## Task B-1.4.

Hitting the target, Karlo hit the number nine several times, and the number eight several times. He hit the number seven half as many times as he hit the number nine, and the number of hits on the number five was three less than the number of hits on the number eight. In total, he scored 200 points.

How many times did Karlo shoot at the target if he never missed and did not hit any other number? (A hit on the number nine scores 9 points, on the number eight 8 points,...)
## Solution.

If Karlo hit the nine $x$ times, then he hit the seven $\frac{x}{2}$ times.

If he hit the eight $y$ times, then he hit the five $y-3$ times.

Thus, he collected a total of $9 x+8 y+7 \cdot \frac{x}{2}+5(y-3)=200$ points.

We are looking for natural (Diophantine) solutions to the equation $\frac{25}{2} x+13 y=215$.

From $25 x+26 y=430$ it follows that $26 y=430-25 x$.

Since the right side of the equation is divisible by 5, the left side of the equation must also be divisible by 5. The number 26 is not divisible by 5, so $y$ must be divisible by 5.

Since $25 x=430-26 y>0$, it follows that $y<17$, so it is sufficient to check for $y=5, y=10$, and $y=15$.

For $y=5$, we have $25 x=430-26 \cdot 5=300$ so $x=12$.

For $y=10$, we have $25 x=430-26 \cdot 10=170$ so $x \notin \mathbb{N}$.

For $y=15$, we have $25 x=430-26 \cdot 15=40$ so $x \notin \mathbb{N}$.

Therefore, the only solution is $x=12, y=5$.

Karlo hit the nine 12 times, the eight 5 times, the seven 6 times, and the five 2 times. In total, he shot $12+5+6+2=25$ times.

Note: For the correct solution, with its verification, but without the proof that it is the only solution, the student receives 2 points.