## Task 4 - 310614

A total of 20 people participated in an excursion. It was observed:

(1) Exactly 5 of the participants were 30 years old or younger.

(2) Of the participants who were older than 30, exactly 10 bought something to drink at the first stop, exactly 12 at the second stop. No participant refrained from buying at both stops.

(3) Exactly 6 of the participants were 40 years old or older, among them exactly 2 did not buy anything to drink at the first stop, and exactly 2 did not buy anything to drink at the second stop.

How many of the participants who were older than 30 but younger than 40 years old bought something to drink at both the first and the second stop?
I. After (1) there were exactly $20-5=15$ participants older than 30 years.

Of them, after (2) exactly $15-10=5$ bought nothing to drink at the first stop and exactly $15-12=3$ bought nothing to drink at the second stop. Since no one abstained from both purchases, these $5+3=8$ participants were all those who did not buy drinks twice. Of the 15 participants over 30 years old, therefore, $15-8=7$ bought drinks twice.

II. After (3), of the 6 participants who were 40 years or older, exactly 2 bought nothing to drink at the first stop and exactly 2 bought nothing to drink at the second stop. It follows accordingly that $6-(2+2)=2$ of these participants bought drinks twice.

From I. and II. it follows: Of the participants who were older than 30 but younger than 40 years, exactly $7-2=5$ bought drinks twice.

Solutions of the I. Round 1991 adopted from [5]

### 3.32.2 II. Round 1991, Class 6