## Task 4 - 300524

In a sports competition, 10 teams are to participate. They should be equipped with monochrome T-shirts and monochrome shorts so that they can be distinguished from each other by the resulting color combinations.

a) What is the smallest number of colors with which this can be achieved?

b) What is the answer if it is additionally required that the two colors of the T-shirt and shorts of each team are different from each other?

Justify both of your answers!
a) With three colors, such as blue, yellow, and red, only 9 different color combinations can be formed, namely $b b, b g, b r, g b, g g, g r, r b, r g, r r$.

To distinguish between 10 teams, three colors are not sufficient. However, if an additional color, such as white, is added, 10 teams can be distinguished, as there are the 9 mentioned color combinations plus, for example, the combination $w w$.

Therefore, the smallest number of colors required to distinguish 10 teams is 4.

b) This also holds true under the condition specified in b).

Justification: With three colors, it is already impossible to equip 10 teams without this condition. Therefore, it is even more impossible when additional combinations are excluded by the condition.

With four colors, however, there are the following combinations available for distinguishing 10 teams:

$$
b g, b r, b w, g b, g r, g m, r b, r g, r w, w b, w g, w r
$$

which is sufficient.

Solutions of the 2nd Round 1990 adopted from [5]

### 2.32 XXXI. Olympiad 1991

### 2.32.1 1st Round 1991, Class 5