1. Viale Marconi is $800 \mathrm{~m}$ long; $200 \mathrm{~m}$ from one end there is a parking meter, and $100 \mathrm{~m}$ from the opposite end there is a clothing store. Astolfo wants to park his car, get a ticket from the parking meter, return to the car to place it on the windshield, visit the store, and finally return to the car. Moreover, he is lazy and therefore wants to walk as little as possible; where should he park to do so? Indicate the set of points on the street that minimize the distance Astolfo has to walk.
(A) The point in front of the parking meter
(B) The point in front of the store
(C) The midpoint between the parking meter and the store

(D) All points on the street

(E) All points on the section of the street between the parking meter and the store.
1. The answer is (E). First, note that the distance between the parking meter and the store is 500 meters. If Astolfo parks at a point between the parking meter and the store, say at a distance $x$ from the parking meter (and therefore a distance $500 m - x$ from the store), he will have to walk a total distance given by $x$ (to go to the parking meter) $+x$ (to return to the car) + $500 m - x$ (to reach the store) $+500 m - x$ (to return to the car), which is $1000 m$, regardless of $x$. If, instead, he parks between the parking meter and the end of Viale Marconi closest to it, say at a distance $y > 0$ from the parking meter, then he will have to walk a distance given by $y$ (to go to the parking meter) $+y$ (to return to the car) $+y$ (to return to the parking meter again) $+500 m$ (to reach the store) $+500 m$ (to return to the parking meter) $+y$ (to return to the car), for a total of $1000 m + 4 y > 1000 m$. Finally, if he parks between the store and the end of Viale Marconi closest to it, at a distance $z > 0$ from the store, the distance he will have to walk is

$$
z + 500 m + 500 m + z + z + z = 1000 m + 4 z > 1000 m
$$

where the various terms correspond to the segments to go from the car to the store, from the store to the parking meter, return from the parking meter to the store, return from the store to the car, and finally return again from the car to the store and vice versa. It is clear, then, that the minimum distance Astolfo is forced to walk is $1000 \mathrm{~m}$, and that it is achieved by all and only the points on the segment between the parking meter and the store.