10.1. There are 40 pencils of four colors - 10 pencils of each color. They were distributed among 10 children so that each received 4 pencils. What is the smallest number of children that can always be selected to ensure that pencils of all colors are found among them, regardless of the distribution of pencils?

(I. Bogdanov, O. Podlipsky)
10.1. Answer. 3 boys.

We will show that it is always possible to choose three boys such that they have pencils of all colors. Since there are 10 pencils of each color and each boy received 4 pencils, at least one boy must have received pencils of at least two different colors. It remains to add to him two boys who have pencils of the remaining two colors.

Now, let's show how to distribute the pencils to the boys so that any two of them together have pencils of no more than three colors. Distribute 4 pencils of the second color to two boys, 4 pencils of the third color to two boys, 4 pencils of the fourth color to two boys, 4 pencils of the first color to one boy, 2 pencils of the first and second colors to one boy, 2 pencils of the first and third colors to one boy, and 2 pencils of the first and fourth colors to one boy.