Inscribed Circle of rhombus $ABCD$ touches $AB,BC,CD,DA$ at $E,F,G,H$. $l_1,l_2$ are two lines that are tangent to the circle. $l_1\cap AB=M,l_1\cap BC=N,l_2\cap CD=P,l_2\cap DA=Q$. Prove that $MQ/\! /NP$.

Suppose we have a pack of $2n$ cards, in the order $1, 2, . . . , 2n$. A perfect shuffle of these cards changes the order to $n+1, 1, n+2, 2, . . ., n- 1, 2n, n$ ; i.e., the cards originally in the first $n$ positions have been moved to the places $2, 4, . . . , 2n$, while the remaining $n$ cards, in their original order, fill the odd positions $1, 3, . . . , 2n - 1.$ Suppose we start with the cards in the above order $1, 2, . . . , 2n$ and then successively apply perfect shuffles. What conditions on the number $n$ are necessary for the cards eventually to return to their original order? Justify your answer. [hide="Remark"] Remark. This problem is trivial. Alternatively, it may be required to find the least number of shuffles after which the cards will return to the original order.[/hide]

Find all primes $p,q$ such that \[p^q-q^p=pq^2-19\]

Find all functions $f: \mathbb R^{+} \to \mathbb R^{+}$ satisfying \begin{align*} f(f(x)+y) = f(y) + x, \qquad \text{for all } x,y \in \mathbb R^{+}. \end{align*} Note that $\mathbb R^{+}$ is the set of all positive real numbers.

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $a_1, a_2, \cdots, a_{2n}$ be $2n$ elements of $\{1, 2, 3, \cdots, 2n-1\}$ ($n>3$) with the sum $a_1+a_2+\cdots+a_{2n}=4n$. Prove that exist some numbers $a_i$ with the sum is $2n$.

Let $a_k$ be the number of nonnegative integers $ n $ with the properties: a) $n\in[0, 10^k)$ has exactly $ k $ digits, such that he zeroes on the first positions of $ n $ are included in the decimal writting. b) the digits of $ n $ can be permutated such that the new number is divisible by $11.$ Show that $a_{2m}=10a_{2m-1}$ for every $m\in\mathbb{N}.$

Let $ABC$ be a triangle with $ \widehat{BAC}\neq 90^\circ $. Let $M$ be the midpoint of $BC$. We choose a variable point $D$ on $AM$. Let $(O_1)$ and $(O_2)$ be two circle pass through $ D$ and tangent to $BC$ at $B$ and $C$. The line $BA$ and $CA$ intersect $(O_1),(O_2)$ at $ P,Q$ respectively. [b]a)[/b] Prove that tangent line at $P$ on $(O_1)$ and $Q$ on $(O_2)$ must intersect at $S$. [b]b)[/b] Prove that $S$ lies on a fix line.

Positive integers $a$ and $b$ are each less than 6. What is the smallest possible value for $2\cdot a-a\cdot b$? $\textbf{(A) }-20\qquad\textbf{(B) }-15\qquad\textbf{(C) }-10\qquad\textbf{(D) }0\qquad\textbf{(E) }2$

Find all pairs $(a, b)$ of positive integers for which $a + b = \phi (a) + \phi (b) + gcd (a, b)$. Here $ \phi (n)$ is the number of numbers $k$ from $\{1, 2,. . . , n\}$ with $gcd (n, k) = 1$.

If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then the sum of the first $4n$ positive integers is $\textbf{(A) }300\qquad\textbf{(B) }350\qquad\textbf{(C) }400\qquad\textbf{(D) }450\qquad \textbf{(E) }600$

In a certain country, every two cities are connected either by an airline route or by a railroad. Prove that one can always choose a type of transportation in such a way that each city can be reached from any other city with at most two transfers.

Let $X$ be a set which consists from $8$ consecutive positive integers. Set $X$ is divided on two disjoint subsets $A$ and $B$ with equal number of elements. If sum of squares of elements from set $A$ is equal to sum of squares of elements from set $B$, prove that sum of elements of set $A$ is equal to sum of elements of set $B$.

A positive integer is written on a blackboard. Players $A$ and $B$ play the following game: in each move one has to choose a proper divisor $m$ of the number $n$ written on the blackboard ($1<m<n$) and replaces $n$ with $n-m$. Player $A$ makes the first move, then players move alternately. The player who can't make a move loses the game. For which starting numbers is there a winning strategy for player $B$?

Determine the largest real number $c$ such that for any $2017$ real numbers $x_1, x_2, \dots, x_{2017}$, the inequality $$\sum_{i=1}^{2016}x_i(x_i+x_{i+1})\ge c\cdot x^2_{2017}$$ holds.

Let $ABC$ be a triangle and let $X$,$Y$,$Z$ be interior points on the sides $BC$, $CA$, $AB$, respectively. Show that the magnified image of the triangle $XYZ$ under a homothety of factor $4$ from its centroid covers at least one of the vertices $A$, $B$, $C$.

In a cone with height $3$ and base radius $4$, let $X$ be a point on the circumference of the base. Let $Y$ be a point on the surface of the cone such that the distance from $Y$ to the vertex of the cone is $2$, and $Y$ is diametrically opposite $X$ with respect to the base of the cone. The length of the shortest path across the surface of the cone from $X$ to $Y$ can be expressed as $\sqrt{a +\sqrt{b}}$, where a and b are positive integers. Find $a +b$.

The diagonals of a parallelogram $ABCD$ meet at point $O$. The tangent to the circumcircle of triangle $BOC$ at $O$ meets ray $CB$ at point $F$. The circumcircle of triangle $FOD$ meets $BC$ for the second time at point $G$. Prove that $AG=AB$.

Let $k>1$ and $n$ be natural numbers and $p=\frac{((n+1)(n+2)…(n+k))}{k!}-1$. Prove that, if $p$ is prime, then $n|k!$.

A quadrilateral whose perimeter is equal to $P$ is inscribed in a circle of radius $R$ and is circumscribed around a circle of radius $r$. Check whether the inequality $P\le \frac{r+\sqrt{r^2+4R^2}}{2}$ holds. Try to find the corresponding inequalities for the $n$-gon ($n \ge 5$) inscribed in a circle of radius $R$ and circumscribed around a circle of radius $r$.

Let $X$ be a set of $11$ integers. Prove that one can find a nonempty subset $\{a_1, a_2, \cdots , a_k \}$ of $X$ such that $3$ divides $k$ and $9$ divides the sum $\sum_{i=1}^{k} 4^i a_i$.

The point $D$ at the side $AB$ of triangle $ABC$ is given. Construct points $E,F$ at sides $BC, AC$ respectively such that the midpoints of $DE$ and $DF$ are collinear with $B$ and the midpoints of $DE$ and $EF$ are collinear with $C.$

Find all positive integers $n$ such that $2^n-n^2+1$ is a perfect square.

Which positive integers can be written in the form \[\frac{\operatorname{lcm}(x, y) + \operatorname{lcm}(y, z)}{\operatorname{lcm}(x, z)}\] for positive integers $x$, $y$, $z$?

Suppose that $a, b, c, k$ are natural numbers with $a, b, c \ge 3$ which fulfill the equation $abc = k^2 + 1$. Show that at least one between $a - 1, b - 1, c -1$ is composite number.