In a theater troupe, there are 60 actors. Any two of them have at least once played in the same play. In each play, no more than 30 actors are involved.

What is the minimum number of plays the theater could have staged?
Example. Let's divide the troupe into four groups of 15 people and hold 6 performances, in each of which some two groups are involved. The number of ways to choose two groups out of four is 4$\cdot$3:2=6.

Estimate. In total, the actors played no more than $30 \cdot 5=150$ roles, so if there are five performances, there will be an actor who played no more than two roles. Then he played in one performance with no more than $29 \cdot 2=58$ colleagues from the remaining 59. Therefore, five performances are insufficient to meet the condition.

## Answer

6 performances.