Problem 6.4. Misha made himself a homemade dartboard during the summer at the cottage. The round board is divided by circles into several sectors - darts can be thrown into it. Points are awarded for hitting a sector as indicated on the board.

Misha threw 8 darts 3 times. The second time he scored twice as many points as the first time, and the third time he scored 1.5 times as many points as the second time. How many points did he score the second time?

![](https://cdn.mathpix.com/cropped/2024_05_06_df4c0436ab68970fab15g-15.jpg?height=363&width=361&top_left_y=1494&top_left_x=546)
Answer: 48.

Solution. The smallest possible score that can be achieved with eight darts is $3 \cdot 8=24$. Then, the second time, Misha scored no less than $24 \cdot 2=48$ points, and the third time, no less than $48 \cdot 1.5=72$.

On the other hand, $72=9 \cdot 8$ is the highest possible score that can be achieved with eight darts. Therefore, the third time, Misha scored exactly 72 points, and the second time, exactly 48.