Question 11
Let $A$, $B$ be two points on a plane and $M$ be the midpoint of $AB$. We firstly choose a point $P$ on the segment $AB$, other than $A$, $B$, $M$. At step $i$ we choose a red point $P_i$ then choose one of $A$ and $B$, call it $X_i$, and reflect $P_i$ with respect to $X_i$ to get $Q_i$, then color the midpoint of $Q_iX_i$ red. Is it possible that after a few steps we color the $M$ by red?