Let $a,b,c,d$ be positive real numbers. Prove that $$\frac{a}{b+2c+3d}+\frac{b}{c+2d+3a}+\frac{c}{d+2a+3b}+\frac{d}{a+2b+3c} \geq \frac{2}{3}.$$

Let $ABCD$ be a parallelogram. Construct the equilateral triangle $DCE$ on the side $DC$ and outside of parallelogram. Let $P$ be an arbitrary point in plane of $ABCD$. Show that $$PA+PB+AD \geq PE.$$

Find all sets of real numbers $\{a_1,a_2,\ldots, _{1375}\}$ such that $$2 \left( \sqrt{a_n - (n-1)}\right) \geq a_{n+1} - (n-1), \quad \forall n \in \{1,2,\ldots,1374\},$$ and $$2 \left( \sqrt{a_{1375}-1374} \right) \geq a_1 +1.$$

Show that there doesn't exist two infinite and separate sets $A,B$ of points such that (i) There are no three collinear points in $A \cup B$, (ii) The distance between every two points in $A \cup B$ is at least $1$, and (iii) There exists at least one point belonging to set $B$ in interior of each triangle which all of its vertices are chosen from the set $A$, and there exists at least one point belonging to set $A$ in interior of each triangle which all of its vertices are chosen from the set $B$.

Find all non-negative integer solutions of the equation $$2^x + 3^y = z^2 .$$

Let $ABCD$ be a convex quadrilateral. Construct the points $P,Q,R,$ and $S$ on continue of $AB,BC,CD,$ and $DA$, respectively, such that $$BP=CQ=DR=AS.$$ Show that if $PQRS$ is a square, then $ABCD$ is also a square.

Let $a_1 \geq a_2 \geq \cdots \geq a_n$ be $n$ real numbers such that $a_1^k +a_2^k + \cdots + a_n^k \geq 0$ for all positive integers $k$. Suppose that $p=\max\{|a_1|,|a_2|, \ldots,|a_n|\}$. Prove that $p=a_1$, and $$(x-a_1)(x-a_2)\cdots(x-a_n)\leq x^n-a_1^n \qquad \forall x>a_1.$$

Let $n$ be a positive integer and suppose that $\phi(n)=\frac{n}{k}$, where $k$ is the greatest perfect square such that $k \mid n$. Let $a_1,a_2,\ldots,a_n$ be $n$ positive integers such that $a_i=p_1^{a_1i} \cdot p_2^{a_2i} \cdots p_n^{a_ni}$, where $p_i$ are prime numbers and $a_{ji}$ are non-negative integers, $1 \leq i \leq n, 1 \leq j \leq n$. We know that $p_i\mid \phi(a_i)$, and if $p_i\mid \phi(a_j)$, then $p_j\mid \phi(a_i)$. Prove that there exist integers $k_1,k_2,\ldots,k_m$ with $1 \leq k_1 \leq k_2 \leq \cdots \leq k_m \leq n$ such that $$\phi(a_{k_{1}} \cdot a_{k_{2}} \cdots a_{k_{m}})=p_1 \cdot p_2 \cdots p_n.$$

Suppose that $S$ is a finite set of real numbers with the property that any two distinct elements of $S$ form an arithmetic progression with another element in $S$. Give an example of such a set with 5 elements and show that no such set exists with more than $5$ elements.

Consider a semicircle of center $O$ and diameter $AB$. A line intersects $AB$ at $M$ and the semicircle at $C$ and $D$ s.t. $MC>MD$ and $MB<MA$. The circumcircles od the $AOC$ and $BOD$ intersect again at $K$. Prove that $MK\perp KO$.

Suppose that $10$ points are given in the plane, such that among any five of them there are four lying on a circle. Find the minimum number of these points which must lie on a circle.

Determine all functions $f : \mathbb N_0 \rightarrow \mathbb N_0 - \{1\}$ such that $$f(n + 1) + f(n + 3) = f(n + 5)f(n + 7) - 1375, \qquad \forall n \in \mathbb N.$$

Let $O$ be the circumcenter and $H$ the orthocenter of an acute-angled triangle $ABC$ such that $BC>CA$. Let $F$ be the foot of the altitude $CH$ of triangle $ABC$. The perpendicular to the line $OF$ at the point $F$ intersects the line $AC$ at $P$. Prove that $\measuredangle FHP=\measuredangle BAC$.

Find all pairs $(p,q)$ of prime numbers such that $$m^{3pq} \equiv m \pmod{3pq} \qquad \forall m \in \mathbb Z.$$