1. Around a round table, 50 schoolchildren are sitting: blondes, brunettes, and redheads. It is known that in any group of schoolchildren sitting in a row, between any two blondes there is at least one brunette, and between any two brunettes - at least one redhead. What is the minimum number of redheads that can sit at this table?
Answer: 17.

Solution. If only blondes were sitting at the table, the number of pairs of neighbors would be equal to the number of blondes. Since there is a brunette between any two blondes, there are no fewer brunettes than blondes sitting at the table. Similarly, it can be proven that there are no fewer redheads than brunettes. Therefore, there are no fewer than $\frac{50}{3}=16 \frac{2}{3}$ redheads sitting at the table, which means there are at least 17. Let's provide an example of seating with exactly 17 redheads:

$$
\text { RCB; RCB; ... RCB; (16 times); RC }
$$

(The letters R, C, and B denote redheads, brunettes, and blondes, respectively).