## Task 4 - 280524

a) In a box, there are 3 green and 4 yellow balls and no others. Kerstin and Steffen are considering how many balls they must take out of the box to ensure that at least one of each color is included. When taking out the balls, they should not look into the box.

Kerstin thinks that they must take out at least 5 balls; this would also be sufficient to ensure the desired outcome. Steffen, on the other hand, believes that 4 balls would already be enough.

Who of the two is right? Justify your decision.

b) Now, let's say the box contains 23 red, 33 blue, 21 black, 21 white, and 2 green balls and no others. State and justify how many balls must be taken out to ensure that 6 of these balls have the same color!

Show that the number you specify is also sufficient!
a) Steffen is not right; for if 4 balls are drawn, it is also possible that these are 4 yellow balls, meaning there is no green ball among them.

Therefore, one must draw at least 5 balls to ensure the desired outcome. This is sufficient, however; because among 5 drawn balls, there cannot be only yellow ones (since there are only 4 yellow ones), and there cannot be only green ones (since there are only 3 green ones). Kerstin is therefore right.

b) 22 balls are not enough; for these could be 5 red, 5 blue, 5 black, 5 white balls, and the two green ones, and then no 6 of these balls would have the same color.

Therefore, one must draw at least 23 balls to achieve the desired outcome.

This is sufficient, however; for if there were no 6 balls of the same color among the drawn balls, then there would be at most 5 red, 5 blue, 5 black, 5 white balls, and at most 2 green ones (since there are no more green ones). Then, however, at most 22 balls would have been drawn, in contradiction to the assumption that 23 balls were drawn.

Solutions of the 2nd round 1988 taken from [5]

### 2.30 XXIX. Olympiad 1989

### 2.30.1 1st Round 1989, Class 5