Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$, such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube.

Given is a triangle $ABC$. Let the points $P$ and $Q$ be on the sides $AB, AC$, respectively, so that $AP=AQ$, and $PQ$ passes through the incenter $I$. Let $(BPI)$ meet $(CQI)$ at $M$, $PM$ meets $BI$ at $D$ and $QM$ meets $CI$ at $E$. Prove that the line $MI$ passes through the midpoint of $DE$.

Initially the numbers $i^3-i$ for $i=2,3 \ldots 2n+1$ are written on a blackboard, where $n\geq 2$ is a positive integer. On one move we can delete three numbers $a, b, c$ and write the number $\frac{abc} {ab+bc+ca}$. Prove that when two numbers remain on the blackboard, their sum will be greater than $16$.

Given is a cube $3 \times 3 \times 3$ with $27$ unit cubes. In each such cube a positive integer is written. Call a $\textit {strip}$ a block $1 \times 1 \times 3$ of $3$ cubes. The numbers are written so that for each cube, its number is the sum of three other numbers, one from each of the three strips it is in. Prove that there are at least $16$ numbers that are at most $60$.

Outside of the trapezoid $ABCD$ with the smaller base $AB$ are constructed the squares $ADEF$ and $BCGH$. Prove that the perpendicular bisector of $AB$ passes through the midpoint of $FH$.

Determine the real numbers $x$, $y$, $z > 0$ for which $xyz \leq \min\left\{4(x - \frac{1}{y}), 4(y - \frac{1}{z}), 4(z - \frac{1}{x})\right\}$

Let $ABC$ be an acute-angled triangle with $BC > AB$, such that the points $A$, $H$, $I$ and $C$ are concyclic (where $H$ is the orthocenter and $I$ is the incenter of triangle $ABC$). The line $AC$ intersects the circumcircle of triangle $BHC$ at point $T$, and the line $BC$ intersects the circumcircle of triangle $AHC$ at point $P$. If the lines $PT$ and $HI$ are parallel, determine the measures of the angles of triangle $ABC$.

Let $ABCDEF$ be a regular hexagon of side length $2$. Let us construct parallels to its sides passing through its vertices and midpoints, which divide the hexagon into $24$ congruent equilateral triangles, whose vertices are called nodes. For each node $X$, we define its trio as the figure formed by three adjacent triangles with vertex $X$, such that their intersection is only $X$ and they are not congruent in pairs. a) Determine the maximum possible area of a trio. b) Show that there exists a node whose trios can cover the entire hexagon, and a node whose trios cannot cover the entire hexagon. c) Determine the total number of triangles associated with the hexagon.

Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$, $b$, $c > M$, such that $n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$ (where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).

Determine all natural numbers $n \geq 2$ with at most four natural divisors, which have the property that for any two distinct proper divisors $d_1$ and $d_2$ of $n$, the positive integer $d_1-d_2$ divides $n$.

Given is a positive integer $n \geq 2$ and three pairwise disjoint sets $A, B, C$, each of $n$ distinct real numbers. Denote by $a$ the number of triples $(x, y, z) \in A \times B \times C$ satisfying $x<y<z$ and let $b$ denote the number of triples $(x, y, z) \in A \times B \times C$ such that $x>y>z$. Prove that $n$ divides $a-b$.

Let the equilateral triangles $ABC$ and $DEF$ be congruent with the centers $O_1$, respectively $O_2$, so that segment $AB$ intersects segments $DE$ and $DF$ at $M, N$, and the segment $AC$ intersects the segments $DF$ and $EF$ at $P$ and $Q$, respectively. We denote by $I$ the intersection point of the bisectors of the angles $EMN$ and $DPQ$ and by $J$ the intersection of the bisectors of the angles $FNM$ and $EQP$. Prove that $IJ$ is the perpendicular bisector of the segment $O_1O_2$.

Let be $a$ be positive real number. Prove that there are no real numbers $b$ and $c$, with $b < c$, so that for any distinct numbers $x, y \in (b, c)$ we have $|\frac{x+y} {x-y}| \leq a$.

Let $a$ and $b$ be two distinct positive integers with the same parity. Prove that the fraction $\frac{a!+b!}{2^a}$ is not an integer.

Suppose that $a, b,$ and $c$ are positive real numbers such that $$a + b + c \ge \frac{1}{a} + \frac{1}{b} + \frac{1}{c}.$$Find the largest possible value of the expression $$\frac{a + b - c}{a^3 + b^3 + abc} + \frac{b + c - a}{b^3 + c^3 + abc} + \frac{c + a - b}{c^3 + a^3 + abc}.$$

Consider a grid with $n{}$ lines and $m{}$ columns $(n,m\in\mathbb{N},m,n\ge2)$ made of $n\cdot m \; 1\times1$ squares called ${cells}$. A ${snake}$ is a sequence of cells with the following properties: the first cell is on the first line of the grid and the last cell is on the last line of the grid, starting with the second cell each has a common side with the previous cell and is not above the previous cell. Define the ${length}$ of a snake as the number of cells it's made of. Find the arithmetic mean of the lengths of all the snakes from the grid.

Let $ABC$ be an acute triangle with $\angle B > \angle C$. On the circle $\mathcal{C}(O, R)$ circumscribed to this triangle points $D, E, J, K, S$ are chosen such that $A, E, J$ and $K$ are on the same side of the line $BC$, the diameter $DE$ is perpendicular on the chord $BC$, $S\in \overarc{EK},\overarc{AE}=\overarc{BJ}=\overarc{CK}=\dfrac{1}{4}\overarc{CE}$ . Let $\{F\}=AC\cap DE, \{M\}=BK\cap AD, \{P\}=BK\cap AC$ and $\{Q\}=CJ\cap BF$. If $\angle SMK =30^{\circ}$ and $\angle AQP = 90^{\circ}$, show that the line $MS$ is tangent to the circumscribed circle of triangle $AOF$.