1. A string is threaded with 150 beads of red, blue, and green. It is known that among any six consecutive beads, there is at least one green, and among any eleven consecutive beads, there is at least one blue. What is the maximum number of red beads that can be on the string?
Answer: 112.

Solution. We can choose $\left[\frac{150}{11}\right]=13$ consecutive blocks of 11 beads each. Since each block contains at least one blue bead, there are at least 13 blue beads on the string. In addition, we can group all the beads into 25 consecutive blocks of 6 beads each. Each block contains at least one green bead, so there are at least 25 of them on the string. Therefore, the number of red beads is no more than $150-25-13=112$.

Let's provide an example where the string contains exactly 112 red beads. Place the green beads at positions that are multiples of 6, and the blue beads at positions

$$
11,22,33,44,55 ; 65,76,87,98,109 ; 119,130,141
$$

Fill the remaining positions with red beads.