6. Maja, Ana, and Dora are training for dance together and have decided to celebrate their birthdays together. They agreed that the total number of guests would be 40.

Maja invited 17 friends. The number of guests invited by Ana alone is 3 more than the number of guests invited by Dora alone.

Since they also have mutual friends, the following happened:

- All three invited 4 mutual friends.
- Maja and Ana invited 5 mutual friends, Ana and Dora 6, and Maja and Dora 7 mutual friends.

How many guests did Ana invite, and how many did Dora invite?
6. The task can be solved using a Venn diagram.

Let $M$ be the set of guests invited by Maja, $A$ the set of guests invited by Ana, and $D$ the set of guests invited by Dora.

The information that all three friends invited 4 common friends is placed in the intersection of all three sets. (Figure 1.)

$1 \text{ POINT}$

![](https://cdn.mathpix.com/cropped/2024_05_30_f210cd8de88dd249b29dg-4.jpg?height=314&width=402&top_left_y=528&top_left_x=290)

D
Figure 1.

![](https://cdn.mathpix.com/cropped/2024_05_30_f210cd8de88dd249b29dg-4.jpg?height=374&width=380&top_left_y=521&top_left_x=1089)

D
Figure 2.

Next, we arrange the common guests.

Since Maja and Ana invited 5 common friends, and all three invited 4 common friends, we conclude that only Maja and Ana invited 5-4=1 common friend. (Figure 2.) $\quad 1$ POINT

![](https://cdn.mathpix.com/cropped/2024_05_30_f210cd8de88dd249b29dg-4.jpg?height=408&width=528&top_left_y=1121&top_left_x=273)

D
Figure 3.

![](https://cdn.mathpix.com/cropped/2024_05_30_f210cd8de88dd249b29dg-4.jpg?height=400&width=611&top_left_y=1125&top_left_x=1065)

D
Figure 4.

Ana and Dora invited 6 common friends, and since all three friends invited 4 common friends, we conclude that only Ana and Dora invited $6-4=2$ common friends. (Figure 3.)

Maja and Dora invited 7 common friends, and since all three friends invited 4 common friends, we conclude that only Maja and Dora invited $7-4=3$ common friends. (Figure 4.)

![](https://cdn.mathpix.com/cropped/2024_05_30_f210cd8de88dd249b29dg-4.jpg?height=500&width=1310&top_left_y=2029&top_left_x=270)

M
A

Figure 6.
Let $x$ be the number of guests invited only by Dora.

From the condition of the task, the number of guests invited only by Ana is $x+3$. (Figure 6.)

1 POINT

Since the total number of guests is 40, it follows:

$9+x+x+3+3+1+2+4=40$

1 POINT

$2x+22=40$

$2x=40-22$

$2x=18$

$x=9$

1 POINT

Dora invited $9+3+4+2=18$ guests.

1 POINT

Ana invited $12+1+4+2=19$ guests.

Note: If the student uses a graphical representation instead of an equation to determine the unknown quantity, the grading should be in accordance with the proposed solution.