Example 24 (23rd All-Soviet Union Mathematical Olympiad) On Sunday, there are 7 children, each of whom comes to the ice cream stand 3 times. It is known that every two of them meet at the stand. Prove: At some moment, at least 3 children meet at the stand.
Problem 942
Official solution
Proof by contradiction. If each meeting involves only two people, and it is known that every 3 people have met at the kiosk, the number of meetings should be times. During the first meeting, two people come to the kiosk, and in the subsequent 20 meetings, each time at least one person is a new arrival to the kiosk, so the total number of arrivals at the kiosk is 22 person-times. On the other hand, each person has been to the kiosk 3 times, and 7 people have gone a total of 21 person-times, which is a contradiction. Thus, the proposition is proved.