Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation?

Let $ABC$ denote a triangle with area $S$. Let $U$ be any point inside the triangle whose vertices are the midpoints of the sides of triangle $ABC$. Let $A'$, $B'$, $C'$ denote the reflections of $A$, $B$, $C$, respectively, about the point $U$. Prove that hexagon $AC'BA'CB'$ has area $2S$.

Find all pairs $(m, n)$ of positive integers for which $4 (mn +1)$ is divisible by $(m + n)^2$.

Let $M$ be a set of six distinct positive integers whose sum is $60$. These numbers are written on the faces of a cube, one number to each face. A move consists of choosing three faces of the cube that share a common vertex and adding $1$ to the numbers on those faces. Determine the number of sets $M$ for which it’s possible, after a finite number of moves, to produce a cube all of whose sides have the same number.