Školjka

%V0 Prove that $\cos{\frac{\pi}{7}}-\cos{\frac{2\pi}{7}}+\cos{\frac{3\pi}{7}}=\frac{1}{2}$

%V0 The variables $a,b,c,d,$ traverse, independently from each other, the set of positive real values. What are the values which the expression $$S= \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}$$ takes?

%V0 If $p$ and $q$ are natural numbers so that $$\frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319},$$ prove that $p$ is divisible with $1979$.

%V0 Let $A$ and $E$ be opposite vertices of an octagon. A frog starts at vertex $A.$ From any vertex except $E$ it jumps to one of the two adjacent vertices. When it reaches $E$ it stops. Let $a_n$ be the number of distinct paths of exactly $n$ jumps ending at $E$. Prove that: $$a_{2n-1}=0, \quad a_{2n}={(2+\sqrt2)^{n-1} - (2-\sqrt2)^{n-1} \over\sqrt2}.$$

%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 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 $n$.