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Let n be the number of people in a party.
The maximum number of friends that 1 person can have is n-1. Because if person has n friends, it means he is a friend with himself.
If all people have different number of friends, it means 1 person has 0 friends, 1 person has 1 friend, ... 1 person has n-1 friends. Person with n-1 friends should be a friend with each other, but a person with 0 friend is not a friend with somebody. Сontradiction. It means some people have the same number of friends.