We have an 8x8 chessboard with 64 squares. Then we have 3x1 dominoes which cover exactly 3 squares. Such dominoes can only be moved parallel to the borders of the chessboard and also only if the passing squares are free. If no dominoes can be moved, then the position is called stable. a. Find the smalles number of covered squares neccessary for a stable position. b. Prove: There exist a stable position with only one square uncovered. c. Find all Squares which are uncoverd in at least one position of b).

[b]6.[/b] Consider a sequence $\{ a_n \}_{n=1}^{\infty}$ such that, for any convergent subsequence $\{ a_{n_k} \}$ of $\{a_n\}$, the sequence $\{ a_{n_k +1} \}$ also is convergent and has the same limit as $\{ a_{n_k}\}$. Prove that the sequence $\{ a_n \}$ is either convergent of has infinitely many accumulation points the set of which is dense in itself. Give an example for the second case. (A sequence $ x_n \to \infty $ or $-\infty$ is considered to be convergente, too) [b](S. 13)[/b]

Let $ABC$ be an acute triangle with circumcircle $\Gamma$. Let the internal angle bisector of $\angle BAC$ intersect $BC$ and $\Gamma$ at $E$ and $N$, respectively. Let $A'$ be the antipode of $A$ on $\Gamma$ and let $V$ be the point where $AA'$ intersects $BC$. Given that $EV=6$, $VA'=7$, and $A'N=9$, compute the radius of $\Gamma$. [i]Proposed by James Lin.[/i]

The sizes of the freshmen class and the sophomore class are in the ratio $5:4$. The sizes of the sophomore class and the junior class are in the ratio $7:8$. The sizes of the junior class and the senior class are in the ratio $9:7$. If these four classes together have a total of $2158$ students, how many of the students are freshmen?

Let $ABC$ be a triangle with $AB<AC$. There are two points $X$ and $Y$ on the angle bisector of $B\widehat AC$ such that $X$ is between $A$ and $Y$ and $BX$ is parallel to $CY$. Let $Z$ be the reflection of $X$ with respect to $BC$. Line $YZ$ cuts line $BC$ at point $P$. If line $BY$ cuts line $CX$ at point $K$, prove that $KA=KP$.

Let $n\ge 2$ be an integer. We call an ordered $n$-tuple of integers primitive if the greatest common divisor of its components is $1$. Prove that for every finite set $H$ of primitive $n$-tuples, there exists a non-constant homogenous polynomial $f(x_1,x_2,\ldots,x_n)$ with integer coefficients whose value is $1$ at every $n$-tuple in $H$. [i]Based on the sixth problem of the 58th IMO, Brazil[/i]

$(NET 3)$ Let $x_1, x_2, x_3, x_4,$ and $x_5$ be positive integers satisfying \[x_1 +x_2 +x_3 +x_4 +x_5 = 1000,\] \[x_1 -x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 +x_2 -x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 +x_3 -x_4 +x_5 > 0,\] \[x_1 -x_2 +x_3 +x_4 -x_5 > 0,\] \[-x_1 +x_2 -x_3 +x_4 +x_5 > 0\] $(a)$ Find the maximum of $(x_1 + x_3)^{x_2+x_4}$ $(b)$ In how many different ways can we choose $x_1, . . . , x_5$ to obtain the desired maximum?

Determine all pairs of positive integers $(a,n)$ with $a\ge n\ge 2$ for which $(a+1)^n+a-1$ is a power of $2$.

Let $f : \left[ 0,1 \right] \to \mathbb R$ be an integrable function such that \[ \int_0^1 f(x) \, dx = \int_0^1 x f(x) \, dx = 1 . \] Prove that \[ \int_0^1 f^2 (x) \, dx \geq 4 . \] [i]Ion Rasa[/i]

Compute the number of real numbers $t$ such that \[t = 50 \sin(t - \lfloor t \rfloor).\] Here $\lfloor \cdot\rfloor$ denotes the greatest integer function. [i]Proposed by David Altizio[/i]

Point $D$ is an arbitary point on side $BC$ of triangle $ABC$. $I$,$I_1$ and$I_2$ are the incenters of triangles $ABC$,$ABD$ and $ACD$ respectively. $M\not=A$ and $N\not=A$ are the intersections of circumcircle of triangle $ABC$ and circumcircles of triangles $IAI_1$ and $IAI_2$ respectively. Prove that regardless of point $D$, line $MN$ goes through a fixed point.

