Prove that the number of incidences of $n$ distinct points on $n$ distinct lines in plane is $\mathcal O (n^{\frac{4}{3}})$. Find a configuration for which $\Omega (n^{\frac{4}{3}})$ incidences happens.

Consider a set of $n$ points in plane. Prove that the number of isosceles triangles having their vertices among these $n$ points is $\mathcal O (n^{\frac{7}{3}})$. Find a configuration of $n$ points in plane such that the number of equilateral triangles with vertices among these $n$ points is $\Omega (n^2)$.

Prove that if $n$ is large enough, among any $n$ points of plane we can find $1000$ points such that these $1000$ points have pairwise distinct distances. Can you prove the assertion for $n^{\alpha}$ where $\alpha$ is a positive real number instead of $1000$?

Prove that from an $n\times n$ grid, one can find $\Omega (n^{\frac{5}{3}})$ points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of $\frac{5}{3}$!

Prove that for each coloring of the points inside or on the boundary of a square with $1391$ colors, there exists a monochromatic regular hexagon.

Suppose $W(k,2)$ is the smallest number such that if $n\ge W(k,2)$, for each coloring of the set $\{1,2,...,n\}$ with two colors there exists a monochromatic arithmetic progression of length $k$. Prove that $W(k,2)=\Omega (2^{\frac{k}{2}})$.

Prove that if $n$ is large enough, then for each coloring of the subsets of the set $\{1,2,...,n\}$ with $1391$ colors, two non-empty disjoint subsets $A$ and $B$ exist such that $A$, $B$ and $A\cup B$ are of the same color.

Prove that if $n$ is large enough, in every $n\times n$ square that a natural number is written on each one of its cells, one can find a subsquare from the main square such that the sum of the numbers is this subsquare is divisible by $1391$.

$P(x)$ is a nonzero polynomial with integer coefficients. Prove that there exists infinitely many prime numbers $q$ such that for some natural number $n$, $q|2^n+P(n)$. Proposed by Mohammad Gharakhani

Prove that there exists infinitely many pairs of rational numbers $(\frac{p_1}{q},\frac{p_2}{q})$ with $p_1,p_2,q\in \mathbb N$ with the following condition: \[|\sqrt{3}-\frac{p_1}{q}|<q^{-\frac{3}{2}}, |\sqrt{2}-\frac{p_2}{q}|< q^{-\frac{3}{2}}.\] Proposed by Mohammad Gharakhani

$p$ is an odd prime number. Prove that there exists a natural number $x$ such that $x$ and $4x$ are both primitive roots modulo $p$. Proposed by Mohammad Gharakhani

$P(x)$ and $Q(x)$ are two polynomials with integer coefficients such that $P(x)|Q(x)^2+1$. a) Prove that there exists polynomials $A(x)$ and $B(x)$ with rational coefficients and a rational number $c$ such that $P(x)=c(A(x)^2+B(x)^2)$. b) If $P(x)$ is a monic polynomial with integer coefficients, Prove that there exists two polynomials $A(x)$ and $B(x)$ with integer coefficients such that $P(x)$ can be written in the form of $A(x)^2+B(x)^2$. Proposed by Mohammad Gharakhani

Let $p$ be a prime number. We know that each natural number can be written in the form \[\sum_{i=0}^{t}a_ip^i (t,a_i \in \mathbb N\cup \{0\},0\le a_i\le p-1)\] Uniquely. Now let $T$ be the set of all the sums of the form \[\sum_{i=0}^{\infty}a_ip^i (0\le a_i \le p-1).\] (This means to allow numbers with an infinite base $p$ representation). So numbers that for some $N\in \mathbb N$ all the coefficients $a_i, i\ge N$ are zero are natural numbers. (In fact we can consider members of $T$ as sequences $(a_0,a_1,a_2,...)$ for which $\forall_{i\in \mathbb N}: 0\le a_i \le p-1$.) Now we generalize addition and multiplication of natural numbers to this set so that it becomes a ring (it's not necessary to prove this fact). For example: $1+(\sum_{i=0}^{\infty} (p-1)p^i)=1+(p-1)+(p-1)p+(p-1)p^2+...$ $=p+(p-1)p+(p-1)p^2+...=p^2+(p-1)p^2+(p-1)p^3+...$ $=p^3+(p-1)p^3+...=...$ So in this sum, coefficients of all the numbers $p^k, k\in \mathbb N$ are zero, so this sum is zero and thus we can conclude that $\sum_{i=0}^{\infty}(p-1)p^i$ is playing the role of $-1$ (the additive inverse of $1$) in this ring. As an example of multiplication consider \[(1+p)(1+p+p^2+p^3+...)=1+2p+2p^2+\cdots\] Suppose $p$ is $1$ modulo $4$. Prove that there exists $x\in T$ such that $x^2+1=0$. Proposed by Masoud Shafaei

