1. A thread is strung with 75 blue, 75 red, and 75 green beads. We will call a sequence of five consecutive beads good if it contains exactly 3 green beads and one each of red and blue. What is the maximum number of good quintets that can be on this thread?
Answer: 123.

Solution. Note that the first and last green beads are included in no more than three good fives, the second and second-to-last - in no more than four fives, and the rest - in no more than five fives. If we add these inequalities, we get $2 \cdot 3 + 2 \cdot 4 + 71 \cdot 5 = 369$ on the right side, and three times the number of good fives on the left side, since each will be counted three times. Therefore, there can be no more than 123 good fives.

Let's provide an example of bead placement that gives exactly 123 good fives:

$$
\text { KSGGG; KSGGG; ...; KSGGG (25 times); KS ... }
$$

(the ellipsis at the end means an arbitrary combination of blue and red beads). The good fives will be those starting at positions $1, 2, \ldots, 123$, and only those.