Let $a, b, c$ be positive real numbers such that $abc = 8$. Prove that $\frac{ab + 4}{a + 2}+\frac{bc + 4}{b + 2}+\frac{ca + 4}{c + 2}\ge 6$.

Let $a,b,c $be positive real numbers.Prove that $\frac{8}{(a+b)^2 + 4abc} + \frac{8}{(b+c)^2 + 4abc} + \frac{8}{(a+c)^2 + 4abc} + a^2 + b^2 + c ^2 \ge \frac{8}{a+3} + \frac{8}{b+3} + \frac{8}{c+3}$.

Find all the pairs of integers $ (m, n)$ such that $ \sqrt {n +\sqrt {2016}} +\sqrt {m-\sqrt {2016}} \in \mathbb {Q}.$

If the non-negative reals $x,y,z$ satisfy $x^2+y^2+z^2=x+y+z$. Prove that $$\displaystyle\frac{x+1}{\sqrt{x^5+x+1}}+\frac{y+1}{\sqrt{y^5+y+1}}+\frac{z+1}{\sqrt{z^5+z+1}}\geq 3.$$When does the equality occur? Proposed by Dorlir Ahmeti, Albania

Let $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{xy+1}{2}}+\sqrt{\frac{yz+1}{2}}+\sqrt{\frac{zx+1}{2}} \ .\] Proposed by Azerbaijan Second Suggested VersionLet $x,y,z$ be positive real numbers such that $x+y+z=\frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Prove that \[x+y+z\geq \sqrt{\frac{x^2+1}{2}}+\sqrt{\frac{y^2+1}{2}}+\sqrt{\frac{z^2+1}{2}} \ .\]

Let $S_n$ be the sum of reciprocal values of non-zero digits of all positive integers up to (and including) $n$. For instance, $S_{13} = \frac{1}{1}+ \frac{1}{2}+ \frac{1}{3}+ \frac{1}{4}+ \frac{1}{5}+ \frac{1}{6}+ \frac{1}{7}+ \frac{1}{8}+ \frac{1}{9}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{1}+ \frac{1}{2}+ \frac{1}{1}+ \frac{1}{3}$ . Find the least positive integer $k$ making the number $k!\cdot S_{2016}$ an integer.

The natural numbers from $1$ to $50$ are written down on the blackboard. At least how many of them should be deleted, in order that the sum of any two of the remaining numbers is not a prime?

A $5 \times 5$ table is called regular f each of its cells contains one of four pairwise distinct real numbers,such that each of them occurs exactly one in every $2 \times 2$ subtable.The sum of all numbers of a regular table is called the total sum of the table.With any four numbers,one constructs all possible regular tables,computes their total sums and counts the distinct outcomes.Determine the maximum possible count.

A splitting of a planar polygon is a finite set of triangles whose interiors are pairwise disjoint, and whose union is the polygon in question. Given an integer $n \ge 3$, determine the largest integer $m$ such that no planar $n$-gon splits into less than $m$ triangles.

Let ${ABC}$ be an acute angled triangle, let ${O}$ be its circumcentre, and let ${D,E,F}$ be points on the sides ${BC,CA,AB}$, respectively. The circle ${(c_1)}$ of radius ${FA}$, centered at ${F}$, crosses the segment ${OA}$ at ${A'}$ and the circumcircle ${(c)}$ of the triangle ${ABC}$again at ${K}$. Similarly, the circle ${(c_2)}$ of radius $DB$, centered at $D$, crosses the segment $\left( OB \right)$ at ${B}'$ and the circle ${(c)}$ again at ${L}$. Finally, the circle ${(c_3)}$ of radius $EC$, centered at $E$, crosses the segment $\left( OC \right)$at ${C}'$ and the circle ${(c)}$ again at ${M}$. Prove that the quadrilaterals $BKF{A}',CLD{B}'$ and $AME{C}'$ are all cyclic, and their circumcircles share a common point. Evangelos Psychas (Greece)

Let ${ABC}$ be a triangle with $\angle BAC={{60}^{{}^\circ }}$. Let $D$ and $E$ be the feet of the perpendiculars from ${A}$ to the external angle bisectors of $\angle ABC$ and $\angle ACB$, respectively. Let ${O}$ be the circumcenter of the triangle ${ABC}$. Prove that the circumcircles of the triangles ${ADE}$and ${BOC}$ are tangent to each other.

