Let $ABC $ be a triangle and $D $ be the feet of $A $-altitude. $E,F $ are defined on segments $AD,BC $,respectively such that $\frac {AE}{DE}=\frac{BF}{CF} $. Assume that $G $ lies on $AF $ such that $BG\perp AF $.Prove that $EF $ is tangent to the circumcircle of $CFG $. Proposed by Mehdi Etesami Fard

Let $n >1$ be a natural number and $T_{n}(x)=x^n + a_{n-1}x^{n-1} + a_{n-2}x^{n-2} + ... + a_1 x^1 + a_0$. Assume that for each nonzero real number $t $ we have $T_{n}(t+\frac {1}{t})=t^n+\frac {1}{t^n} $. Prove that for each $0\le i \le n-1 $ $gcd (a_i,n) >1$. Proposed by Morteza Saghafian

An infinite network is partitioned with dominos. Prove there exist three other tilings with dominos, have neither common domino with the existing tiling nor with each other. Clarifications for network: It means an infinite board consisting of square cells.

Let $a,b $ be two relatively prime positive integers.Also let $m,n $ be positive integers with $n> m $. Prove that $lcm [am+b,a (m+1)+b,...,an+b]\ge (n+1)\cdot \binom {n}{m}$ Proposed by Navid Safaei

Edges of a planar graph $G$ are colored either with blue or red. Prove that there is a vertex like $v$ such that when we go around $v$ through a complete cycle, edges with the endpoint at $v$ change their color at most two times. Clarifications for complete cycle: If all the edges with one endpoint at $v$ are $(v,u_1),(v,u_2),\ldots,(v,u_k)$ such that $u_1,u_2,\ldots,u_k$ are clockwise with respect to $v$ then in the sequence of $(v,u_1),(v,u_2),\ldots,(v,u_k),(v,u_1)$ there are at most two $j$ such that colours of $(v,u_j),(v,u_{j+1})$ ($j \mod k$) differ.

Let $ABCD $ be cyclic quadrilateral with circumcircle $\omega $ and $M $ be any point on $\omega $. Let $E $ and $F $ be the intersection of $AB,CD $ and $AD,BC $ respectively. $ME $ intersects lines $AD,BC $ at $P,Q $ and similarly $MF$ intersects lines $AB,CD $ at $R,S $. Let the lines $PS $ and $RQ $ meet at $X $. Prove that as $M $ varies over $\omega $ $MX $ passes through fixed point. Proposed by Mehdi Etesami Fard