Determine the smallest possible value of $| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |$, where $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ sets simultaneously satisfying the following conditions: $(i)$ $| A_{i}\cap A_{j} | = 1$ for all $1\leq i < j\leq 5$, i.e. any two distinct sets contain exactly one element in common; $(ii)$ $A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing$ for all $1\leq i<j<k<l\leq 5$, i.e. any four different sets contain no common element. Where $| S |$ means the number of elements of $S$.

Given an isosceles triangle $ABC$ with base $AB$. Point $K$ is taken on the extension of the side $AC$ (beyond the point $C$ ) so that $\angle KBC = \angle ABC$. Denote $S$ the intersection point of angle - bisectors of $\angle BKC$ and $\angle ACB$. Lines $AB$ and $KS$ intersect at point $L$, lines $BS$ and $CL$ intersect at point $M$ . Prove that line $KM$ passes through the midpoint of the segment $BC$.

For all $a,b,c\in R^+ $ such that $a+b+c=1$ and $$ ( \frac{1}{(a+b)^2}+\frac{1}{(b+c)^2}+\frac{1}{(c+a)^2} )(a-bc)(b-ac)(c-ab)\le M \cdot abc.$$Find min $M$.

Prove that there are infinitely many primes representable in the form $m^2+mn+n^2$ for some integers $m,n$ .