8.1. On Fyodor's bookshelf, there are volumes of works by various authors. When a friend borrowed two volumes of Pushkin, Fyodor noticed that among the remaining books, he had read at least half of them in full. After the friend returned the two volumes of Pushkin and borrowed two volumes of Lermontov, Fyodor realized that among the remaining books, he had read less than one third of them in full. What is the greatest number of volumes that could have been on the shelf at Fyodor's?
Answer: 12

Solution. Let's number all of Fyodor's books from 1 to $\mathrm{n}$.

![](https://cdn.mathpix.com/cropped/2024_05_06_01174ccc998f2eafc50eg-10.jpg?height=289&width=254&top_left_y=438&top_left_x=907)

Let the first two volumes be Pushkin's, $s_{1}$ - the number of them read by Fyodor; the last two - Lermontov's, $s_{2}$ - the number of them read by Fyodor. $S$ - the number of read volumes among the books from 3 to $n-2$.

According to the condition,

$S+s_{2} \geqslant \frac{n-2}{2}, S+s_{1}<\frac{n-2}{3}$,

$3 S+3 s_{1}<n-2 \leqslant 2 S+2 s_{2}$,

$3 S+3 s_{1}<2 S+2 s_{2}$

$S<2 s_{2}-3 s_{1} \leqslant 4$

Thus, $S$ does not exceed 3. Then the maximum possible $S=3$. From this, $n \leqslant 12$. Example for 12 (1 read, 0 - not read): 001110000011.