Školjka

%V0 Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$ x f(y) + f( x f(y) ) - x f( f(y) ) - f(xy) = 2x + f(y) - f(x + y) $$ holds for all $x, y \in \mathbb{R}$.

%V0 We consider dissections of regular $n$-gons into $n - 2$ triangles by $n - 3$ diagonals which do not intersect inside the $n$-gon. A [i]bicoloured triangulation[/i] is such a dissection of an $n$-gon in which each triangle is coloured black or white and any two triangles which share an edge have different colours. We call a positive interger $n \geq 4$ [i]triangulable[/i] if every regular $n$-gon has a bicoloured triangulation such that for each vertex $A$ of the $n$-gon the number of black triangles of which $A$ is a vertex is greater that the number of white triangles of which $A$ is a vertex. Find all triangulable numbers.

%V0 Let $ABC$ be a triangle with $AB < AC$ and incentre $I$. Let $E$ be the point on the side $AC$ such that $AE = AB$. Let $G$ be the point on the line $EI$ such that $\angle IBG$ = $\angle CBA$ and such that $E$ and $G$ lie on opposite sides of $I$. Prove that the line $AI$, the perpendicular to $AE$ at $E$, and the bisector of the angle $\angle BGI$ are concurrent.

%V0 For integers $n \geq k \geq 0$ we define the [i]bibinomial coefficient[/i] $\displaystyle \left(\!\!\binom{n}{k}\!\!\right)$ by $$ \left(\!\!\binom{n}{k}\!\!\right) = \frac{n!!}{k!!(n - k)!!} \text{.} $$ Determine all pairs $(n, k)$ of integers with $n \geq k \geq 0$ such that the corresponding bibinomial coefficient is an integer. [i]Remark. The double factorial $n!!$ is defined to be the product of all even positive integers up to $n$ if $n$ is even and the product of all odd positive integers up to $n$ if $n$ is odd. So e.g. $0!! = 1$, $4!! = 2 \cdot 4 = 8$, and $7!! = 1 \cdot 3 \cdot 5 \cdot 7 = 105$.[/i]

%V0 Determine the lowest possible value of the expression $$ \frac{1}{a + x} + \frac{1}{a + y} + \frac{1}{b + x} + \frac{1}{b + y} \text{,} $$ where $a$, $b$, $x$, and $y$ are positive real numbers satisfying the inequalities $$ \frac{1}{a + x} \geq \frac12, \quad \frac{1}{a + y} \geq \frac12, \quad \frac{1}{b + x} \geq \frac12, \quad \text{and} \ \frac{1}{b + y} \geq 1 \text{.} $$

%V0 Determine all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$ x f(xy) + xy f(x) \geq f(x^2) f(y) + x^2y $$ holds for all $x, y \in \mathbb{R}$.

%V0 Let $K$ and $L$ be positive integers. On a board consisting of $2K \times 2L$ unit squares an ant starts in the lower left corner square and walks to the upper right corner square. In each step it goes horizontally or vertically to a neighbourning square. It never visits a square twice. At the end some squares may remain unvisited. In some cases the collection of all unvisited squares forms a single rectangle. In such cases, we call this rectangle [i]MEMOrable[/i]. Determine the number of different MEMOrable rectangles. [i]Remark. Rectangles are different unless they consist of exactly the same squares.[/i]

%V0 In Happy City there are $2014$ citizens called $A_1, A_2, \ldots, A_{2014}$. Each of them is either [i]happy[/i] or [i]unhappy[/i] at any moment in time. The mood of any citizen $A$ changes (from being unhappy to being happy or vice versa) if and only if some other happy citizen smiles at $A$. On Monday morning there were $N$ happy citizens in the city. The following happened on Monday during the day: citizen $A_1$ smiled at citizen $A_2$, then $A_2$ smiled at $A_3$, etc., and, finally, $A_{2013}$ smiled at $A_{2014}$. Nobody smiled at anyone else apart from this. Exactly the same repeated on Tuesday, Wednesday and Thursday. There were exactly $2000$ happy citizens on Thursday evening. Determine the largest possible value of $N$.

%V0 Let $ABC$ be a triangle with $AB < AC$. Its incircle with centre $I$ touches the sides $BC$, $CA$, and $AB$ in the points $D$, $E$, and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$. Prove that $D$ is the incentre of the triangle $XYZ$.

%V0 Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$. Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$. Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively. Prove that the points $B$, $C$, $N$, and $L$ are concyclic.

%V0 A finite set of positive integers $A$ is called [i]meanly[/i] if for each of its nonempty subsets the arithmetic mean of its elements is also a positive integer. In other words, A is meanly if $\frac{1}{k}(a_1 + \ldots + a_k)$ is an integer whenever $k \geq 1$ and $a_1, \ldots, a_k \in A$ are distinct. Given a positive integer $n$, determine the least possible sum of the elements of a meanly $n$-element set.

%V0 Determine all quadruples $(x, y, z, t)$ of positive integers such that $$ 20^x + 14^{2y} = (x + 2y + z)^{zt} \text{.} $$