Školjka

%V0 Let $a$, $b$, $c$ be positive real numbers such that $abc = 1$. Prove that $$\frac {1}{a^{3}\left(b + c\right)} + \frac {1}{b^{3}\left(c + a\right)} + \frac {1}{c^{3}\left(a + b\right)}\geq \frac {3}{2}.$$

%V0 Let $A,B,C,D$ be four distinct points on a line, in that order. The circles with diameters $AC$ and $BD$ intersect at $X$ and $Y$. The line $XY$ meets $BC$ at $Z$. Let $P$ be a point on the line $XY$ other than $Z$. The line $CP$ intersects the circle with diameter $AC$ at $C$ and $M$, and the line $BP$ intersects the circle with diameter $BD$ at $B$ and $N$. Prove that the lines $AM,DN,XY$ are concurrent.

%V0 Let $ABCDEF$ be a convex hexagon with $AB = BC = CD$ and $DE = EF = FA$, such that $\angle BCD = \angle EFA = \frac {\pi}{3}$. Suppose $G$ and $H$ are points in the interior of the hexagon such that $\angle AGB = \angle DHE = \frac {2\pi}{3}$. Prove that $AG + GB + GH + DH + HE \geq CF$.

%V0 Determine all integers $n > 3$ for which there exist $n$ points $A_{1},\cdots ,A_{n}$ in the plane, no three collinear, and real numbers $r_{1},\cdots ,r_{n}$ such that for $1\leq i < j < k\leq n$, the area of $\triangle A_{i}A_{j}A_{k}$ is $r_{i} + r_{j} + r_{k}$.

%V0 At a meeting of $12k$ people, each person exchanges greetings with exactly $3k+6$ others. For any two people, the number who exchange greetings with both is the same. How many people are at the meeting?

%V0 Find the maximum value of $x_{0}$ for which there exists a sequence $x_{0},x_{1}\cdots ,x_{1995}$ of positive reals with $x_{0} = x_{1995}$, such that $$x_{i - 1} + \frac {2}{x_{i - 1}} = 2x_{i} + \frac {1}{x_{i}},$$ for all $i = 1,\cdots ,1995$.