## Task 2 - 210812

In a GST shooting competition with air rifles, Falk and Heiko each fired 5 shots. The following hits were recorded on the targets: Exactly once the 3, twice the 5, twice the 6, twice the 7, once the 8, once the 9, and once the 10. Furthermore, it is known:

(1) Falk scored nine times as many points with his last four shots as with his first shot.

(2) Falk hit the 9.

Can the following two questions be answered definitively based on this information?

a) Which of the two boys achieved the better result?

b) Which of the two boys hit the 10?
a) Since the hits $3,5,5,6,6,7,7,8,9,10$ were achieved, the sum of the rings for four shots could have been at most $7+8+9+10=34(*)$.

If Falk had scored a 5 (or a higher ring number) with his first shot, then the sum of the other four ring numbers would have to be equal to (or greater than) 45 due to (1), which contradicts (*).

Therefore, Falk must have scored the ring number 3 with his first shot, thus achieving 30 rings. Heiko, due to $3+5+5+6+6+7+7+8+9+10=66$ and $66-30=36$, was the better marksman with his ring number of 36.

b) According to a), Falk shot the 3 and, due to (2), the 9. If he had also shot the 10, the remaining $30-3-9-10=8$ rings would have to be represented as the sum of two hits from the ring numbers 5, 5, 6, 6, 7, 7, 8.

However, this is not possible (the two smallest already add up to 10). Consequently, Heiko shot the 10.