Školjka

%V0 Let $\mathbb{R} ^{+}$ denote the set of all positive real numbers. Find all functions $\mathbb{R} ^{+} \to \mathbb{R} ^{+}$ such that $$f(x+f(y)) = yf(xy+1)$$ holds for all $x, y \in \mathbb{R} ^{+}$.

%V0 Let $N$ be a positive integer. A set $S \subset \{ 1, 2, \cdots, N \}$ is called allowed if it does not contain three distinct elements $a, b, c$ such that $a$ divides $b$ and $b$ divides $c$. Determine the largest possible number of elements in an allowed set $S$.

%V0 In a given trapezium $ABCD$ with $AB$ parallel to $CD$ and $AB > CD$, the line $BD$ bisects the angle $\angle ADC$. The line through $C$ parallel to $AD$ meets the segments $BD$ and $AB$ in $E$ and $F$, respectively. Let $O$ be the circumcenter of the triangle $BEF$. Suppose that $\angle ACO = 60^{\circ}$. Prove the equality $$CF = AF + FO .$$

%V0 The sequence $\{ a_n \} _ { n \ge 0 }$ is defined by $a_0 = 2 , a_1 = 4$ and $$a_{n+1} = \frac{a_n a_{n-1}}{2} + a_n + a_{n-1}$$ for all positive integers $n$. Determine all prime numbers $p$ for which there exists a positive integer $m$ such that $p$ divides the number $a_m - 1$.

%V0 Find all triplets $(x,y,z)$ of real numbers such that $$2x^3 + 1 = 3zx$$ $$2y^3 + 1 = 3xy$$ $$2z^3 + 1 = 3yz$$

%V0 Let $a,b$ and $c$ be positive real numbers with $abc = 1$. Prove that $$\sqrt{ 9 + 16a^2}+\sqrt{ 9 + 16b^2}+\sqrt{ 9 + 16c^2} \ge 3 +4(a+b+c)$$

%V0 Let $n$ be a positive integer. Consider words of length $n$ composed of letters from the set $\{ M, E, O \}$. Let $a$ be the number of such words containing an even number (possibly 0) of blocks $ME$ and an even number (possibly 0) blocks of $MO$ . Similarly let $b$ the number of such words containing an odd number of blocks $ME$ and an odd number of blocks $MO$. Prove that $a>b$.

%V0 Let $p>2$ be a prime number. For any permutation $\pi = ( \pi(1) , \pi(2) , \cdots , \pi(p) )$ of the set $S = \{ 1, 2, \cdots , p \}$, let $f( \pi )$ denote the number of multiples of $p$ among the following $p$ numbers: $$\pi(1) , \pi(1) + \pi(2) , \cdots , \pi(1) + \pi(2) + \cdots + \pi(p)$$ Determine the average value of $f( \pi)$ taken over all permutations $\pi$ of $S$.

%V0 Let $K$ be the midpoint of the side $AB$ of a given triangle $ABC$. Let $L$ and $M$ be points on the sides $AC$ and $BC$, respectively, such that $\angle CLK = \angle KMC$. Prove that the perpendiculars to the sides $AB, AC,$ and $BC$ passing through $K,L,$ and $M$, respectively, are concurrent.

%V0 Let $ABCD$ be a convex quadrilateral with no pair of parallel sides, such that $\angle ABC = \angle CDA$. Assume that the intersections of the pairs of neighbouring angle bisectors of $ABCD$ form a convex quadrilateral $EFGH$. Let $K$ be the intersection of the diagonals of $EFGH$. Prove that the lines $AB$ and $CD$ intersect on the circumcircle of the triangle $BKD$.

%V0 Find all triplets $(x,y,z)$ of positive integers such that $$x^y + y^x = z^y$$ $$x^y + 2012 = y^{z+1}$$

%V0 For any positive integer $n$ let $d(n)$ denote the number of positive divisors of $n$. Do there exist positive integers $a$ and $b$, such that $d(a)=d(b)$ and $d(a^2 ) = d(b^2 )$, but $d(a^3 ) \ne d(b^3 )$ ?