a) An ant is at vertex $A$ of a rectangle $ABCD$ and moves towards $B$ along the side of the rectangle. After a second movement, it goes in the same way to $C$. After three movements, in the same manner, it reaches $D$ and on the fourth, maintaining the path along the side, it returns to $A$. After 2018 similar movements to the previous ones, at which vertex will the ant be?

b) Ants Albormiga and Clarormiga start from vertices $A$ and $C$ of a rectangle, respectively, moving with constant speeds (even during the $90^{\circ}$ turns at each vertex). The first one moves clockwise and the second one counterclockwise. From $A$ to the first meeting, Albormiga traveled 360 meters. From the first to the second meeting, Clarormiga walked 300 meters. What is the perimeter of the rectangle?

c) Three ants are located at distinct vertices of a rectangle, leaving the other vertex free. An ant can move only along a line parallel to the line determined by the points where the other two ants are located, moving one at a time. Is it possible, after several movements of the ants, for all of them to be at the midpoint of three distinct sides of the rectangle?
Solution

a) Notice that every 4 moves, the ant returns to vertex $A$ (initial). Thus, since $2018=4 \cdot 504+2$, we can conclude that it completed 504 times the perimeter of the rectangle and then made two more moves, arriving at $C$.

b) Since they started from opposite points on the rectangle, they each traveled half the perimeter $2p$ until their first meeting. The distance $x$ traveled by Clarormiga plus the 360 meters traveled by Albormiga equals half the perimeter, that is,

$$
\frac{2 p}{2}=x+360 \Rightarrow 2 p=2 x+720
$$

Until the second meeting, together they covered $2p$. With constant speed, the time until the second meeting is double compared to the first. Thus, since Clarormiga traveled 300 meters between the meetings, in the first part, she moved half that distance, equivalent to 150 meters, which is the value of $x$. Finally, substituting, we get $2 p=2 \cdot 150+720=1020$ meters.

c) The ants $A, B$, and $C$ in their initial positions, if interpreted as points, can be connected to form a triangle with half the area of the rectangle. When $C$ moves along the line parallel to $A B$, the area of triangle $A B C$ does not change, the same happening with $A$ (along the line parallel to $B C$) and $B$ (along the line parallel to $A C$). If it were possible for all of them to stop at the midpoints of three distinct sides, this new triangle would have an area equal to one quarter of the area of the rectangle, and since the area is invariant, this cannot happen.