SILK-ROAD 2011

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 $BC$ .

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