6.5. 101 people bought 212 balloons of four colors, and each of them bought at least one balloon, but no one had two balloons of the same color. The number of people who bought 4 balloons is 13 more than the number of people who bought 2 balloons. How many people bought only one balloon? Provide all possible answers and prove that there are no others.
# Solution:

Method 1. First, exclude the 13 people who bought 4 balloons. There will be 88 people left, who bought a total of 160 balloons, with each person buying between 1 and 4 balloons, and the number of people who bought 2 and 4 balloons is now equal. Suppose now that each person who bought 4 balloons gives one balloon to someone who bought 2. Now everyone has either 1 or 3 balloons, and the total number of people and balloons remains the same. Suppose now that everyone releases one balloon into the sky; 72 balloons will remain, with each person having either 2 balloons or none. Therefore, there are now 36 people with balloons, and the remaining 52 people have no balloons. These are the people who initially had one balloon.

Method 2. Let \( x_{i} (1 \leqslant i \leqslant 4) \) be the number of people who bought exactly \( i \) balloons. According to the problem, we have the system

\[
\left\{\begin{aligned}
x_{1}+x_{2}+x_{3}+x_{4} & =101 \\
x_{1}+2 x_{2}+3 x_{3}+4 x_{4} & =212 \\
x_{4}-x_{2} & =13
\end{aligned}\right.
\]

By adding the third equation to the first, we get \( x_{1}+x_{3}+2 x_{4}=114 \) (so \( x_{3}+2 x_{4}=114-x_{1} \)), and by adding twice the third equation to the second, we find that \( x_{1}+3 x_{3}+6 x_{4}=238 \). Then

\[
238=x_{1}+3\left(x_{3}+2 x_{4}\right)=x_{1}+3\left(114-x_{1}\right)=342-2 x_{1} \text{. }
\]

From this, \( x_{1}=52 \).

Answer: 52 people.

Note: By solving the system from the second method, all possible distributions of balloons among people can be found: \( x_{1}=52, x_{2}=a, x_{3}=36-2 a, x_{4}=a+13 \) (where \( a \) is an integer from the interval \([0 ; 18]\)).

Recommendations for checking:

| included in the work | points |
| :--- | :--- |
| Correct and fully justified answer | 7 points |
| There are arithmetic errors in the correct solution process, possibly leading to an incorrect answer | -1 point for each error |
| The system of equations describing the conditions of the problem is correctly set up, but the problem is not fully solved | 3 points |
| A correct specific case (or cases, but not all) of the distribution of balloons among people is provided | 1 point |
| Answer without justification and/or incorrect answer | 0 points |