Školjka

%V0 Prove that for every natural number $n$, and for every real number $x \neq \frac{k\pi}{2^t}$ ($t=0,1, \dots, n$; $k$ any integer) $$\frac{1}{\sin{2x}}+\frac{1}{\sin{4x}}+\dots+\frac{1}{\sin{2^nx}}=\cot{x}-\cot{2^nx}$$

%V0 Find all positive integers $n$ such that the set $\{n,n+1,n+2,n+3,n+4,n+5\}$ can be partitioned into two subsets so that the product of the numbers in each subset is equal.

%V0 a.) For which $n>2$ is there a set of $n$ consecutive positive integers such that the largest number in the set is a divisor of the least common multiple of the remaining $n-1$ numbers? b.) For which $n>2$ is there exactly one set having this property?

%V0 Prove that if $n$ is a positive integer such that the equation $$x^3-3xy^2+y^3=n$$ has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

%V0 Given a set $M$ of $1985$ positive integers, none of which has a prime divisor larger than $26$, prove that the set has four distinct elements whose geometric mean is an integer.

%V0 An $n \times n$ matrix whose entries come from the set $S = \{1, 2, \ldots , 2n - 1\}$ is called a silver matrix if, for each $i = 1, 2, \ldots , n$, the $i$-th row and the $i$-th column together contain all elements of $S$. Show that: (a) there is no silver matrix for $n = 1997$; (b) silver matrices exist for infinitely many values of $n$.