In the diagram with seven cells, draw stars so that all the following conditions are met:

1. There are a total of 21 stars.
2. Each cell contains at least one star.
3. The cells labeled $A, B, C$ together contain 8 stars.
4. The cells labeled $A$ and $B$ together contain fewer stars than the cell labeled $C$.
5. The cell labeled $B$ contains more stars than the cell labeled $A$.
6. The circle contains a total of 15 stars, the triangle contains a total of 12 stars, and the rectangle contains a total of 14 stars.

(E. Semerádová)

![](https://cdn.mathpix.com/cropped/2024_04_17_bc4a95ca96fb1c2e7459g-2.jpg?height=597&width=646&top_left_y=455&top_left_x=705)

Hint. First determine the number of stars in the cells $A, B, C$.
From the second and fifth conditions, it follows that there is at least 1 star in cell $A$ and at least 2 stars in cell $B$. Therefore, there are at least 3 stars in cells $A$ and $B$ combined. From the third and fourth conditions, it follows that there are no more than 3 stars in these two cells combined. Therefore, there are exactly 3 stars in cells $A$ and $B$ combined, and there are 5 stars in cell $C$:

$$
A=1, \quad B=2, \quad C=5
$$

We will also label the other cells with letters as shown in the following image:

![](https://cdn.mathpix.com/cropped/2024_04_17_bc4a95ca96fb1c2e7459g-2.jpg?height=594&width=646&top_left_y=1688&top_left_x=705)

Each of the cells $A, B$, and $C$ is shared by two of the three shapes mentioned in the sixth condition (for example, cell $A$ belongs to the circle and the triangle). Cell $D$ is shared by all three shapes. The remaining cells $E, F$, and $G$ belong to different shapes. The sum of the stars in the circle, triangle, and rectangle is $15+12+14=41$, and in this sum, the stars in cells $A, B, C$ are counted twice, the stars in cell $D$ are counted three times, and the stars in cells $E, F, G$ are counted once. According to the first condition, there are a total of 21 stars, and in this sum, the stars in each cell are counted once. The difference of 20 stars therefore corresponds to the sum of the stars in cells $A, B, C$ (which is a total of 8) and twice the number of stars in cell $D$. Therefore, there must be 6 stars in cell $D$:

$$
D=(20-8): 2=6 .
$$

The number of stars in the remaining cells can now be calculated according to the information in the sixth condition:

$$
\begin{aligned}
& 15=A+C+D+E, \quad \text { so } \quad E=15-1-5-6=3, \\
& 12=A+B+D+F, \quad \text { so } \quad F=12-1-5-2=3, \\
& 14=B+C+D+G, \quad \text { so } \quad G=14-2-5-6=1 .
\end{aligned}
$$

![](https://cdn.mathpix.com/cropped/2024_04_17_bc4a95ca96fb1c2e7459g-3.jpg?height=596&width=642&top_left_y=1044&top_left_x=707)

Another solution. As in the previous solution, we derive the number of stars in cells $A, B$, and $C$:

$$
A=1, \quad B=2, \quad C=5 .
$$

According to the information in the sixth condition, we find that

$$
\begin{aligned}
& 15=A+C+D+E, \quad \text { so } \quad D+E=15-1-5=9, \\
& 12=A+B+D+F, \quad \text { so } \quad D+F=12-1-2=9, \\
& 14=B+C+D+G, \quad \text { so } \quad D+G=14-2-5=7 .
\end{aligned}
$$

From this, we see that the number of stars in cells $E$ and $F$ is the same, and it is 2 more than the number of stars in cell $G$. Now we can substitute the number of stars in any of the cells $D$, $E, F, G$, express the number of stars in the remaining three cells from the previous expressions, and verify whether the total sum $A+B+C+D+E+F+G$ is equal to 21. We substitute for $G$, keeping in mind that there must be at least one star in each cell:

| $G$ | $E=F$ | $D$ | sum |
| :---: | :---: | :---: | :---: |
| $\mathbf{1}$ | $\mathbf{3}$ | $\mathbf{6}$ | $\mathbf{2 1}$ |
| 2 | 4 | 5 | 23 |
| 3 | 5 | 4 | 25 |
| 4 | 6 | 3 | 27 |
| 5 | 7 | 2 | 29 |
| 6 | 8 | 1 | 31 |

The only valid option is highlighted in the first row.