1. A necklace consists of 100 red and a certain number of blue beads. It is known that in any segment of the necklace containing 10 red beads, there are at least 7 blue ones. What is the minimum number of blue 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: 78.

Solution. Note that any segment of the necklace containing 16 beads has no more than 9 red and no fewer than 7 blue beads (otherwise, it would contain 10 red beads and no more than 6 blue ones). Fix a blue bead in the necklace. The 11 consecutive segments of 16 beads adjacent to it do not cover the entire necklace, as these segments contain no more than 99 red beads out of 100. Therefore, the total number of blue beads is no less than $11 \cdot 7+1=78$.

Let's provide an example of a necklace containing exactly 78 blue beads. Consider the block

$$
\text { B = CB CB CB CB CB CB CB; RR, }
$$

consisting of 7 blue and 9 red beads. Then the desired necklace has the form

$$
\text { B, B, .., B (11 times); CR. }
$$