## Task 4 - 040524

During a performance at the "Theater der Jungen Welt" in Leipzig, some seats remained empty. Alfred counted 17, while Annerose counted 16 empty seats.

Heinz said that Alfred must have definitely miscounted.

How could Heinz justify his statement, knowing that the theater has 520 seats and that there were 68 more girls than boys in attendance?
If the number of boys is odd, then the number of girls is also odd. The sum of two odd numbers is, however, an even number.

If the number of boys is even, then the number of girls is also even. The sum of two even numbers is also an even number. The number of free seats must thus be even, because the difference between two even numbers is again an even number.

Or: If $a$ is the number of boys, then $2a + 68$ is the number of boys and girls in the theater. The number of free seats must thus be even.

Solutions of the II. Round 1964 taken from [5]

### 2.6 V. Olympiad 1965

### 2.6.1 I. Round 1965, Class 5