y$ be two complex numbers so that: $\frac{\pi }{3} \le \arg (x)-\arg (y) \le \frac{2 \pi }{3}$ Prove that for any $z \in \mathbb{C}$ we have: $|z|+|z-y|+|z-x| \ge | \frac{\sqrt{3}}{2} x +(y-\frac{x}{2})i|$

Let $P(x)$ be a non-zero polynomial with real coefficient so that $P(0)=0$.Prove that for any positive real number $M$ there exist a positive integer $d$ so that for any monic polynomial $Q(x)$ with degree at least $d$ the number of integers $k$ so that $|P(Q(k))| \le M$ is at most equal to the degree of $Q$.