In how many ways can 10001 be written as the sum of two primes? $ \textbf{(A)}0\qquad\textbf{(B)}1\qquad\textbf{(C)}2\qquad\textbf{(D)}3\qquad\textbf{(E)}4 $

There is an $n*n$ table with some unit cells colored black and the others are white. In each step , Amin takes a $row$ with exactly one black cell in it , and color all cells in that black cell's $column$ red. While Ali , takes a $column$ with exactly one black cell in it , and color all cells in that black cell's $row$ red. Prove that Amin can color all the cells red , iff Ali can do so.

Sides $AB$ and $CD$ of quadrilateral $ABCD$ intersect at point $E$. On the diagonals$ AC$ and $BD$ points $M$ and $N$ are taken, respectively, so that $AM / AC = BN / BD = k$. Find the area of ​​a triangle $EMN$ if the area of ​​$ABCD$ is $S$.

Let $R$ be a set of exactly $6$ elements. A set $F$ of subsets of $R$ is called an $S$-family over $R$ if and only if it satisfies the following three conditions: (i) For no two sets $X, Y$ in $F$ is $X \subseteq Y$ ; (ii) For any three sets $X, Y,Z$ in $F$, $X \cup Y \cup Z \neq R,$ (iii) $\bigcup_{X \in F} X = R$

Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$? $ \textbf{(A) }60 \qquad \textbf{(B) }65 \qquad \textbf{(C) }70 \qquad \textbf{(D) }75 \qquad \textbf{(E) }80 \qquad $

Consider the following operation on positive real numbers written on a blackboard: Choose a number $ r$ written on the blackboard, erase that number, and then write a pair of positive real numbers $ a$ and $ b$ satisfying the condition $ 2 r^2 \equal{} ab$ on the board. Assume that you start out with just one positive real number $ r$ on the blackboard, and apply this operation $ k^2 \minus{} 1$ times to end up with $ k^2$ positive real numbers, not necessarily distinct. Show that there exists a number on the board which does not exceed kr.

Let set $ T \equal{} \{1,2,3,4,5,6,7,8\}$. Find the number of all nonempty subsets $ A$ of $ T$ such that $ 3|S(A)$ and $ 5\nmid S(A)$, where $ S(A)$ is the sum of all the elements in $ A$.

It is known that circle $\Gamma_1(O_1)$ has center at $O_1$, circle $\Gamma_2(O_2)$ has center at $O_2$, and both intersect at points $C$ and $D$. It is also known that points $P$ and $Q$ lie on circles $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$, respectively. ). A line $\ell$ passes through point $D$ and intersects $\Gamma_1(O_1)$ and $\Gamma_2(O_2)$ at points $A$ and $B$, respectively. The lines $PD$ and $AC$ meet at point $M$, and the lines $QD$ and $BC$ meet at point $N$. Let $O$ be center outer circle of triangle $ABC$. Prove that $OD$ is perpendicular to $MN$ if and only if a circle can be found which passes through the points $P, Q, M$ and $N$.

It is known that a circle can be inscribed in the quadrilateral $ABCD$, in addition $\angle A = \angle C$. Prove that $AB = BC$, $CD = DA$. (Olena Artemchuk)

Let $a_1, a_2,\cdots , a_n$ and $b_1, b_2,\cdots , b_n$ be (not necessarily distinct) positive integers. We continue the sequences as follows: For every $i>n$, $a_i$ is the smallest positive integer which is not among $b_1, b_2,\cdots , b_{i-1}$, and $b_i$ is the smallest positive integer which is not among $a_1, a_2,\cdots , a_{i-1}$. Prove that there exists $N$ such that for every $i>N$ we have $a_i=b_i$ or for every $i>N$ we have $a_{i+1}=a_i$.

A strange calculator has only two buttons with positive itegers, each of them consisting of two digits. It displays the number 1 at the beginning. Whenever a button with number $N$ is pressed, the calculator replaces the displayed number $X$ with the number $X\cdot N$ or $X+N$. Multiplication and addition alternate, multiplication is the first. (For example,if the number 10 is on the 1st button, the number 20 is on the 2nd button, and we consecutively press the 1st, 2nd, 1st and 1st button, we get the results $1\cdot 10=10$, $10+20=30$, $30\cdot 10=300$, and $300+10=310$.) Decide whether there exist particular values of the two-digit nubers on the buttons such that one can display infinitely many numbers (without cleaning the display, i.e. you must keep going and get infinitel many numbers) ending with (a) $2015$, (b) $5813$. [i]Proposed by Michal Rolínek and Peter Novotný[/i]

