Yatsenko I.V.

a) In Mexico City, to limit the traffic flow, two days of the week are established for each private car when it cannot leave the city streets. The family needs to have at least ten cars available every day. What is the smallest number of cars the family can manage with, if its members can choose the prohibited days for their cars themselves?

b) In Mexico City, one day of the week is established for each private car when it cannot leave the city streets. A wealthy family of ten people has bribed the police, and for each car, they name two days, one of which the police choose as the non-driving day. What is the smallest number of cars the family needs to buy so that each day every family member can drive independently, if the designation of non-driving days for cars goes sequentially?
a) 13 cars are not enough. Indeed, the total number of prohibitions is 26, and there are 7 days in a week, so on one of the days, at least four cars will be "resting," meaning only 9 can go out.

14 cars are sufficient: prohibit four cars from Monday and Tuesday, four from Wednesday and Thursday, two from Friday and Saturday, two from Saturday and Sunday, and two from Friday and Sunday.

b) 11 cars are not enough: on one of the days of the week, at least two cars will be "resting."

We will show that 12 cars are enough. When submitting the "application" for the next car, the family will indicate the pair of days for which there is currently a prohibition on no more than one car. This can be continued until there are 6 days with two prohibitions, meaning this can be done for each of the 12 cars. As a result, on each day of the week, the family will have no more than two non-driving cars.

## Answer

a) 14; b) 12 cars.

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