Fixed points $B$ and $C$ are on a fixed circle $\omega$ and point $A$ varies on this circle. We call the midpoint of arc $BC$ (not containing $A$) $D$ and the orthocenter of the triangle $ABC$, $H$. Line $DH$ intersects circle $\omega$ again in $K$. Tangent in $A$ to circumcircle of triangle $AKH$ intersects line $DH$ and circle $\omega$ again in $L$ and $M$ respectively. Prove that the value of $\frac{AL}{AM}$ is constant. Proposed by Mehdi E'tesami Fard

Let the Nagel point of triangle $ABC$ be $N$. We draw lines from $B$ and $C$ to $N$ so that these lines intersect sides $AC$ and $AB$ in $D$ and $E$ respectively. $M$ and $T$ are midpoints of segments $BE$ and $CD$ respectively. $P$ is the second intersection point of circumcircles of triangles $BEN$ and $CDN$. $l_1$ and $l_2$ are perpendicular lines to $PM$ and $PT$ in points $M$ and $T$ respectively. Prove that lines $l_1$ and $l_2$ intersect on the circumcircle of triangle $ABC$. Proposed by Nima Hamidi

Cosider ellipse $\epsilon$ with two foci $A$ and $B$ such that the lengths of it's major axis and minor axis are $2a$ and $2b$ respectively. From a point $T$ outside of the ellipse, we draw two tangent lines $TP$ and $TQ$ to the ellipse $\epsilon$. Prove that \[\frac{TP}{TQ}\ge \frac{b}{a}.\] Proposed by Morteza Saghafian

The incircle of triangle $ABC$ for which $AB\neq AC$, is tangent to sides $BC,CA$ and $AB$ in points $D,E$ and $F$ respectively. Perpendicular from $D$ to $EF$ intersects side $AB$ at $X$, and the second intersection point of circumcircles of triangles $AEF$ and $ABC$ is $T$. Prove that $TX\perp TF$. Proposed By Pedram Safaei

Two fixed lines $l_1$ and $l_2$ are perpendicular to each other at a point $Y$. Points $X$ and $O$ are on $l_2$ and both are on one side of line $l_1$. We draw the circle $\omega$ with center $O$ and radius $OY$. A variable point $Z$ is on line $l_1$. Line $OZ$ cuts circle $\omega$ in $P$. Parallel to $XP$ from $O$ intersects $XZ$ in $S$. Find the locus of the point $S$. Proposed by Nima Hamidi

We've colored edges of $K_n$ with $n-1$ colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist? Proposed by Morteza Saghafian

Suppose $s,k,t\in \mathbb N$. We've colored each natural number with one of the $k$ colors, such that each color is used infinitely many times. We want to choose a subset $\mathcal A$ of $\mathbb N$ such that it has $t$ disjoint monochromatic $s$-element subsets. What is the minimum number of elements of $A$? Proposed by Navid Adham

In a tree with $n$ vertices, for each vertex $x_i$, denote the longest paths passing through it by $l_i^1,l_i^2,...,l_i^{k_i}$. $x_i$ cuts those longest paths into two parts with $(a_i^1,b_i^1),(a_i^2,b_i^2),...,(a_i^{k_i},b_i^{k_i})$ vertices respectively. If $\max_{j=1,...,k_i} \{a_i^j\times b_i^j\}=p_i$, find the maximum and minimum values of $\sum_{i=1}^{n} p_i$. Proposed by Sina Rezaei

a) Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $3$-element subset of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $3$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $3$-element subsets of it are green. b) Prove that for all $m,n\in \mathbb N$ there exists a natural number $a$ such that if we color every $k$-element subset ($k>3$) of the set $\mathcal A=\{1,2,3,...,a\}$ using $2$ colors red and green, there exists an $m$-element subset of $\mathcal A$ such that all $k$-element subsets of it are red or there exists an $n$-element subset of $\mathcal A$ such that all $k$-element subsets of it are green.

Suppose $0<m_1<...<m_n$ and $m_i \equiv i (\mod 2)$. Prove that the following polynomial has at most $n$ real roots. ($\forall 1\le i \le n: a_i \in \mathbb R$). \[a_0+a_1x^{m_1}+a_2x^{m_2}+...+a_nx^{m_n}.\]

Suppose $N\in \mathbb N$ is not a perfect square, hence we know that the continued fraction of $\sqrt{N}$ is of the form $\sqrt{N}=[a_0,\overline{a_1,a_2,...,a_n}]$. If $a_1\neq 1$ prove that $a_i\le 2a_0$.

