Školjka

%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 $n$.

%V0 We consider a point $P$ in a plane $p$ and a point $Q \not\in p$. Determine all the points $R$ from $p$ for which $$\frac{QP+PR}{QR}$$ is maximum.

%V0 When $4444^{4444}$ is written in decimal notation, the sum of its digits is $A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $B.$ ($A$ and $B$ are written in decimal notation.)

%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 sets of four real numbers $x_1, x_2, x_3, x_4$ such that the sum of any one and the product of the other three is equal to 2.

%V0 Solve the equation $\cos^2{x}+\cos^2{2x}+\cos^2{3x}=1$