1. A necklace consists of 50 blue and a certain number of red beads. It is known that in any segment of the necklace containing 8 blue beads, there are at least 4 red ones. What is the minimum number of red beads that can be in this necklace? (The beads in the necklace are arranged cyclically, meaning the last one is adjacent to the first.)
Answer: 29.

Solution. Note that any segment of the necklace consisting of 11 beads contains no more than 7 blue and no fewer than 4 red beads (otherwise, it would contain 8 blue beads and no more than 3 red ones). Fix a red bead in the necklace. The 7 consecutive segments of 11 beads adjacent to it do not cover the entire necklace, as these segments contain no more than 49 blue beads out of 50. Therefore, the total number of red beads is no less than $7 \cdot 4 + 1 = 29$.

Let's provide an example of a necklace containing exactly 29 red beads. Consider the block

$$
\text{B = RC RC RC RC; CCC,}
$$

consisting of 4 red and 7 blue beads. Then the desired necklace has the form

$$
\text{B, B, .., B (7 times); RC.}
$$