Let $\{x_n\}$ and $\{y_n\}$ be two sequences of natural numbers defined as follow: $x_1 = 1, \,\,\, x_2 = 1, \,\,\, x_{n+2} = x_{n+1} + 2x_n$ for $n = 1, 2, 3, ...$ $y_1 = 1, \,\,\, y_2 = 7, \,\,\, y_{n+2} = 2y_{n+1} + 3y_n$ for $n = 1, 2, 3, ...$ Prove that, except for the case $x_1 = y_1 = 1$, there is no natural value that occurs in the two sequences.

Find a number with $3$ digits, knowing that the sum of its digits is $9$, their product is $24$ and also the number read from right to left is $\frac{27}{38}$ of the original.

Let $F$ be a point inside a convex pentagon $ABCDE$, and let $a_{1}$, $a_{2}$, $a_{3}$, $a_{4}$, $a_{5}$ denote the distances from $F$ to the lines $AB$, $BC$, $CD$, $DE$, $EA$, respectively. The points $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ are chosen on the inner bisectors of the angles $A$, $B$, $C$, $D$, $E$ of the pentagon respectively, so that $AF_{1} = AF$ , $BF_{2} = BF$ , $CF_{3} = CF$ , $DF_{4} = DF$ and $EF_{5} = EF$ . If the distances from $F_{1}$, $F_{2}$, $F_{3}$, $F_{4}$, $F_{5}$ to the lines $EA$, $AB$, $BC$, $CD$, $DE$ are $b_{1}$, $b_{2}$, $b_{3}$, $b_{4}$, $b_{5}$, respectively. Prove that $a_{1} + a_{2} + a_{3} + a_{4} + a_{5} \leq b_{1} + b_{2} + b_{3} + b_{4} + b_{5}$

Find all functions $ f : \mathbb{R} \rightarrow \mathbb{R} $ such that \[ f( xf(x) + 2y) = f(x^2)+f(y)+x+y-1 \] holds for all $ x, y \in \mathbb{R}$.

Let $ABC$ be an acute-angled, not equilateral triangle, where vertex $A$ lies on the perpendicular bisector of the segment $HO$, joining the orthocentre $H$ to the circumcentre $O$. Determine all possible values for the measure of angle $A$. (U.S.A. - 1989 IMO Shortlist)

Let $Q_n$ be the set of all $n$-tuples $x=(x_1,\ldots,x_n)$ with $x_i \in \{0,1,2 \}$, $i=1,2,\ldots,n$. A triple $(x,y,z)$ (where $x=(x_1,x_2,\ldots,x_n)$, $y=(y_1,y_2,\ldots,y_n)$, $z=(z_1,z_2,\ldots,z_n)$) of distinct elements of $Q_n$ is called a [i]good[/i] triple, if there exists at least one $i \in \{1,2, \ldots, n \}$, for which $\{x_i,y_i,z_i \}=\{0,1,2 \}$. A subset $A$ of $Q_n$ will be called a [i]good[/i] subset, if any three elements of $A$ form a [i]good[/i] triple. Prove that every [i]good[/i] subset of $Q_n$ contains at most $2 \cdot \left(\frac{3}{2}\right)^n$ elements.

How many integers between $80$ and $100$ are prime? $\textbf{(A) } 3\qquad\textbf{(B) } 4\qquad\textbf{(C) } 5\qquad\textbf{(D) } 6\qquad\textbf{(E) } 7$

A merchant offers a large group of items at $30\%$ off. Later, the merchant takes $20\%$ off these sale prices and claims that the final price of these items is $50\%$ off the original price. The total discount is $\text{(A)}\ 35\% \qquad \text{(B)}\ 44\% \qquad \text{(C)}\ 50\% \qquad \text{(D)}\ 56\% \qquad\text{(E)}\ 60\%$

A [i]site[/i] is any point $(x, y)$ in the plane such that $x$ and $y$ are both positive integers less than or equal to 20. Initially, each of the 400 sites is unoccupied. Amy and Ben take turns placing stones with Amy going first. On her turn, Amy places a new red stone on an unoccupied site such that the distance between any two sites occupied by red stones is not equal to $\sqrt{5}$. On his turn, Ben places a new blue stone on any unoccupied site. (A site occupied by a blue stone is allowed to be at any distance from any other occupied site.) They stop as soon as a player cannot place a stone. Find the greatest $K$ such that Amy can ensure that she places at least $K$ red stones, no matter how Ben places his blue stones. [i]Proposed by Gurgen Asatryan, Armenia[/i]

Point $O$ is the circumcenter of acute-angled triangle $ABC$, points $A_1,B_1, C_1$ are the bases of its altitudes. Points $A', B', C'$ lying on lines $OA_1, OB_1, OC_1$ respectively are such that quadrilaterals $AOBC', BOCA', COAB'$ are cyclic. Prove that the circumcircles of triangles $AA_1A', BB_1B', CC_1C'$ have a common point.

