1. A necklace consists of 80 beads of red, blue, and green colors. It is known that on any segment of the necklace between two blue beads, there is at least one red bead, and on any segment of the necklace between two red beads, there is at least one green bead. What is the minimum number of green beads that can be in this necklace? (The beads in the necklace are arranged cyclically, that is, the last one is adjacent to the first.)
Answer: 27.

Solution. If the blue beads are arranged in a circle, the number of pairs of adjacent beads is equal to the number of beads. Since there is a red bead between any two blue beads, there are no fewer red beads in the necklace than blue ones. Similarly, it can be proven that there are no fewer green beads than red ones. Therefore, there are no fewer than $\frac{80}{3}=26 \frac{2}{3}$ green beads, which means there are at least 27. An example of a necklace containing exactly 27 green beads is as follows:

GKR 3 GKR ... ; GKR (26 times); GK.