## Task 2 - 260612

For the execution of a field game, it was necessary that each participant had a writing instrument with them. Only the following types of writing instruments were brought by the participants: ballpoint pens, red pencils, and green pencils; none of these three types were duplicated by any participant. Specifically, the following was observed:

(1) There were a total of 100 participants in the field game.

(2) Exactly 20 of the participants had a ballpoint pen but no red pencil.

(3) Exactly 15 of the participants had a ballpoint pen but no green pencil.

(4) Exactly 5 of the participants had a ballpoint pen but neither a red pencil nor a green pencil.

(5) Exactly 65 of the participants had no ballpoint pen.

(6) Exactly 55 of the participants had no red pencil.

(7) Exactly 40 of the participants had no green pencil.

(8) Exactly 15 of the participants had neither a red pencil nor a green pencil.

a) Determine from these data the number of participants who brought at least one writing instrument!

b) Were the writing instruments brought sufficient to provide each of the 100 participants with a writing instrument, given appropriate distribution?
a) Among the 15 participants mentioned in (8) without red and green pencils, there were also the 5 participants mentioned in (4) (without red and green pencils, but) with a ballpoint pen; the other 10 consequently had no writing instrument at all.

From (8) and (1) it also follows: All 85 participants not mentioned in (8) had a writing instrument (namely at least one of the devices red or green pencil). Therefore, only the 10 mentioned participants had no writing instrument.

This shows: Exactly 90 of the participants brought at least one writing instrument.

b) Among the 20 participants mentioned in (2) who all had a ballpoint pen but no red pencil, there were also the 5 participants mentioned in (4) (with a ballpoint pen, without a red pencil, and) without a green pencil; the other 15 consequently brought a green pencil in addition to the ballpoint pen. This leads to the conclusion:

The writing instruments brought were sufficient to supply all participants. It was sufficient, for example, to distribute 10 of the mentioned green pencils to the participants without a brought writing instrument as determined in a).