Four girls played a series of matches at a camp. When asked how many matches they won, they answered very evasively:

"If we added up the number of their victories for every pair of girls, we would get the numbers $8, 10, 12, 12, 14$ and 16."

Determine how many victories each of the girls won.

(M. Volfová)
Let's denote the number of victories of each girl as $a, b, c, d$. If any two girls had the same number of victories, there would have to be at least two pairs of identical numbers among the sums in the problem (for example, if $a=b$, then $a+c=b+c$ and also $a+d=b+d$). However, this is not true, so the numbers $a, b, c, d$ are all different.

If we arrange these numbers in ascending order $a<b<c<d$, then the following holds:

$$
a+b<a+c<a+d<b+d<c+d
$$

and simultaneously

$$
a+c<b+c<b+d \text {. }
$$

Given that there are two numbers of the same value in the problem, it must be that $a+d=b+c$. According to the problem, we can set up the following system of equations:

$$
\begin{aligned}
& a+b=8 \\
& a+c=10, \\
& b+c=12 \\
& a+d=12 \\
& b+d=14, \\
& c+d=16
\end{aligned}
$$

From the 1st and 2nd equations, it follows that $c=b+2$. Substituting into the 3rd equation, we get $b+b+2=12$, so $b=5$. Substituting $b$ into the 1st, 3rd, and 5th equations, we obtain the values of the other unknowns: $a=3, c=7$, and $d=9$.

We did not use the 4th and 6th equations in solving the system, so we need to verify their validity by substituting the calculated values: $3+9=12, 7+9=16$.

The individual girls won $3, 5, 7$, and $9$ victories.

Evaluation. 2 points for setting up the system of equations and justifying that the system is uniquely determined except for the labeling and ordering of the unknowns; 4 points for solving the system (if the verification of the equations not used in the solution is not performed, deduct 1 point).

Note. It is not necessary to set up and solve a system of equations to solve the problem; the entire process can be expressed verbally. The evaluation of such a solution is similar to the one described above.