MEMO 2012

Let $K$ be the midpoint of the side $AB$ of a given triangle $ABC$ . Let $L$ and $M$ be points on the sides $AC$ and $BC$ , respectively, such that $\angle CLK = \angle KMC$ . Prove that the perpendiculars to the sides $AB, AC,$ and $BC$ passing through $K,L,$ and $M$ , respectively, are concurrent.

Let $ABCD$ be a convex quadrilateral with no pair of parallel sides, such that $\angle ABC = \angle CDA$ . Assume that the intersections of the pairs of neighbouring angle bisectors of $ABCD$ form a convex quadrilateral $EFGH$ . Let $K$ be the intersection of the diagonals of $EFGH$ . Prove that the lines $AB$ and $CD$ intersect on the circumcircle of the triangle $BKD$ .

Find all triplets $(x,y,z)$ of positive integers such that $$x^y + y^x = z^y$$$$x^y + 2012 = y^{z+1}$$

For any positive integer $n$ let $d(n)$ denote the number of positive divisors of $n$ . Do there exist positive integers $a$ and $b$ , such that $d(a)=d(b)$ and $d(a^2 ) = d(b^2 )$ , but $d(a^3 ) \ne d(b^3 )$ ?