Let $ ABC$ be an equilateral triangle, take line $ t $ such that $ t\parallel BC $ and $ t $ passes through $ A $. Let point $ D $ be on side $ AC $ , the bisector of angle $ ABD $ intersects line $ t $ in point $ E $. Prove that $ BD = CD + AE $.

Find the number of 4-digit numbers with distinct digits chosen from the set $\{0,1,2,3,4,5\}$ in which no two adjacent digits are even.

Jorn wants to cheat at the role play: he intends to cheat the sides to re-label its two octahedra, so that each of the numbers from $1$ to $16$ has the same probability as the sum of the dice occurs. So that the game master does not notice this so easily, he only wants to use numbers from $0$ to $8$ , if necessary several times or not at all. Is this possible?

In the quadrilateral $ABCD$ the side $AB$ has length $7, BC$ length $14, CD$ length $26$, and $DA$ length $23$. Show that the diagonals are perpendicular. You may assume that the quadrilateral is convex (all internal angles are less than $180^o$).

For a sequence, one can perform the following operation: select three adjacent terms $a,b,c,$ and change it into $b,c,a.$ Determine all the possible positive integers $n\geq 3,$ such that after finite number of operation, the sequence $1,2,\cdots, n$ can be changed into $n,n-1,\cdots,1$ finally.

Mathematical clubs are popular among the inhabitants of the same city. Every two of them they have at least one member in common. Prove that we can give the people of the city compasses and rulers so that only one inhabitant gets both, while each club will to have both a ruler and a compass at the full participation of its members.

Suppose that $n - 1$ items $A_1,A_2,...,A_{n-1}$ have already been arranged in the increasing order, and that another item $A_n$ is to be inserted to preserve the order. What is the expected number of comparisons necessary to insert $A_n$?

The sphere inscribed in a tetrahedron $ABCD$ touches face $ABC$ at point $H$. Another sphere touches face $ABC$ at $O$ and the planes containing the other three faces at points exterior to the faces. Prove that if $O$ is the circumcenter of triangle $ABC$, then $H$ is the orthocenter of that triangle.

Let $ a,b,c\in\mathbb{C}^* $ pairwise distinct, having the same absolute value, and satisfying: $$ a^2+b^2+c^2-ab-bc-ca=0. $$ Prove that $ a,b,c $ represents the affixes of the vertices of a right or equilateral triangle.

A triangle $ABC$ and spheres are given in space $S_1$ and $S_2$, each of which passes through points $A, B$ and $C$. For points $M$ spheres $S_1$ not lying in the plane of triangle $ABC$ are drawn lines $MA, MB$ and $MC$, intersecting the sphere $S_2$ for the second time at points $A_1,B_1$ and $C_1$, respectively. Prove that the planes passing through points $A_1, B_1$ and $C_1$, touch a fixed sphere or pass through a fixed point.

Let $ABCD$ be a rhombus with the condition $\angle ABC > 90^o$. The circle $\Gamma_B$ with center at $B$ goes through $C$, and the circle $\Gamma_C$ with center at $C$ goes through $B$. Denote by $E$ one of the intersection points of $\Gamma_B$ and $\Gamma_C$. The line $ED$ intersects intersects $\Gamma_B$ again at $F$. Find the value of $\angle AFB$.

Let $ n$ and $ m$ be given positive integers. In one move, a chess piece called an $ (n,m)$-crocodile goes $ n$ squares horizontally or vertically and then goes $ m$ squares in a perpendicular direction. Prove that the squares of an infinite chessboard can be painted in black and white so that this chess piece always moves from a black square to a white one or vice-versa.

For every natural number $n$, let $Q(n)$ denote the sum of the decimal digits of $n$. Prove that there are infinitely many positive integers $k$ with $Q(3^k) \ge Q(3^{k+1})$.

Circles $\omega_1$ and $\omega_2$ are externally tangent to each other at $P$. A random line $\ell$ cuts $\omega_1$ at $A$ and $C$ and $\omega_2$ at $B$ and $D$ (points $A,C,B,D$ are in this order on $\ell$). Line $AP$ meets $\omega_2$ again at $E$ and line $BP$ meets $\omega_1$ again at $F$. Prove that the radical axis of circles $(PCD)$ and $(PEF)$ is parallel to $\ell$. \\ \\ [i](Vlad Robu)[/i]

Solve for $x$: $2x+7=21$ $\textbf{(A) } 5\qquad\textbf{(B) } 6\qquad\textbf{(C) } 7\qquad\textbf{(D) } 8\qquad\textbf{(E) } 9$