## Task 2.

In one organization, there are three committees. Each person belongs to exactly one committee. For any two people who belong to different committees, in the remaining committee, there are exactly 10 people who both of these two people know, and exactly 10 people whom neither of these two people knows. Acquaintances are mutual. How many people are there in all three committees combined?
## First Solution.

Let the committees be $A, B, C$ and let there be $a, b, c$ people in them, respectively. Imagine the people as points in a plane such that no three are collinear. Connect every two people who know each other with a blue line, and every two who do not know each other with a red line. Meanwhile, people who are in the same committee are not connected at all.

First, count the monochromatic triangles (those with all three sides the same color). For every pair of people $x \in A, y \in B$, there are exactly 10 monochromatic triangles. According to the problem's condition, if $\overline{x y}$ is a blue line, then there are exactly 10 points $z \in C$ such that the lines $\overline{x z}$ and $\overline{y z}$ are blue, and similarly for red lines. Thus, the number of monochromatic triangles is $10ab$. Similarly, we find that the number of monochromatic triangles is also $10bc$ and $10ca$, from which we conclude that $a=b=c$.

The total number of triangles is $a^{3}$, and the number of monochromatic triangles is $10 a^{2}$. It remains to count the bichromatic triangles. For every pair $x \in A, y \in B$, we get exactly 10 bichromatic triangles such that the lines $\overline{x z}$ and $\overline{y z}$ are the same color, i.e., we have $10 a^{2}$ such triangles, those where the lines $\overline{y x}$ and $\overline{z x}$ are the same color are also $10 a^{2}$, and those where $\overline{x y}$ and $\overline{z y}$ are the same color are again $10 a^{2}$. Thus, the number of bichromatic triangles is $30 a^{2}$.

Finally, we have $a^{3}=10 a^{2}+30 a^{2}$, so $a=40$, which means there are a total of 120 people in all three committees combined.