Consider a $ n \times n$ checkerboard with $ n > 1, n \in \mathbb{N}.$ How many possibilities are there to put $ 2n \minus{} 2$ identical pebbles on the checkerboard (each on a different field/place) such that no two pebbles are on the same checkerboard diagonal. Two pebbles are on the same checkerboard diagonal if the connection segment of the midpoints of the respective fields are parallel to one of the diagonals of the $ n \times n$ square.

Consider $ 8$ points that are a knight’s move away from the origin (i.e., the eight points $\{(2, 1)$ , $(2, -1)$ , $(1, 2)$ , $(1, -2)$ , $(-1, 2)$ , $(-1, -2)$ , $(-2, 1)$, $(-2, -1)\}$). Each point has probability $\frac12$ of being visible. What is the expected value of the area of the polygon formed by points that are visible? (If exactly $0, 1, 2$ points appear, this area will be zero.)

Given a point $M$ inside a right tetrahedron. Prove that at least one tetrahedron edge is seen from the $M$ in an angle, that has a cosine not greater than $-1/3$. (e.g. if $A$ and $B$ are the vertices, corresponding to that edge, $cos(\widehat{AMB}) \le -1/3$)

The operation $\#$ is defined by $x\#y=\frac{x-y}{xy}$. For how many real values $a$ is $a\#\left(a\#2\right)=1$? $\text{(A) }0\qquad\text{(B) }1\qquad\text{(C) }2\qquad\text{(D) }4\qquad\text{(E) infinitely many}$

There are 2000 cities in a country and no roads. Prove that some cities can be connected by a road such that there would be 2 cities with 1 road passing through them, there would be 2 cities with 2 roads passim through them,...,there would be 2 cities with 1000 roads passing through them. [I]Proposed by F. Bakharev[/i]

Can a regular octahedron be inscribed in a cube in such a way that all vertices of the octahedron are on cube's edges? (4)

A natural number is called [i]triangular[/i] if it may be presented in the form $\frac{n(n+1)}2$. Find all values of $a$ $(1\le a\le9)$ for which there exist a triangular number all digit of which are equal to $a$.

The numbers $25$ and $76$ have the property that when squared in base 10, their squares also end in the same two digits. A positive integer that has at most $3$ digits when expressed in base 21 and also has the property that its base $21$ square ends in the same $3$ digits is called amazing. Find the sum of all amazing numbers. Express your answer in base $21$.

In $\triangle{ABC}$, $AX=XY=YB=BC$, and $m\angle{ABC}=120^{\circ}$. What is $m\angle{BAC}$? [asy] pair A, B, C, X, Y; A = origin; X = dir(30); Y = X + dir(0); B = Y + dir(60); C = B + dir(330); draw(A--B--C--cycle); draw(X--Y--B); label("$A$",A,W); label("$B$",B,N); label("$C$",C,E); label("$X$",X,NW); label("$Y$",Y,SE); [/asy] $\text{(A) }15\qquad\text{(B) }20\qquad\text{(C) }25\qquad\text{(D) }30\qquad\text{(E) }35$

A graph $G$ has $n$ vertices ($n>1$). For each edge $e$ let $c(e)$ be the number of vertices of the largest complete subgraph containing $e$. Prove that the inequality (the summation is over all edges of $G$): \[\sum_{e} \frac{c(e)}{c(e)-1}\le \frac{n^2}{2}.\]

Prove that the product of three consecutive natural numbers, the middle of which is the cube of a natural number, is divisible by $ 504 $ .

$AL$ is internal bisector of scalene $\triangle ABC$ ($L \in BC$). $K$ is chosen on segment $AL$. Point $P$ lies on the same side with respect to line $BC$ as point $A$ such that $\angle BPL = \angle CKL$ and $\angle CPL = \angle BKL$. $M$ is midpoint of segment $KP$, and $D$ is foot of perpendicular from $K$ on $BC$. Prove that $\angle AMD = 180^\circ - |\angle ABC - \angle ACB|$. [i]Proposed by Mykhailo Shtandenko and Fedir Yudin[/i]

Prove that $\frac 12 \cdot \sqrt{4\sin^2 36^{\circ} - 1}=\cos 72^\circ$.

$4n$ first grade students at Songkhla Primary School, including $2n$ boys and $2n$ girls, participate in a taekwondo tournament where every pair of students compete against each other exactly once. The tournament is scored as follows: $\bullet$ In a match between two boys or between two girls, a win is worth $3$ points, a draw $1$ point, and a loss $0$ points. $\bullet$ In a math between a boy and a girl, if the boy wins, he receives $2$ points, else he receives $0$ points. If the girl wins, she receives $3$ points, if she draws, she receives $2$ points, and if she loses, she receives $0$ points. After the tournament, the total score of each student is calculated. Let $P$ be the number of matches ending in a draw, and let $Q$ be the total number of matches. Suppose that the maximum total score is $4n - 1$. Find $P/Q$.

Let $ABC$ be a triangle. Points $D$ and $E$ lie on sides $BC$ and $CA$, respectively, such that $BD=AE$. Segments $AD$ and $BE$ meet at $P$. The bisector of angle $BCA$ meet segments $AD$ and $BE$ at $Q$ and $R$, respectively. Prove that $\frac{PQ}{AD}=\frac{PR}{BE}$.

If for numbers $a$, $b$ and $c$ holds $a : b=4:3$ and $b : c=2:5$, find the value $$(3a-2b):(b+2c)$$

Find $a\in Z$ such that the equation $2x^2+2ax+a-1=0$ has integer solutions, which should be found.

Let $T$ be a $30-60-90$ triangle with hypotenuse of length $20$. Three circles, each externally tangent to the other two, have centers at the three vertices of $T$. The area of the union of the circles intersected with $T$ is $(m + n \sqrt{3}) \pi$ for rational numbers $m$ and $n$. Find $m + n$.

If $0 < v <\frac{\pi}{2}$ and $\tan v = 2v$, decide whether $sinv < \frac{20}{21}$.

The point $P$ is outside the circle $\omega$ with center $O$. Lines $\ell_1$ and $\ell_2$ pass through a point $P$, $\ell_1$ touches the circle $\omega$ at the point $A$ and $\ell_2$ intersects $\omega$ at the points $B$ and $C$. Tangent to the circle $\omega$ at points $B$ and $C$ intersect at point $Q$. Let $K$ be the point of intersection of the lines $BC$ and $AQ$. Prove that $(OK) \perp (PQ)$.

Let $ I,J $ be two intervals, $ \varphi :J\longrightarrow\mathbb{R} $ be a continuous function whose image doesn't contain $ 0, $ and $ f,g:I\longrightarrow J $ be two differentiable functions such that $ f'=\varphi\circ f,g'=\varphi\circ g $ and such that the image of $ f-g $ contains $ 0. $ Show that $ f $ and $ g $ are the same function.

Determine all triples of positive integers $a$, $ b$ and $c$ such that their least common multiple is equal to their sum.

Let $P$ be a point in the plane of $\triangle ABC$, and $\gamma$ a line passing through $P$. Let $A', B', C'$ be the points where the reflections of lines $PA, PB, PC$ with respect to $\gamma$ intersect lines $BC, AC, AB$ respectively. Prove that $A', B', C'$ are collinear.

Let $ABCD$ a convex quadrilateral. Suppose that the circumference with center $B$ and radius $BC$ is tangent to $AD$ in $F$ and the circumference with center $A$ and radius $AD$ is tangent to $BC$ in $E$. Prove that $DE$ and $CF$ are perpendicular.

Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$. .