Define the operation "$\circ$" by $x \circ y=4x-3y+xy$, for all real numbers $x$ and $y$. For how many real numbers $y$ does $3 \circ y=12$? $ \textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \text{more than}\ 4 $

Find all quadrilaterals $ABCD$ such that all four triangles $DAB$, $CDA$, $BCD$ and $ABC$ are similar to one-another. [i]Proposed by Morteza Saghafian[/i]

Given a scalene triangle $\vartriangle ABC$ with orthocenter $H$ and circumcenter $O$. The exterior angle bisector of $\angle BAC$ intersects circumcircle of $\vartriangle ABC$ at $N \ne A$. Let $D$ be another intersection of $HN$ and the circumcircle of $\vartriangle ABC$. The line passing through $O$, which is parallel to $AN$, intersects $AB,AC$ at $E, F$, respectively. Prove that $DH$ bisects the angle $\angle EDF$. [img]https://3.bp.blogspot.com/-F1mFwojG_I0/XnYNR8ofqSI/AAAAAAAALeo/zge24WF0EO8umPAaXprKAeXJHAj7pr6tQCK4BGAYYCw/s1600/imoc2019g5.png[/img]

Is there any real number $\alpha$ for which there exist two functions $f,g: \mathbb{N} \to \mathbb{N}$ such that $$\alpha=\lim_{n \to \infty} \frac{f(n)}{g(n)},$$ but the function which associates to $n$ the $n$-th decimal digit of $\alpha$ is not recursive?

Find all pairs of positive integers $(a,b)$ such that $a-b$ is a prime number and $ab$ is a perfect square.

On sides $BC$, $CA$, $AB$ of triangle $ABC$, points $A_1$, $B_1$, $C_1$ are chosen, respectively, so that the medians $A_1A_2$, $B_1B_2$, $C_1C_2$ of the triangle $A_1B_1C_1$ are respectively parallel to straight lines $AB$, $BC$, $CA$. Determine in what ratio points $A_1$, $B_1$, $C_1$ divide the sides of the triangle $ABC$.

In Pascal's Triangle, each entry is the sum of the two entries above it. The first few rows of the triangle are shown below. \[\begin{array}{c@{\hspace{8em}} c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{6pt}}c@{\hspace{4pt}}c@{\hspace{2pt}} c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{2pt}}c@{\hspace{3pt}}c@{\hspace{6pt}} c@{\hspace{6pt}}c@{\hspace{6pt}}c} \vspace{4pt} \text{Row 0: } & & & & & & & 1 & & & & & & \\\vspace{4pt} \text{Row 1: } & & & & & & 1 & & 1 & & & & & \\\vspace{4pt} \text{Row 2: } & & & & & 1 & & 2 & & 1 & & & & \\\vspace{4pt} \text{Row 3: } & & & & 1 & & 3 & & 3 & & 1 & & & \\\vspace{4pt} \text{Row 4: } & & & 1 & & 4 & & 6 & & 4 & & 1 & & \\\vspace{4pt} \text{Row 5: } & & 1 & & 5 & &10& &10 & & 5 & & 1 & \\\vspace{4pt} \text{Row 6: } & 1 & & 6 & &15& &20& &15 & & 6 & & 1 \end{array}\] In which row of Pascal's Triangle do three consecutive entries occur that are in the ratio $3: 4: 5$?

A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$.

For all $x,y,z\in \mathbb{R}\backslash \{1\}$, such that $xyz=1$, prove that \[ \frac{x^2}{(x-1)^2}+\frac{y^2}{(y-1)^2}+\frac{z^2}{(z-1)^2}\ge 1 \]

Wiebke and Stefan play the following game on a rectangular sheet of paper. They start with a rectangle with $60$ rows and $40$ columns and cut it in turns into smaller rectangles. The cuttings must be made along the gridlines, and a player in turn may cut only one smaller rectangle. By that, Stefan makes only vertical cuts, while Wiebke makes only horizontal cuts. A player who cannot make a regular move loses the game. (a) Who has a winning strategy if Stefan makes the first move? (b) Who has a winning strategy if Wiebke makes the first move?

A set $S$ has $7$ elements. Several $3$-elements subsets of $S$ are listed, such that any $2$ listed subsets have exactly $1$ common element. What is the maximum number of subsets that can be listed?

Three circle of unit radius passing through the point $P$ and one of the points of $A, B$ and $C$ each. What can be the radius of the circumcircle of the triangle $ABC$?

Let $ABCD$ be a parallelogram. Let $E$ and $F$ be the midpoints of sides $AB$ and $BC$ respectively. The lines $EC$ and $FD$ intersect at $P$ and form four triangles $APB, BPC, CPD, DPA$. If the area of the parallelogram is $100$, what is the maximum area of a triangles among these four triangles?

A cake in the shape of a rectangular prism has dimensions $6\text{ cm}\times14\text{ cm}\times21\text{ cm}$. It is cut into $1764$ equally sized cubes such that each cube is $1\text{ cm}^3$. Andy the ant starts at one corner of the cake and eats through the cake in a straight line to the opposite corner of the cake. How many of the $1\text{ cm}^3$ cubes does Andy bite through?