## Task B-2.7.

Luka proposed to Mili, Niki, and Tini that they solve tasks together. By that time, each of the girls had already solved a certain number of tasks, and these numbers were different for each of them. The girls decided to accept Luka's proposal only if he could figure out how many tasks each of the three had already solved. The sum of the number of tasks solved by Mili and Niki and the product of those numbers is 14. For Mili and Tini, that sum is 11, and for Niki and Tini, it is 19. What numbers does Luka need to find?
## First solution.

Let $m$ be the number of problems solved by Mila, $n$ be the number of problems solved by Nika, and $t$ be the number of problems solved by Tina.

Luka needs to solve the system

$$
\begin{gathered}
m+n+m n=14 \\
m+t+m t=11 \\
n+t+n t=19
\end{gathered}
$$

which can be rewritten as

$$
\begin{gathered}
m+n+m n+1=15 \\
m+t+m t+1=12 \\
n+t+n t+1=20
\end{gathered}
$$

or equivalently

$$
\begin{gathered}
(m+1)(n+1)=15 \\
(m+1)(t+1)=12 \\
(n+1)(t+1)=20
\end{gathered}
$$

From the first two equations, we get $n+1=\frac{5}{4}(t+1)$,

and substituting this into the third equation gives $(t+1)^{2}=16$.

Thus, $t+1=4$ or $t+1=-4$ 1 point

but since $m, n$, and $t$ are natural numbers, $t+1=4$, so $t=3$.

Then from $(n+1)(t+1)=20$ we get $n=4$, and from $(m+1)(t+1)=12$ we get $m=2$.