$x,y,z$ are three real numbers inequal to zero satisfying $x+y+z=xyz$. Prove that $$ \sum (\frac{x^2-1}{x})^2 \geq 4$$ Proposed by Amin Fathpour

Prove that there are no functions $f,g:\mathbb{R}\rightarrow \mathbb{R}$ such that $\forall x,y\in \mathbb{R}:$ $ f(x^2+g(y)) -f(x^2)+g(y)-g(x) \leq 2y$ and $f(x)\geq x^2$. Proposed by Mohammad Ahmadi

Does there exist an irreducible two variable polynomial $f(x,y)\in \mathbb{Q}[x,y]$ such that it has only four roots $(0,1),(1,0),(0,-1),(-1,0)$ on the unit circle.

$p(x)\in \mathbb{C}[x]$ is a polynomial such that: $\forall z\in \mathbb{C}, |z|=1\Longrightarrow p(z)\in \mathbb{R}$ Prove that $p(x)$ is constant.

Find all polynomials $p(x)\in\mathbb{R}[x]$ such that for all $x\in \mathbb{R}$: $p(5x)^2-3=p(5x^2+1)$ such that: $a) p(0)\neq 0$ $b) p(0)=0$

$a_1,a_2,\dots ,a_n>0$ are positive real numbers such that $\sum_{i=1}^{n} \frac{1}{a_i}=n$ prove that: $\sum_{i<j} \left(\frac{a_i-a_j}{a_i+a_j}\right)^2\le\frac{n^2}{2}\left(1-\frac{n}{\sum_{i=1}^{n}a_i}\right)$

Prove that there are infinitely natural numbers $n$ such that $n$ can't be written as a sum of two positive integers with prime factors less than $1394$.

$M_0 \subset \mathbb{N}$ is a non-empty set with a finite number of elements. Ali produces sets $ M_1,M_2,...,M_n $ in the following order: In step $n$, Ali chooses an element of $M_{n-1} $ like $b_n$ and defines $M_n$ as $$M_n = \left \{ b_nm+1 \vert m\in M_{n-1} \right \}$$ Prove that at some step Ali reaches a set which no element of it divides another element of it.

Let $p>5$ be a prime number and $A=\{b_1,b_2,\dots,b_{\frac{p-1}{2}}\}$ be the set of all quadratic residues modulo $p$, excluding zero. Prove that there doesn't exist any natural $a,c$ satisfying $(ac,p)=1$ such that set $B=\{ab_1+c,ab_2+c,\dots,ab_{\frac{p-1}{2}}+c\}$ and set $A$ are disjoint modulo $p$. This problem was proposed by Amir Hossein Pooya.

$a,b,c,d,k,l$ are positive integers such that for every natural number $n$ the set of prime factors of $n^k+a^n+c,n^l+b^n+d$ are same. prove that $k=l,a=b,c=d$.

$p>30$ is a prime number. Prove that one of the following numbers is in form of $x^2+y^2$. $$ p+1 , 2p+1 , 3p+1 , .... , (p-3)p+1$$

Let $ABCD$ be the trapezoid such that $AB\parallel CD$. Let $E$ be an arbitrary point on $AC$. point $F$ lies on $BD$ such that $BE\parallel CF$. Prove that circumcircles of $\triangle ABF,\triangle BED$ and the line $AC$ are concurrent.

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $K$ be the midpoint of $AH$. point $P$ lies on $AC$ such that $\angle BKP=90^{\circ}$. Prove that $OP\parallel BC$.

Let $ABC$ be a triangle. consider an arbitrary point $P$ on the plain of $\triangle ABC$. Let $R,Q$ be the reflections of $P$ wrt $AB,AC$ respectively. Let $RQ\cap BC=T$. Prove that $\angle APB=\angle APC$ if and if only $\angle APT=90^{\circ}$.

Let $ABC$ be a triangle with incenter $I$. Let $K$ be the midpoint of $AI$ and $BI\cap \odot(\triangle ABC)=M,CI\cap \odot(\triangle ABC)=N$. points $P,Q$ lie on $AM,AN$ respectively such that $\angle ABK=\angle PBC,\angle ACK=\angle QCB$. Prove that $P,Q,I$ are collinear.

Let $ABC$ be a triangle with orthocenter $H$ and circumcenter $O$. Let $R$ be the radius of circumcircle of $\triangle ABC$. Let $A',B',C'$ be the points on $\overrightarrow{AH},\overrightarrow{BH},\overrightarrow{CH}$ respectively such that $AH.AA'=R^2,BH.BB'=R^2,CH.CC'=R^2$. Prove that $O$ is incenter of $\triangle A'B'C'$.