MEMO 2014

Let $ABC$ be a triangle with $AB < AC$ . Its incircle with centre $I$ touches the sides $BC, CA,$ and $AB$ in the points $D, E,$ and $F$ respectively. The angle bisector $AI$ intersects the lines $DE$ and $DF$ in the points $X$ and $Y$ respectively. Let $Z$ be the foot of the altitude through $A$ with respect to $BC$ .

Prove that $D$ is the incentre of the triangle $XYZ$ .

Let the incircle $k$ of the triangle $ABC$ touch its side $BC$ at $D$ . Let the line $AD$ intersect $k$ at $L \neq D$ and denote the excentre of $ABC$ opposite to $A$ by $K$ . Let $M$ and $N$ be the midpoints of $BC$ and $KM$ respectively.

Prove that the points $B, C, N,$ and $L$ are concyclic.

A finite set of positive integers $A$ is called meanly if for each of its nonempy subsets the arithmetic mean of its elements is also a positive integer. In other words, $A$ is meanly if $\frac{1}{k}(a_1 + \dots + a_k)$ is an integer whenever $k \ge 1$ and $a_1, \dots, a_k \in A$ are distinct.

Given a positive integer $n$ , determine the least possible sum of the elements of a meanly $n$ -element set.

Determine all quadruples $(x,y,z,t)$ of positive integers such that $$20^x + 14^{2y} = (x + 2y + z)^{zt}.$$