2. In the parliament of a certain state, there are 2016 deputies, who are divided into 3 factions: "blues," "reds," and "greens." Each deputy either always tells the truth or always lies. Each deputy was asked the following three questions:
1) Are you a member of the "blues" faction?
2) Are you a member of the "reds" faction?
3) Are you a member of the "greens" faction?

1208 deputies answered affirmatively to the first question, 908 deputies answered affirmatively to the second question, and 608 deputies answered affirmatively to the third question. In which faction are there more deputies who lie than those who tell the truth, and by how many?
# Solution.

Let the number of deputies telling the truth in the "blue," "red," and "green" factions be $r_{1}, r_{2},$ and $r_{3}$ respectively, and the number of deputies lying in the "blue," "red," and "green" factions be $l_{1}, l_{2},$ and $l_{3}$ respectively.

According to the problem:

$\left\{\begin{array}{l}r_{1}+r_{2}+r_{3}+l_{1}+l_{2}+l_{3}=2016, \\ r_{1}+l_{2}+l_{3}=1208, \\ r_{2}+l_{1}+l_{3}=908, \\ r_{3}+l_{1}+l_{2}=608 .\end{array}\right.$

Let $l_{1}-r_{1}=a, l_{2}-r_{2}=b, l_{3}-r_{3}=c$.

Subtracting the third equation from the second, we get $b-a=300$. Or $a=b-300$.

Subtracting the fourth equation from the third, we get $c-b=300$. Or $c=b+300$.

Subtracting the first equation from the sum of the second, third, and fourth equations, we get

$l_{1}+l_{2}+l_{3}=708$. From this, $r_{1}+r_{2}+r_{3}=1308$.

Then $a+b+c=-600$. Or $3 b=-600$.

Thus, $b=-200, a=-500, c=100$. That is, there are 100 more liars in the "green" faction.

Answer: There are 100 more people lying in the "green" faction.

## Recommendations for checking.

The mathematical model of the problem is correctly formulated - 3 points.