Suppose $p$ is a prime number and $a,b,c \in \mathbb Q^+$ are rational numbers; a) Prove that $\mathbb Q(\sqrt[p]{a}+\sqrt[p]{b})=\mathbb Q(\sqrt[p]{a},\sqrt[p]{b})$. b) If $\sqrt[p]{b} \in \mathbb Q(\sqrt[p]{a})$, prove that for a nonnegative integer $k$ we have $\sqrt[p]{\frac{b}{a^k}}\in \mathbb Q$. c) If $\sqrt[p]{a}+\sqrt[p]{b}+\sqrt[p]{c} \in \mathbb Q$, then prove that numbers $\sqrt[p]{a},\sqrt[p]{b}$ and $\sqrt[p]{c}$ are rational.

Suppose $f(z)=z^n+a_1z^{n-1}+...+a_n$ for which $a_1,a_2,...,a_n\in \mathbb C$. Prove that the following polynomial has only one positive real root like $\alpha$ \[x^n+\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|\] and the following polynomial has only one positive real root like $\beta$ \[x^n-\Re(a_1)x^{n-1}-|a_2|x^{n-2}-...-|a_n|.\] And roots of the polynomial $f(z)$ satisfy $-\beta \le \Re(z) \le \alpha$.

Let $p$ be an odd prime number and let $a_1,a_2,...,a_n \in \mathbb Q^+$ be rational numbers. Prove that \[\mathbb Q(\sqrt[p]{a_1}+\sqrt[p]{a_2}+...+\sqrt[p]{a_n})=\mathbb Q(\sqrt[p]{a_1},\sqrt[p]{a_2},...,\sqrt[p]{a_n}).\]

Let $G$ be a simple undirected graph with vertices $v_1,v_2,...,v_n$. We denote the number of acyclic orientations of $G$ with $f(G)$. a) Prove that $f(G)\le f(G-v_1)+f(G-v_2)+...+f(G-v_n)$. b) Let $e$ be an edge of the graph $G$. Denote by $G'$ the graph obtained by omiting $e$ and making it's two endpoints as one vertex. Prove that $f(G)=f(G-e)+f(G')$. c) Prove that for each $\alpha >1$, there exists a graph $G$ and an edge $e$ of it such that $\frac{f(G)}{f(G-e)}<\alpha$. Proposed by Morteza Saghafian

Suppose $S$ is a convex figure in plane with area $10$. Consider a chord of length $3$ in $S$ and let $A$ and $B$ be two points on this chord which divide it into three equal parts. For a variable point $X$ in $S-\{A,B\}$, let $A'$ and $B'$ be the intersection points of rays $AX$ and $BX$ with the boundary of $S$. Let $S'$ be those points $X$ for which $AA'>\frac{1}{3} BB'$. Prove that the area of $S'$ is at least $6$. Proposed by Ali Khezeli

Prove that for each $n \in \mathbb N$ there exist natural numbers $a_1<a_2<...<a_n$ such that $\phi(a_1)>\phi(a_2)>...>\phi(a_n)$. Proposed by Amirhossein Gorzi

We have $n$ bags each having $100$ coins. All of the bags have $10$ gram coins except one of them which has $9$ gram coins. We have a balance which can show weights of things that have weight of at most $1$ kilogram. At least how many times shall we use the balance in order to find the different bag? Proposed By Hamidreza Ziarati

We call the three variable polynomial $P$ cyclic if $P(x,y,z)=P(y,z,x)$. Prove that cyclic three variable polynomials $P_1,P_2,P_3$ and $P_4$ exist such that for each cyclic three variable polynomial $P$, there exists a four variable polynomial $Q$ such that $P(x,y,z)=Q(P_1(x,y,z),P_2(x,y,z),P_3(x,y,z),P_4(x,y,z))$. Solution by Mostafa Eynollahzade and Erfan Salavati

a) Prove that $a>0$ exists such that for each natural number $n$, there exists a convex $n$-gon $P$ in plane with lattice points as vertices such that the area of $P$ is less than $an^3$. b) Prove that there exists $b>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $bn^2$. c) Prove that there exist $\alpha,c>0$ such that for each natural number $n$ and each $n$-gon $P$ in plane with lattice points as vertices, the area of $P$ is not less than $cn^{2+\alpha}$. Proposed by Mostafa Eynollahzade

The city of Bridge Village has some highways. Highways are closed curves that have intersections with each other or themselves in $4$-way crossroads. Mr.Bridge Lover, mayor of the city, wants to build a bridge on each crossroad in order to decrease the number of accidents. He wants to build the bridges in such a way that in each highway, cars pass above a bridge and under a bridge alternately. By knowing the number of highways determine that this action is possible or not. Proposed by Erfan Salavati

a) Does there exist an infinite subset $S$ of the natural numbers, such that $S\neq \mathbb{N}$, and such that for each natural number $n\not \in S$, exactly $n$ members of $S$ are coprime with $n$? b) Does there exist an infinite subset $S$ of the natural numbers, such that for each natural number $n\in S$, exactly $n$ members of $S$ are coprime with $n$? Proposed by Morteza Saghafian