CZ-SK-MO 2018

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 $$\frac{1}{|x^2+2yz|}, \frac{1}{|y^2+2zx|}, \frac{1}{|z^2+2xy|}$$ 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 $$\frac{a^2+b^2-c^2}{a+b-c}, \frac{b^2+c^2-a^2}{b+c-a}, \frac{c^2+a^2-b^2}{c+a-b}$$ 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.