On every face of a cube we draw one of the two diagonals. Some of these diagonals share a vertex. Let $N$ denote the number of pairs of diagonals that have a vertex in common (each diagonal can occur in several different pairs). Find the greatest and the smallest possible values of $N$.
In a quadrilateral $ABCD$ let $K$ be a point inside the triangle $ABD$ such that triangles $ABD$ and $KCD$ are similar. Prove that triangles $BCD$ and $AKD$ are similar as well.
The sides of an equilateral triangle $ABC$ measure $4$ cm. Denote the orthogonal projections of the midpoint $D$ of the side $AB$ onto the sides $BC$ and $AC$ by $E$ and $F$. Find the area of the triangle $DEF$.
At the National Mathematical Olympiad the students were given 4 problems. Each solution was awarded with an integral number of points between 0 and 7. There were 42 participants and exactly half of them achieved at least 50\% of the points. To win the award one had to get at least 22 points and one sixth of the contestants achieved this. Together, the students who did not get the award had three times as many points as those who did. Prove that there were at least 6 contestants such that each of them got at least 25\% of the points but none of them got more than 50\%.
Let $ABCD$ be a convex quadrilateral such that the triangle $BCD$ is acute and $|AB| = |AD|$. Denote the intersection of the bisector of the angle $\angle CAD$ with the side $CD$ by $K$ and the intersection of the bisector of the angle $\angle BAC$ with the side $BC$ by $L$. Let $K'$ and $L'$ be the orthogonal projections of $K$ and $L$ onto the sides $BC$ and $CD$, respectively. Prove that $B$, $D$, $L'$ and $K'$ are concyclic.
Let $a_1, a_2, \dots, a_n$ be positive integers. Assume that for some positive integer $k$, $k \lt n$, the following is true: if we choose any $k$ of the given $n$ numbers their sum is divisible by $n$. Prove that $a_1 + a_2 + \dots + a_n$ is also divisible by $n$.
Let $ABC$ be an isosceles triangle with the apex at $C$ and let $A'$ be the foot of the altitude from $A$. Assuming that $|CA'| = \frac{1}{2}|AB|$, prove that the triangle $ABC$ is equilateral.
Let $D$ be a point on the side $AB$ of an acute triangle $ABC$ such that the triangle $BCD$ is also acute. Denote the orthocentre of the triangle $BCD$ by $H$. Prove: if points $A$, $D$, $H$ and $C$ are concyclic, then the triangle $ABC$ is isosceles.
Maja can write positive integers onto a blackboard according to two rules. For every number $n$ that is already on the board, she can write down $3n+13$. Whenever one of the numbers on the board is a perfect square, she can also write down its square root.
a. Using only this two rules, can she get the number $55$ assuming she starts with $256$?
b. Using only this two rules, can she get the number $256$ assuming she starts with $55$?
Let $ABC$ be an acute triangle. Denote its orthocentre by $H$ and let $A'$, $B'$ and $C'$ be the feet of the altitudes from $A$, $B$ and $C$. Let $P$ be the midpoint of $AH$, let $Q$ be the intersection of lines $B'P$ and $AB$ and denote the intersection of segments $A'C'$ and $BB'$ by $R$. Prove that the line $QR$ is perpendicular to the side $BC$.
On every square of an $n \times n$ chessboard we write one of the numbers $1$ or $-1$. Let $a_k$ be the product of all numbers in the $k^{\text{th}}$ row and $b_l$ be the product of all numbers in the $l^{\text{th}}$ column. Assuming $n = 2007$, can we choose the numbers in such a way that the sum
$$
a_1 + a_2 + \dots + a_n + b_1 + b_2 + \dots + b_n
$$
will be zero? What about $n = 2008$?