A trapezoid $ABCD$ ($AB || CF$,$AB > CD$) is circumscribed.The incircle of the triangle $ABC$ touches the lines $AB$ and $AC$ at the points $M$ and $N$,respectively.Prove that the incenter of the trapezoid $ABCD$ lies on the line $MN$.

Let ${ABC}$ be an acute angled triangle whose shortest side is ${BC}$. Consider a variable point ${P}$ on the side ${BC}$, and let ${D}$ and ${E}$ be points on ${AB}$ and ${AC}$, respectively, such that ${BD=BP}$ and ${CP=CE}$. Prove that, as ${P}$ traces ${BC}$, the circumcircle of the triangle ${ADE}$ passes through a fixed point.

Let $ABC$ be an acute angled triangle with orthocenter ${H}$ and circumcenter ${O}$. Assume the circumcenter ${X}$ of ${BHC}$ lies on the circumcircle of ${ABC}$. Reflect $O$ across ${X}$ to obtain ${O'}$, and let the lines ${XH}$and ${O'A}$ meet at ${K}$. Let $L,M$ and $N$ be the midpoints of $\left[ XB \right],\left[ XC \right]$ and $\left[ BC \right]$, respectively. Prove that the points $K,L,M$ and ${N}$ are concyclic.

Given an acute triangle ${ABC}$, erect triangles ${ABD}$ and ${ACE}$ externally, so that ${\angle ADB= \angle AEC=90^o}$ and ${\angle BAD= \angle CAE}$. Let ${{A}_{1}}\in BC,{{B}_{1}}\in AC$ and ${{C}_{1}}\in AB$ be the feet of the altitudes of the triangle ${ABC}$, and let $K$ and ${K,L}$ be the midpoints of $[ B{{C}_{1}} ]$ and ${BC_1, CB_1}$, respectively. Prove that the circumcenters of the triangles $AKL,{{A}_{1}}{{B}_{1}}{{C}_{1}}$ and ${DEA_1}$ are collinear. (Bulgaria)

Let ${AB}$ be a chord of a circle ${(c)}$ centered at ${O}$, and let ${K}$ be a point on the segment ${AB}$ such that ${AK<BK}$. Two circles through ${K}$, internally tangent to ${(c)}$ at ${A}$ and ${B}$, respectively, meet again at ${L}$. Let ${P}$ be one of the points of intersection of the line ${KL}$ and the circle ${(c)}$, and let the lines ${AB}$ and ${LO}$ meet at ${M}$. Prove that the line ${MP}$ is tangent to the circle ${(c)}$. Theoklitos Paragyiou (Cyprus)

Determine the largest positive integer $n$ that divides $p^6 - 1$ for all primes $p > 7$.

Find the maximum number of natural numbers $x_1,x_2, ... , x_m$ satisfying the conditions: a) No $x_i - x_j , 1 \le i < j \le m$ is divisible by $11$, and b) The sum $x_2x_3 ...x_m + x_1x_3 ... x_m + \cdot \cdot \cdot + x_1x_2... x_{m-1}$ is divisible by $11$.

Find all positive integers $n$ such that the number $A_n =\frac{ 2^{4n+2}+1}{65}$ is a) an integer, b) a prime.

Find all triplets of integers $(a,b,c)$ such that the number $$N = \frac{(a-b)(b-c)(c-a)}{2} + 2$$is a power of $2016$. (A power of $2016$ is an integer of form $2016^n$,where n is a non-negative integer.)

Determine all four-digit numbers $\overline{abcd} $ such that $(a + b)(a + c)(a + d)(b + c)(b + d)(c + d) =\overline{abcd} $: