Suppose that $ f(x)\in\mathbb Z[x]$ be an irreducible polynomial. It is known that $ f$ has a root of norm larger than $ \frac32$. Prove that if $ \alpha$ is a root of $ f$ then $ f(\alpha^3+1)\neq0$.

Find the smallest real $ K$ such that for each $ x,y,z\in\mathbb R^{ + }$: \[ x\sqrt y + y\sqrt z + z\sqrt x\leq K\sqrt {(x + y)(y + z)(z + x)} \]

Let $ (b_0,b_1,b_2,b_3)$ be a permutation of the set $ \{54,72,36,108\}$. Prove that $ x^5+b_3x^3+b_2x^2+b_1x+b_0$ is irreducible in $ \mathbb Z[x]$.

Let $ x,y,z\in\mathbb R^{+}$ and $ x+y+z=3$. Prove that: \[ \frac{x^3}{y^3+8}+\frac{y^3}{z^3+8}+\frac{z^3}{x^3+8}\geq\frac19+\frac2{27}(xy+xz+yz)\]

Prove that the following polynomial is irreducible in $ \mathbb Z[x,y]$: \[ x^{200}y^5+x^{51}y^{100}+x^{106}-4x^{100}y^5+x^{100}-2y^{100}-2x^6+4y^5-2\]

Let $ k>1$ be an integer. Prove that there exists infinitely many natural numbers such as $ n$ such that: \[ n|1^n+2^n+\dots+k^n\]

Prove that there exists infinitely many primes $ p$ such that: \[ 13|p^3+1\]

Let $ P$ be a regular polygon. A regular sub-polygon of $ P$ is a subset of vertices of $ P$ with at least two vertices such that divides the circumcircle to equal arcs. Prove that there is a subset of vertices of $ P$ such that its intersection with each regular sub-polygon has even number of vertices.

Let $ u$ be an odd number. Prove that $ \frac{3^{3u}-1}{3^u-1}$ can be written as sum of two squares.

Find all polynomials $ f\in\mathbb Z[x]$ such that for each $ a,b,x\in\mathbb N$ \[ a+b+c|f(a)+f(b)+f(c)\]

Prove that the number of pairs $ \left(\alpha,S\right)$ of a permutation $ \alpha$ of $ \{1,2,\dots,n\}$ and a subset $ S$ of $ \{1,2,\dots,n\}$ such that \[ \forall x\in S: \alpha(x)\not\in S\] is equal to $ n!F_{n + 1}$ in which $ F_n$ is the Fibonacci sequence such that $ F_1 = F_2 = 1$

Prove that the number permutations $ \alpha$ of $ \{1,2,\dots,n\}$ s.t. there does not exist $ i<j<n$ s.t. $ \alpha(i)<\alpha(j+1)<\alpha(j)$ is equal to the number of partitions of that set.

Prove that for each $ n$: \[ \sum_{k=1}^n\binom{n+k-1}{2k-1}=F_{2n}\]

Let $ S$ be a sequence that: \[ \left\{ \begin{array}{cc} S_0=0\hfill\\ S_1=1\hfill\\ S_n=S_{n-1}+S_{n-2}+F_n& (n>1) \end{array} \right.\] such that $ F_n$ is Fibonacci sequence such that $ F_1=F_2=1$. Find $ S_n$ in terms of Fibonacci numbers.

$ n$ people decide to play a game. There are $ n-1$ ropes and each of its two ends are in hand of one of the players, in such a way that ropes and players form a tree. (Each person can hold more than rope end.) At each step a player gives one of the rope ends he is holding to another player. The goal is to make a path of length $ n-1$ at the end. But the game regulations change before game starts. Everybody has to give one of his rope ends only two one of his neighbors. Let $ a$ and $ b$ be minimum steps for reaching to goal in these two games. Prove that $ a=b$ if and only if by removing all players with one rope end (leaves of the tree) the remaining people are on a path. (the remaining graph is a path.)

Prove that for $ n > 0$ and $ a\neq0$ the polynomial $ p(z) = az^{2n + 1} + bz^{2n} + \bar bz + \bar a$ has a root on unit circle

Let $ g,f: \mathbb C\longrightarrow\mathbb C$ be two continuous functions such that for each $ z\neq 0$, $ g(z)=f(\frac1z)$. Prove that there is a $ z\in\mathbb C$ such that $ f(\frac1z)=f(-\bar z)$

For each $ c\in\mathbb C$, let $ f_c(z,0)=z$, and $ f_c(z,n)=f_c(z,n-1)^2+c$ for $ n\geq1$. a) Prove that if $ |c|\leq\frac14$ then there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is bounded. b) Prove that if $ c>\frac14$ is a real number there is a neighborhood $ U$ of origin such that for each $ z\in U$ the sequence $ f_c(z,n),n\in\mathbb N$ is unbounded.

