## Task 1 - 330831

At the Sprachfix School in Lernhausen, there are 120 students. Each of them learns at least one of the languages English, Latin, French. The reporter Schreibklug learns the following facts:

(1) For exactly 102 of the 120 students: Each of these 102 students learns at least one of the languages English, Latin.

(2) For exactly 75 of the 120 students: Each of these 75 students learns at least one of the languages Latin, French.

(3) Exactly 18 of the 120 students learn only Latin.

(4) The number of students who learn exactly the two languages English and Latin is 9 more than the number of students who learn exactly the two languages French and Latin.

(5) None of the 120 students learn both English and French.

Schreibklug wants to report how many of the students learn exactly one of the three languages and how many of the students learn exactly two of the three languages. Are these two numbers uniquely determined by the information (1) to (5)? If so, determine these two numbers!
In the left image, the students learning English, Latin, and French are summarized by each circle. Additionally, the possible language combinations are visible there. The numbers in the right image indicate the student numbers for these. From (3) and (5), the entries 18 and 0 follow in particular. Considering these, we further get:

Due to the total number 120, we have $a+b+c+d+18=120$ (6), due to (1) we have $a+b+d+18=102$ (7), due to (2) we have $c+b+d+18=75$ (8), and due to (4) we have $b=9+d$ (9).

From (6) and (7), it follows that $c=18$ (10), and from (6) and (8), it follows that $a=45$ (11).

Thus, from (7) or (8), we get $b+d=39$.

The desired numerical values are therefore uniquely determined; they are:

A total of $a+c+18=81$ students learn exactly one of the three languages, and a total of $b+d=39$ students learn exactly two of the three languages.