# 7.1. Condition:

On the Christmas tree, there is a garland of 100 bulbs. It is known that the first and third bulbs are yellow. Moreover, among any five consecutive bulbs, exactly two are yellow and exactly three are blue. Father Frost can see only part of the garland from behind the tree. Help him find out the color and order of the bulbs in the 97th, 98th, 99th, and 100th positions.

## Options for matching:

Blue color

98

99

Yellow color

100

#
# Answer:

$97 ; 99 ; 100$ - Blue color

98 - Yellow color

## Solution.

Consider the first five light bulbs. From the condition, it follows that blue light bulbs are in the 2nd, 4th, and 5th positions. Note that the color of the sixth light bulb must be the same as the first, since for the first five light bulbs and for the light bulbs from 2 to 6, the condition of the problem must be satisfied, and the light bulbs from 2 to 5 are common to the first and sixth. Thus, if the first and sixth light bulbs are of different colors, the condition will not be met for one of the sets of five. Similarly, the 2nd and 7th, 3rd and 8th, 4th and 9th, 5th and 10th, etc. (i.e., any two light bulbs with five others between them) must be the same color. This means that the first five light bulbs shine in the same colors as the next five. Then we can notice that the sets of five light bulbs - from 1 to 5, from 6 to 10, from 11 to 15, ..., should have the same colors as the first. The colors of the first five light bulbs are known: yellow, blue, yellow, blue, blue. Therefore, the last five will have the following colors: 96 - yellow, 97 - blue, 98 - yellow, 99 - blue, 100 - blue. From this, we get the answer.