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.

Let $a,b,c,d$ be integers such that the number $a-b+c-d$ is odd and it divides the number $a^2-b^2+c^2-d^2$. Show that, for every positive integer $n$, $a-b+c-d$ divides $a^n-b^n+c^n-d^n$.

Let $ABC$ be a triangle, $P$ and $Q$ the intersections of the parallel line to $BC$ that passes through $A$ with the external angle bisectors of angles $B$ and $C$, respectively. The perpendicular to $BP$ at $P$ and the perpendicular to $CQ$ at $Q$ meet at $R$. Let $I$ be the incenter of $ABC$. Show that $AI = AR$.

Show that, for every positive integer $n$, there exist $n$ consecutive positive integers such that none is divisible by the sum of its digits. (Alternative Formulation: Call a number good if it's not divisible by the sum of its digits. Show that for every positive integer $n$ there are $n$ consecutive good numbers.)