Acute-angled triangle $ABC$ is inscribed in a circle with center at $O$; $\stackrel \frown {AB} = 120$ and $\stackrel \frown {BC} = 72$. A point $E$ is taken in minor arc $AC$ such that $OE$ is perpendicular to $AC$. Then the ratio of the magnitudes of angles $OBE$ and $BAC$ is: $\textbf{(A)}\ \dfrac{5}{18} \qquad \textbf{(B)}\ \dfrac{2}{9} \qquad \textbf{(C)}\ \dfrac{1}{4} \qquad \textbf{(D)}\ \dfrac{1}{3} \qquad \textbf{(E)}\ \dfrac{4}{9}$

$3$ builders are scheduled to build a house in $60$ days. However, they suffer from a bout of procrastination and thus do nothing for the first $50$ days. Panicked, they realize in order to build the house on time, they must hire more workers [i]and[/i] work twice as fast as they would have originally. If the new workers they hire also will work at the doubled rate, how many new workers will they need to hire? Assume each builder works at the same rate as the others and they do not get in each other's way.

The vertices of a regular nonagon are colored such that $1)$ adjacent vertices are different colors and $2)$ if $3$ vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)

For which real numbers $a$ does the sequence $(u_n )$ defined by the initial condition $u_0 =a$ and the recursion $u_{n+1} =2u_n - n^2$ have $u_n >0$ for all $n \geq 0?$

There are 2023 employees in the office, each of them knowing exactly $1686$ of the others. For any pair of employees they either both know each other or both don’t know each other. Prove that we can find $7$ employees each of them knowing all $6$ others.

Real numbers $x, y$ satisfy the inequality $x^2 + y^2 \le 2$. Orove that $xy + 3 \ge 2x + 2y$

Let $O$ be the circumcenter of triangle $ABC$ and let $AD$ be the height from $A$ ($D\in BC$). Let $M,N,P$ and $Q$ be the midpoints of $AB,AC,BD$ and $CD$ respectively. Let $\mathcal{C}_1$ and $\mathcal{C}_2$ be the circumcircles of triangles $AMN$ and $POQ$. Prove that $\mathcal{C}_1\cap \mathcal{C}_2\cap AD\neq \emptyset$.

Let $a$, $b$, $c$ be positive real numbers satisfying $a^2<bc$. Prove that $b^3+ac^2>ab(a+c)$.

There are $ n$ markers, each with one side white and the other side black. In the beginning, these $ n$ markers are aligned in a row so that their white sides are all up. In each step, if possible, we choose a marker whose white side is up (but not one of the outermost markers), remove it, and reverse the closest marker to the left of it and also reverse the closest marker to the right of it. Prove that, by a finite sequence of such steps, one can achieve a state with only two markers remaining if and only if $ n \minus{} 1$ is not divisible by $ 3$. [i]Proposed by Dusan Dukic, Serbia[/i]

For which integers $n \ge 8$ is $n^{\frac{1}{n-7}}$ an integer?

Let $G$ is a graph and $x$ is a vertex of $G$. Define the transformation $\varphi_{x}$ over $G$ as deleting all incident edges with respect of $x$ and drawing the edges $xy$ such that $y\in G$ and $y$ is not connected with $x$ with edge in the beginning of the transformation. A graph $H$ is called $G-$[i]attainable[/i] if there exists a sequece of such transformations which transforms $G$ in $H.$ Let $n\in\mathbb{N}$ and $4|n.$ Prove that for each graph $G$ with $4n$ vertices and $n$ edges there exists $G-$[i]attainable[/i] graph with at least $9n^{2}/4$ triangles.

Find all nondecreasing functions $f:\mathbb R\to \mathbb R$ such that, for all $x,y\in \mathbb R$, $$f(f(x))+f(y)=f(x+f(y))+1.$$ [i]Proposed by Carl Schildkraut[/i]

Let $ABC$ be a triangle with all angles $\leq 120^{\circ}$. Let $F$ be the Fermat point of triangle $ABC$, that is, the interior point of $ABC$ such that $\angle AFB = \angle BFC = \angle CFA = 120^\circ$. For each one of the three triangles $BFC$, $CFA$ and $AFB$, draw its Euler line - that is, the line connecting its circumcenter and its centroid. Prove that these three Euler lines pass through one common point. [i]Remark.[/i] The Fermat point $F$ is also known as the [b]first Fermat point[/b] or the [b]first Toricelli point[/b] of triangle $ABC$. [i]Floor van Lamoen[/i]

Does there exist a convex polyhedron which has a triangular section (by a plane not passing through the vertices) and each vertex of the polyhedron belonging to (a) no less than $ 5$ faces? (b) exactly $5$ faces? (G. Galperin)