7.4. A biologist sequentially placed 150 beetles into ten jars. Moreover, in each subsequent jar, he placed more beetles than in the previous one. The number of beetles in the first jar is no less than half the number of beetles in the tenth jar. How many beetles are in the sixth jar?
Answer: In the sixth jar, there are -16 beetles.

Solution. Let there be $x$ beetles in the first jar, then in the second jar there are no fewer than $x+1$ beetles, in the third jar no fewer than $x+2$ beetles, and so on. Thus, in the tenth jar there are no fewer than $x+9$ beetles. Therefore, the total number of beetles is no less than $10 x+45$. Considering that a total of 150 beetles were distributed, we get: $x \leqslant 10$.

On the other hand, in the tenth jar there should be no more than $2 x$ beetles, in the ninth jar no more than $2 x-1$ beetles, and so on. This means that in the first jar there are no more than $2 x-9$ beetles, and the total number of beetles is no more than $20 x-45$. Since a total of 150 beetles were distributed, then $x \geqslant 10$.

Thus, there are exactly 10 beetles in the first jar, and 19 or 20 in the last jar. Let's find the sum of eleven consecutive numbers starting from ten: $10+11+\ldots+19+20=165$. Since there should be a total of 150 beetles, the jar with 15 beetles is missing. Therefore, the distribution is uniquely determined: 10, 11, 12, 13, 14, 16, 17, 18, 19, and 20 beetles from the first to the tenth jar, respectively. Thus, in the sixth jar, there are 16 beetles.

Having proved that $x \leqslant 10$, we can continue the reasoning differently. Since in the tenth jar there are no fewer than $x+9$ beetles, and $x+9 \leqslant 2 x$, then $x \geqslant 9$. Then consider two cases: $x=9$ and $x=10$, estimating the number of beetles in the tenth jar.

## Evaluation Criteria:

+ The correct answer and a fully justified solution are provided

$\pm$ The correct answer and generally correct estimates of the number of beetles both "from above" and "from below" are provided, but they contain some gaps

$\pm$ Correct and justified estimates of the number of beetles both "from above" and "from below" are provided, the distribution of beetles in jars is correctly found, but the answer to the question of the problem is incorrect or missing

干 Only one of the two required estimates is correctly performed

干 The distribution of beetles in jars is correctly indicated, but it is not justified

- Only the answer is provided
- The problem is not solved or is solved incorrectly