## Task 3 - 261213

At an international conference, exactly two representatives from the People's Republic of Hungary, the CSSR, the People's Republic of Poland, and the GDR participated. It is known about these eight participants:

(1) Each of these participants speaks, in addition to their native language, at least one language from the other three mentioned countries. (The languages from the four countries were Hungarian, Czech, Polish, and German; other languages from these countries, such as Slovak or Sorbian, were not present.)

(2) Each of the four languages is spoken by participants from exactly three of the mentioned countries.

(3) Each participant who speaks Polish also speaks Hungarian, but not German.

(4) The language skills of the two Polish participants differ with respect to the four mentioned languages.

a) Determine how many of the mentioned participants speak German and from which countries these participants came.

b) Determine which languages the two participants from the People's Republic of Hungary spoke.
Let the countries be denoted by the initial letters of their names with $U, T, P, D$, and the languages correspondingly with $u, t, p, d$.

In the following table, the fact that a participant masters a certain language is marked with "+", non-mastery is marked with "-". From (1), (3), and (4), it follows that with a suitable choice of the order of the two Polish participants,

|  | $\mathrm{U}$ |  | $\mathrm{T}$ |  | $\mathrm{P}$ |  | $\mathrm{D}$ |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  | 1. | 2. | 1. | 2. | 1. | 2. | 1. | 2. |
| $\mathrm{u}$ | + | + |  |  | + | + |  |  |
| $\mathrm{t}$ |  |  | + | + | + | - |  |  |
| $\mathrm{p}$ |  |  |  |  | + | + |  |  |
| $\mathrm{d}$ |  |  |  |  | - | - | + | + |

From (3), it follows that the participants who speak German do not speak Polish, in particular, the representatives of the GDR do not speak Polish. Since Polish is spoken by representatives of three countries according to (2), at least one participant from the Hungarian People's Republic and the CSSR must speak Polish, which we assume without loss of generality to be the first representative of these countries.

Because of (3), these do not speak German, but Hungarian. The same argument shows for the German language that the second representatives of Hungary and the CSSR speak German, from which it follows according to (3) that these do not speak Polish.

The current result is illustrated by the following table:

|  | $\mathrm{U}$ |  | $\mathrm{T}$ |  | $\mathrm{P}$ |  | $\mathrm{D}$ |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
|  | 1. | 2. | 1. | 2. | 1. | 2. | 1. | 2. |
| $\mathrm{u}$ | + | + | + |  | + | + |  |  |
| $\mathrm{t}$ |  |  | + | + | + | - |  |  |
| $\mathrm{p}$ | + | - | + | - | + | + | - | - |
| $\mathrm{d}$ | - | + | - | + | - | - | + | + |

Thus, the result for a) is: Exactly four of the participants speak German: two representatives of the GDR and one each from the Hungarian People's Republic and the CSSR.

Further, for b): According to (2), Hungarian is spoken by representatives of exactly three countries, so the participants from the GDR do not speak Hungarian. From (1), it follows that they speak Czech (since they do not speak Polish either). According to (2), it is therefore impossible for a participant from Hungary to speak Czech. The first participant from the Hungarian People's Republic thus speaks exactly Hungarian and Polish, and the second exactly Hungarian and German.

Adapted from $[5]$