Maths Olympiad Prep

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Geometry Difficulty 6.2 National Olympiad Find the answer Italy

Problem:

Viale Marconi is 800 m800~\mathrm{m} long; 200 m200~\mathrm{m} from one end there is a parking meter, 100 m100~\mathrm{m} from the opposite end there is a clothing shop. Astolfo wants to park his car, get a ticket from the parking meter, go back to the car to display it on the windshield, visit the shop and finally return to the car. Moreover he is lazy and therefore wants to walk as little as possible; where should he park to achieve this? Indicate the set of points on the avenue that minimize the distance Astolfo has to travel on foot.

Pick one

Solution

Solution:

The answer is (E). Let us first note that the distance between the parking meter and the shop is 500 m500~\mathrm{m}. If Astolfo parks at a point between the parking meter and the shop, say at distance xx from the parking meter (and hence distance 500 m−x500~\mathrm{m}-x from the shop), then he will have to walk a total distance given by xx (to go to the parking meter) +x+x (to return to the car) +500 m−x+500~\mathrm{m}-x (to reach the shop) +500 m−x+500~\mathrm{m}-x (to return to the car), that is 1000 m1000~\mathrm{m}, regardless of xx.

If instead he parks between the parking meter and the end of Viale Marconi closest to it, say at distance y>0y>0 from the parking meter, then he will have to walk a distance given by yy (to go to the parking meter) +y+y (to return to the car) +y+y (to go once again to the parking meter) +500 m+500~\mathrm{m} (to reach the shop) +500 m+500~\mathrm{m} (to return to the parking meter) +y+y (to return to the car), for a total of 1000 m+4y>1000 m1000~\mathrm{m}+4y>1000~\mathrm{m}.

Finally, if he parks between the shop and the end of Viale Marconi closest to it, at distance z>0z>0 from the shop, the path he will have to walk will have length
z+500 m+500 m+z+z+z=1000 m+4z>1000 m z+500~\mathrm{m}+500~\mathrm{m}+z+z+z=1000~\mathrm{m}+4z>1000~\mathrm{m}
where the various terms correspond to the stretches from the car to the shop, from the shop to the parking meter, back from the parking meter to the shop, back from the shop to the car, and finally once more from the car to the shop and back.

We thus see that the minimum distance Astolfo is forced to walk is 1000 m1000~\mathrm{m}, and that it is achieved by all and only the points of the stretch between the parking meter and the shop.

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Source: MathNet, licensed CC-BY-4.0. Statement translated into English from it; metadata (topic, difficulty) added by this project.