Let $ ABC$ be a triangle with $ BC > AC > AB$. Let $ A',B',C'$ be feet of perpendiculars from $ A,B,C$ to $ BC,AC,AB$, such that $ AA' = BB' = CC' = x$. Prove that: a) If $ ABC\sim A'B'C'$ then $ x = 2r$ b) Prove that if $ A',B'$ and $ C'$ are collinear, then $ x = R + d$ or $ x = R - d$. (In this problem $ R$ is the radius of circumcircle, $ r$ is radius of incircle and $ d = OI$)

Let $ l_a,l_b,l_c$ be three parallel lines passing through $ A,B,C$ respectively. Let $ l_a'$ be reflection of $ l_a$ into $ BC$. $ l_b'$ and $ l_c'$ are defined similarly. Prove that $ l_a',l_b',l_c'$ are concurrent if and only if $ l_a$ is parallel to Euler line of triangle $ ABC$.

Let $ ABCD$ be a quadrilateral, and $ E$ be intersection points of $ AB,CD$ and $ AD,BC$ respectively. External bisectors of $ DAB$ and $ DCB$ intersect at $ P$, external bisectors of $ ABC$ and $ ADC$ intersect at $ Q$ and external bisectors of $ AED$ and $ AFB$ intersect at $ R$. Prove that $ P,Q,R$ are collinear.

Let $ ABC$ be an isosceles triangle with $ AB=AC$, and $ D$ be midpoint of $ BC$, and $ E$ be foot of altitude from $ C$. Let $ H$ be orthocenter of $ ABC$ and $ N$ be midpoint of $ CE$. $ AN$ intersects with circumcircle of triangle $ ABC$ at $ K$. The tangent from $ C$ to circumcircle of $ ABC$ intersects with $ AD$ at $ F$. Suppose that radical axis of circumcircles of $ CHA$ and $ CKF$ is $ BC$. Find $ \angle BAC$.

Let $ D,E,F$ be tangency point of incircle of triangle $ ABC$ with sides $ BC,AC,AB$. $ DE$ and $ DF$ intersect the line from $ A$ parallel to $ BC$ at $ K$ and $ L$. Prove that the Euler line of triangle $ DKL$ passes through Feuerbach point of triangle $ ABC$.

Police want to arrest on the famous criminals of the country whose name is Kaiser. Kaiser is in one of the streets of a square shaped city with $ n$ vertical streets and $ n$ horizontal streets. In the following cases how many police officers are needed to arrest Kaiser? a) Each police officer has the same speed as Kaiser and every police officer knows the location of Kaiser anytime. b) Kaiser has an infinite speed (finite but with no bound) and police officers can only know where he is only when one of them see Kaiser. Everybody in this problem (including police officers and Kaiser) move continuously and can stop or change his path.

Consider six arbitrary points in space. Every two points are joined by a segment. Prove that there are two triangles that can not be separated.

a) Prove that there are two polynomials in $ \mathbb Z[x]$ with at least one coefficient larger than 1387 such that coefficients of their product is in the set $ \{-1,0,1\}$. b) Does there exist a multiple of $ x^2-3x+1$ such that all of its coefficient are in the set $ \{-1,0,1\}$

=A subset $ S$ of $ \mathbb R^2$ is called an algebraic set if and only if there is a polynomial $ p(x,y)\in\mathbb R[x,y]$ such that \[ S = \{(x,y)\in\mathbb R^2|p(x,y) = 0\} \] Are the following subsets of plane an algebraic sets? 1. A square 2. A closed half-circle

a) Suppose that $ RBR'B'$ is a convex quadrilateral such that vertices $ R$ and $ R'$ have red color and vertices $ B$ and $ B'$ have blue color. We put $ k$ arbitrary points of colors blue and red in the quadrilateral such that no four of these $ k+4$ point (except probably $ RBR'B'$) lie one a circle. Prove that exactly one of the following cases occur? 1. There is a path from $ R$ to $ R'$ such that distance of every point on this path from one of red points is less than its distance from all blue points. 2. There is a path from $ B$ to $ B'$ such that distance of every point on this path from one of blue points is less than its distance from all red points. We call these two paths the blue path and the red path respectively. Let $ n$ be a natural number. Two people play the following game. At each step one player puts a point in quadrilateral satisfying the above conditions. First player only puts red point and second player only puts blue points. Game finishes when every player has put $ n$ points on the plane. First player's goal is to make a red path from $ R$ to $ R'$ and the second player's goal is to make a blue path from $ B$ to $ B'$. b) Prove that if $ RBR'B'$ is rectangle then for each $ n$ the second player wins. c) Try to specify the winner for other quadrilaterals.

There are five research labs on Mars. Is it always possible to divide Mars to five connected congruent regions such that each region contains exactly on research lab.

A graph is called a self-intersecting graph if it is isomorphic to a graph whose every edge is a segment and every two edges intersect. Notice that no edge contains a vertex except its two endings. a) Find all $ n$'s for which the cycle of length $ n$ is self-intersecting. b) Prove that in a self-intersecting graph $ |E(G)|\leq|V(G)|$. c) Find all self-intersecting graphs.

In an old script found in ruins of Perspolis is written: This script has been finished in a year whose 13th power is 258145266804692077858261512663 You should know that if you are skilled in Arithmetics you will know the year this script is finished easily. Find the year the script is finished. Give a reason for your answer.