OIM 2012

Let $ABCD$ be a rectangle. Construct equilateral triangles $BCX$ and $DCY$ , in such a way that both of these triangles share some of their interior points with some interior points of the rectangle. Line $AX$ intersects line $CD$ on $P$ , and line $AY$ intersects line $BC$ on $Q$ . Prove that triangle $APQ$ is equilateral.

A positive integer is called shiny if it can be written as the sum of two not necessarily distinct integers $a$ and $b$ which have the same sum of their digits. For instance, $2012$ is shiny, because $2012 = 2005 + 7$ , and both $2005$ and $7$ have the same sum of their digits. Find all positive integers which are not shiny (the dark integers).

Let $n$ to be a positive integer. Given a set $\{ a_1, a_2, \ldots, a_n \}$ of integers, where $a_i \in \{ 0, 1, 2, 3, \ldots, 2^n -1 \},$ $\forall i$ , we associate to each of its subsets the sum of its elements; particularly, the empty subset has sum of its elements equal to $0$ . If all of these sums have different remainders when divided by $2^n$ , we say that $\{ a_1, a_2, \ldots, a_n \}$ is $n$ -complete.

For each $n$ , find the number of $n$ -complete sets.