In a group of people, there are some mutually friendly pairs. For positive integer $k\ge3$ we say the group is $k$-great, if every (unordered) $k$-tuple of people from the group can be seated around a round table it the way that all pairs of neighbors are mutually friendly. (Since this was the 67th year of CZE/SVK MO,) show that if the group is 6-great, then it is 7-great as well. Bonus (not included in the competition): Determine all positive integers $k\ge3$ for which, if the group is $k$-great, then it is $(k+1)$-great as well.

Let $x,y,z$ be real numbers such that the numbers $$\frac{1}{|x^2+2yz|},\quad\frac{1}{|y^2+2zx|},\quad\frac{1}{|z^2+2xy|}$$are lengths of sides of a (non-degenerate) triangle. Determine all possible values of $xy+yz+zx$.

In triangle $ABC$ let be $D$ an intersection of $BC$ and the $A$-angle bisector. Denote $E,F$ the circumcenters of $ABD$ and $ACD$ respectively. Assuming that the circumcenter of $AEF$ lies on the line $BC$ what is the possible size of the angle $BAC$ ?

Let $a,b,c$ be integers which are lengths of sides of a triangle, $\gcd(a,b,c)=1$ and all the values $$\frac{a^2+b^2-c^2}{a+b-c},\quad\frac{b^2+c^2-a^2}{b+c-a},\quad\frac{c^2+a^2-b^2}{c+a-b}$$are integers as well. Show that $(a+b-c)(b+c-a)(c+a-b)$ or $2(a+b-c)(b+c-a)(c+a-b)$ is a perfect square.

Let $ABCD$ an isosceles trapezoid with the longer base $AB$. Denote $I$ the incenter of $\Delta ABC$ and $J$ the excenter relative to the vertex $C$ of $\Delta ACD$. Show that the lines $IJ$ and $AB$ are parallel.

Determine the least positive integer $n$ with the following property – for every 3-coloring of numbers $1,2,\ldots,n$ there are two (different) numbers $a,b$ of the same color such that $|a-b|$ is a perfect square.